Eureka Math Precalculus Module 2

$Eureka Math Precalculus Module 2$

FL State Adoption Bid # 3707

Teacher Edition Eureka Math Precalculus Module 2

Special thanks go to the GordRn A. Cain Center and to the Department of Mathematics at Louisiana State University for their support in the development of Eureka Math.

For a free Eureka Math Teacher Resource Pack, Parent Tip Sheets, and more please visit www.Eureka.tools

Published by Great Minds®.

Copyright © 2018Great Minds® . No part of this work may be reproduced, sold, or commercialized, in whole or in part, without written permission from Great Minds® . Noncommercial use is licensed pursuant to a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license; for more information, go to http://greatminds.org/copyright. Great Minds and Eureka Math are registered trademarks of Great Minds®.

Printed in the U.S.A. This book may be purchased from the publisher at eureka-math.org. 10 9 8 7 6 5 4 3 2 1

ISBN 978-1-64054-387-4

G12-M2-UTE-1.3.0-05.2018 ƵƌĞŬĂDĂƚŚ͗^ƚŽƌǇŽĨ&ƵŶĐƚŝŽŶƐŽŶƚƌŝďƵƚŽƌƐ DŝŵŝůŬŝƌĞ͕>ĞĂĚtƌŝƚĞƌͬĚŝƚŽƌ͕ůŐĞďƌĂ/ DŝĐŚĂĞůůůǁŽŽĚ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ dŝĂŚůƉŚŽŶƐŽ͕WƌŽŐƌĂŵDĂŶĂŐĞƌͶƵƌƌŝĐƵůƵŵWƌŽĚƵĐƚŝŽŶ ĂƚƌŝŽŶĂŶĚĞƌƐŽŶ͕WƌŽŐƌĂŵDĂŶĂŐĞƌͶ/ŵƉůĞŵĞŶƚĂƚŝŽŶ^ƵƉƉŽƌƚ ĞĂƵĂŝůĞǇ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ^ĐŽƚƚĂůĚƌŝĚŐĞ͕>ĞĂĚDĂƚŚĞŵĂƚŝĐŝĂŶĂŶĚ>ĞĂĚƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ŚƌŝƐƚŽƉŚĞƌĞũĂƌ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ŶĚƌĞǁĞŶĚĞƌ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ŽŶŶŝĞĞƌŐƐƚƌĞƐƐĞƌ͕DĂƚŚƵĚŝƚŽƌ ŚƌŝƐůĂĐŬ͕DĂƚŚĞŵĂƚŝĐŝĂŶĂŶĚ>ĞĂĚtƌŝƚĞƌ͕ůŐĞďƌĂ// 'ĂŝůƵƌƌŝůů͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ĂƌůŽƐĂƌƌĞƌĂ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ĞƚŚŚĂŶĐĞ͕^ƚĂƚŝƐƚŝĐŝĂŶ͕ƐƐĞƐƐŵĞŶƚĚǀŝƐŽƌ͕^ƚĂƚŝƐƚŝĐƐ ŶĚƌĞǁŚĞŶ͕ĚǀŝƐŝŶŐDĂƚŚĞŵĂƚŝĐŝĂŶ DĞůǀŝŶĂŵĂŽůĂŽ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ tĞŶĚǇĞŶĞƐƚĞŶ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ :ŝůůŝŶŝǌ͕WƌŽŐƌĂŵŝƌĞĐƚŽƌ >Žƌŝ&ĂŶŶŝŶŐ͕DĂƚŚƵĚŝƚŽƌ :ŽĞ&ĞƌƌĂŶƚĞůůŝ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ůůĞŶ&Žƌƚ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ <ĂƚŚǇ&ƌŝƚǌ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ dŚŽŵĂƐ'ĂĨĨĞǇ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ^ŚĞƌŝ'ŽŝŶŐƐ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ WĂŵ'ŽŽĚŶĞƌ͕>ĞĂĚtƌŝƚĞƌͬĚŝƚŽƌ͕'ĞŽŵĞƚƌǇĂŶĚWƌĞĐĂůĐƵůƵƐ ^ƚĞĨĂŶŝĞ,ĂƐƐĂŶ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ^ŚĞƌƌŝ,ĞƌŶĂŶĚĞǌ͕DĂƚŚƵĚŝƚŽƌ Žď,ŽůůŝƐƚĞƌ͕DĂƚŚƵĚŝƚŽƌ WĂƚƌŝĐŬ,ŽƉĨĞŶƐƉĞƌŐĞƌ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ :ĂŵĞƐ<ĞǇ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ :ĞƌĞŵǇ<ŝůƉĂƚƌŝĐŬ͕DĂƚŚĞŵĂƚŝĐƐĚƵĐĂƚŽƌ͕ůŐĞďƌĂ// :ĞŶŶǇ<ŝŵ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ƌŝĂŶ<Žƚǌ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ,ĞŶƌǇ<ƌĂŶĞŶĚŽŶŬ͕>ĞĂĚtƌŝƚĞƌͬĚŝƚŽƌ͕^ƚĂƚŝƐƚŝĐƐ zǀŽŶŶĞ>Ăŝ͕DĂƚŚĞŵĂƚŝĐŝĂŶ͕'ĞŽŵĞƚƌǇ ŽŶŶŝĞ>ĂƵŐŚůŝŶ͕DĂƚŚƵĚŝƚŽƌ ƚŚĞŶĂ>ĞŽŶĂƌĚŽ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ :ĞŶŶŝĨĞƌ>ŽĨƚŝŶ͕WƌŽŐƌĂŵDĂŶĂŐĞƌͶWƌŽĨĞƐƐŝŽŶĂůĞǀĞůŽƉŵĞŶƚ :ĂŵĞƐDĂĚĚĞŶ͕DĂƚŚĞŵĂƚŝĐŝĂŶ͕>ĞĂĚtƌŝƚĞƌ͕'ĞŽŵĞƚƌǇ EĞůůDĐŶĞůůǇ͕WƌŽũĞĐƚŝƌĞĐƚŽƌ ĞŶDĐĂƌƚǇ͕DĂƚŚĞŵĂƚŝĐŝĂŶ͕>ĞĂĚtƌŝƚĞƌ͕'ĞŽŵĞƚƌǇ ^ƚĂĐŝĞDĐůŝŶƚŽĐŬ͕ŽĐƵŵĞŶƚWƌŽĚƵĐƚŝŽŶDĂŶĂŐĞƌ ZŽďĞƌƚDŝĐŚĞůŝŶ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ŚŝŚDŝŶŐ,ƵĂŶŐ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ WŝĂDŽŚƐĞŶ͕>ĞĂĚtƌŝƚĞƌͬĚŝƚŽƌ͕'ĞŽŵĞƚƌǇ :ĞƌƌǇDŽƌĞŶŽ͕^ƚĂƚŝƐƚŝĐŝĂŶ ŚƌŝƐDƵƌĐŬŽ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ^ĞůĞŶĂKƐǁĂůƚ͕>ĞĂĚtƌŝƚĞƌͬĚŝƚŽƌ͕ůŐĞďƌĂ/͕ůŐĞďƌĂ//͕ĂŶĚWƌĞĐĂůĐƵůƵƐ ZŽǆǇWĞĐŬ͕DĂƚŚĞŵĂƚŝĐŝĂŶ͕>ĞĂĚtƌŝƚĞƌ͕^ƚĂƚŝƐƚŝĐƐ EŽĂŵWŝůůŝƐĐŚĞƌ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ dĞƌƌŝĞWŽĞŚů͕DĂƚŚƵĚŝƚŽƌ ZŽďZŝĐŚĂƌĚƐŽŶ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ <ƌŝƐƚĞŶZŝĞĚĞů͕DĂƚŚƵĚŝƚdĞĂŵ>ĞĂĚ ^ƉĞŶĐĞƌZŽďǇ͕DĂƚŚƵĚŝƚŽƌ tŝůůŝĂŵZŽƌŝƐŽŶ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ ůĞǆ^ĐǌĞƐŶĂŬ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ DŝĐŚĞů^ŵŝƚŚ͕DĂƚŚĞŵĂƚŝĐŝĂŶ͕ůŐĞďƌĂ// ,ĞƐƚĞƌ^ƵƚƚŽŶ͕ƵƌƌŝĐƵůƵŵtƌŝƚĞƌ :ĂŵĞƐdĂŶƚŽŶ͕ĚǀŝƐŝŶŐDĂƚŚĞŵĂƚŝĐŝĂŶ ^ŚĂŶŶŽŶsŝŶƐŽŶ͕>ĞĂĚtƌŝƚĞƌͬĚŝƚŽƌ͕^ƚĂƚŝƐƚŝĐƐ ƌŝĐtĞďĞƌ͕DĂƚŚĞŵĂƚŝĐƐĚƵĐĂƚŽƌ͕ůŐĞďƌĂ// ůůŝƐŽŶtŝƚĐƌĂĨƚ͕DĂƚŚƵĚŝƚŽƌ ĂǀŝĚtƌŝŐŚƚ͕DĂƚŚĞŵĂƚŝĐŝĂŶ͕'ĞŽŵĞƚƌǇ Board of Trustees Lynne Munson, President and Executive Director of Great Minds Nell McAnelly, Chairman, Co-Director Emeritus of the Gordon A. Cain Center for STEM Literacy at Louisiana State University William Kelly, Treasurer, Co-Founder and CEO at ReelDx Jason Griffiths, Secretary, Director of Programs at the National Academy of Advanced Teacher Education Pascal Forgione, Former Executive Director of the Center on K-12 Assessment and Performance Management at ETS Lorraine Griffith, Title I Reading Specialist at West Buncombe Elementary School in Asheville, North Carolina Bill Honig, President of the Consortium on Reading Excellence (CORE) Richard Kessler, Executive Dean of Mannes College the New School for Music Chi Kim, Former Superintendent, Ross School District Karen LeFever, Executive Vice President and Chief Development Officer at ChanceLight Behavioral Health and Education Maria Neira, Former Vice President, New York State United Teachers This page intentionally left blank ^dKZzK&&hEd/KE^ DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ

WZ>h>h^EsEdKW/^ͻDKh>Ϯ Table of Contents1 sĞĐƚŽƌƐĂŶĚDĂƚƌŝĐĞƐ

Module Overview ...... 3 Topic A: Networks and Matrices...... 15 Lesson 1: Introduction to Networks ...... 16 Lesson 2: Networks and Matrix Arithmetic ...... 33 Lesson 3: Matrix Arithmetic in Its Own Right ...... 46 Topic B: Linear Transformations of Planes and Space ...... 64 Lesson 4: Linear Transformations Review ...... 66 Lesson 5: Coordinates of Points in Space ...... 81 Lesson 6: Linear Transformations as Matrices ...... 106 Lesson 7: Linear Transformations Applied to Cubes ...... 122 Lessons 8–9: Composition of Linear Transformations ...... 138 Lesson 10: Matrix Multiplication Is Not Commutative ...... 167 Lesson 11: Matrix Addition Is Commutative ...... 177 Lesson 12: Matrix Multiplication Is Distributive and Associative ...... 187 Lesson 13: Using Matrix Operations for Encryption ...... 199 Mid-Module Assessment and Rubric ...... 215 Topics A through B (assessment 1 day, return 1 day, remediation or further applications 2 days) Topic C: Systems of Linear Equations...... 239 Lesson 14: Solving Equations Involving Linear Transformations of the Coordinate Plane ...... 241 Lesson 15: Solving Equations Involving Linear Transformations of the Coordinate Space ...... 255 Lesson 16: Solving General Systems of Linear Equations ...... 270 Topic D: Vectors in Plane and Space ...... 282 Lesson 17: Vectors in the Coordinate Plane ...... 285

1Each lesson is ONE day, and ONE day is considered a 45-minute period.

DŽĚƵůĞϮ: Vectors and Matrices 1

Lesson 18: Vectors and Translation Maps ...... 303 Lesson 19: Directed Line Segments and Vectors ...... 326 Lesson 20: Vectors and Stone Bridges ...... 344 Lesson 21: Vectors and the Equation of a Line ...... 362 Lesson 22: Linear Transformations of Lines ...... 376 Lessons 23–24: Why Are Vectors Useful? ...... 389 Topic E: First-Person Video Games—Projection Matrices ...... 416 Lesson 25: First-Person Computer Games...... 418 Lesson 26: Projecting a 3-D Object onto a 2-D Plane ...... 436 Lesson 27: Designing Your Own Game ...... 448 End-of-Module Assessment and Rubric ...... 462 Topics C through E (assessment 1 day, return 1 day, remediation or further applications 3 days)

2 DŽĚƵůĞϮ: Vectors and Matrices

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

Precalculus and Advanced Topics ͻ Module 2 sĞĐƚŽƌƐĂŶĚDĂƚƌŝĐĞƐ

OVERVIEW In Module 1, students learned that throughout the 1800s, mathematicians encountered a number of disparate situations that seemed to call for displaying information via tables and performing arithmetic operations on those tables. One such context arose in Module 1, where students saw the utility of representing linear transformations in the two-dimensional coordinate plane via matrices. Students viewed matrices as representing transformations in the plane and developed an understanding of multiplication of a matrix by a vector as a transformation acting on a point in the plane. This module starts with a second context for matrix representation, networks. In Topic A, students look at incidence relationships in networks and encode information about them via high- dimensional matrices. Questions on counting routes, the results of combining networks, payoffs, and other applications, provide context and use for matrix manipulations: matrix addition and subtraction, matrix product, and multiplication of matrices by scalars. The question naturally arises as to whether there is a geometric context for higher-dimensional matrices as there is for 2×2 matrices. Topic B explores this question, extending the concept of a linear transformation from Module 1 to linear transformations in three- (and higher-) dimensional space. The geometric effect of matrix operations—matrix product, matrix sum, and scalar multiplication—are examined, and students come to see, geometrically, that matrix multiplication for square matrices is not a commutative operation but that it still satisfies the associative and distributive properties. The geometric and arithmetic roles of the zero matrix and identity matrix are discussed, and students see that a multiplicative inverse to a square matrix exists precisely when the determinant of the matrix (given by the area of the image of the unit square in two- dimensional space, or the volume of the image of the unit cube in three-dimensional space) is nonzero. This work is phrased in terms of matrix operations on vectors, seen as matrices with one column. Topic C provides a third context for the appearance of matrices via the study of systems of linear equations. Students see that a system of linear equations can be represented as a single matrix equation in a vector variable, and that one can solve the system with the aid of the multiplicative inverse to a matrix if it exists. Topic D opens with a formal definition of a vector (the motivation and context for it is well in place at this point) and then explores the arithmetical work for vector addition, subtraction, scalar multiplication, and vector magnitude along with the geometrical frameworks for these operations. Students also solve problems involving velocity and other quantities that can be represented by vectors. Parametric equations are introduced in Topic D allowing students to connect their prior work with functions to vectors.

DŽĚƵůĞϮ: Vectors and Matrices 3

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

The module ends with Topic E where students apply the knowledge developed in this module to understand how first-person video games use matrix operations to project three-dimensional objects onto two- dimensional screens and animate those images to give the illusion of motion.

Focus Standards

Represent and model with vector quantities.

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their .(ݒ ,||ܞ|| ,|ܞ| ,ܞ ,.magnitudes (e.g

(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

(+) Add and subtract vectors.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Understand vector subtraction ܞെܟ as ܞ +(െܟ), where െܟ is the additive inverse of ܟ, with the same magnitude as ܟ and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

(+) Multiply a vector by a scalar.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as .(ݒ௫, ݒ௬)=(ܿݒ௫, ܿݒ௬)ܿ

ݒ. Compute the|ܿ| = ||ܞܿ|| using ܞܿ Compute the magnitude of a scalar multiple for ܞ is either along ܞܿ knowing that when |ܿ|ݒ്0, the direction of ܞܿ direction of (ܿ >0) or against ܞ (for ܿ <0). Perform operations on matrices and use matrices in applications

(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

4 DŽĚƵůĞϮ: Vectors and Matrices

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

(+) Add, subtract, and multiply matrices of appropriate dimensions.

(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

^ŽůǀĞƐǇƐƚĞŵƐŽĨeƋƵĂƚŝŽŶƐ

(+) Represent a system of linear equations as a single matrix equation in a vector variable.

(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension ͵ൈ͵ or greater).

&ŽƵŶĚĂƚŝŽŶĂů^ƚĂŶĚĂƌĚƐ

ZĞĂƐŽŶƋƵĂŶƚŝƚĂƚŝǀĞůǇĂŶĚƵƐĞƵŶŝƚƐƚŽƐŽůǀĞƉƌŽďůĞŵƐ͘

ƾ Define appropriate quantities for the purpose of descriptive modeling.

WĞƌĨŽƌŵĂƌŝƚŚŵĞƚŝĐŽƉĞƌĂƚŝŽŶƐǁŝƚŚĐŽŵƉůĞǆŶƵŵďĞƌƐ͘

ଶ Know there is a complex number ݅ such that ݅ ൌെͳ, and every complex number has the form ܽ൅ܾ݅ with ܽ and ܾ real. ଶ Use the relation ݅ ൌെͳ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. hƐĞĐŽŵƉůĞǆŶƵŵďĞƌƐŝŶƉŽůǇŶŽŵŝĂůŝĚĞŶƚŝƚŝĞƐ ĂŶĚĞƋƵĂƚŝŽŶƐ͘

Solve quadratic equations with real coefficients that have complex solutions. ଶ Extend polynomial identities to the complex numbers. For example, rewrite ݔ ൅Ͷ as (+) .ሺݔ ൅ ʹ݅ሻሺݔ െ ʹ݅ሻ

DŽĚƵůĞϮ: Vectors and Matrices 5

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

/ŶƚĞƌƉƌĞƚƚŚĞƐƚƌƵĐƚƵƌĞŽĨĞǆƉƌĞƐƐŝŽŶƐ͘

ƾ Interpret expressions that represent a quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret ܲሺͳ ൅ ݎሻ݊ as the product of ܲ and a factor not depending on ܲ. tƌŝƚĞĞǆƉƌĞƐƐŝŽŶƐŝŶĞƋƵŝǀĂůĞŶƚĨŽƌŵƐƚŽƐŽůǀĞƉƌŽďůĞŵƐ͘

Choose and produce an equivalent form of an expression to reveal, and explain properties of the quantity represented by the expression.ƾ

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions. ଵଶ௧ For example, the expression ͳǤͳͷ௧ can be rewritten as ൫ͳǤͳͷଵȀଵଶ൯ ൎ ͳǤͲͳʹଵଶ௧to reveal the approximate equivalent monthly interest rate if the annual rate is ͳͷΨ.

ƌĞĂƚĞĞƋƵĂƚŝŽŶƐƚŚĂƚĚĞƐĐƌŝďĞŶƵŵďĞƌƐŽƌƌĞůĂƚŝŽŶƐŚŝƉƐ͘

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law ܸൌܫܴ to highlight resistance ܴ. hŶĚĞƌƐƚĂŶĚƐŽůǀŝŶŐĞƋƵĂƚŝŽŶƐĂƐĂƉƌŽĐĞƐƐŽĨƌĞĂƐŽŶŝŶŐĂŶĚĞǆƉůĂŝŶƚŚĞƌĞĂƐŽŶŝŶŐ͘

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

^ŽůǀĞĞƋƵĂƚŝŽŶƐĂŶĚŝŶĞƋƵĂůŝƚŝĞƐŝŶŽŶĞǀĂƌŝĂďůĞ͘

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

6 DŽĚƵůĞϮ: Vectors and Matrices

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

^ŽůǀĞƐǇƐƚĞŵƐŽĨĞƋƵĂƚŝŽŶƐ͘

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

ǆƚĞŶĚƚŚĞĚŽŵĂŝŶŽĨƚƌŝŐŽŶŽŵĞƚƌŝĐĨƵŶĐƚŝŽŶƐƵƐŝŶŐƚŚĞƵŶŝƚĐŝƌĐůĞ͘

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for ߨȀ͵, ߨȀͶ and ߨȀ͸, and use the unit circle to express the values of sine, cosine, and tangent .for ߨെݔ, ߨ൅ݔ, and ʹߨ െ ݔ in terms of their values for ݔ, where ݔ is any real number

WƌŽǀĞĂŶĚĂƉƉůǇƚƌŝŐŽŶŽŵĞƚƌŝĐŝĚĞŶƚŝƚŝĞƐ͘

Prove the Pythagorean identity ଶሺߠሻ ൅ ଶሺߠሻ ൌͳ and use it to find ሺߠሻ, ሺߠሻ, or ሺߠሻ given ሺߠሻ, ሺߠሻ, or ሺߠሻ and the quadrant of the angle.

ǆƉĞƌŝŵĞŶƚǁŝƚŚƚƌĂŶƐĨŽƌŵĂƚŝŽŶƐŝŶƚŚĞƉůĂŶĞ͘

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. dƌĂŶƐůĂƚĞďĞƚǁĞĞŶƚŚĞŐĞŽŵĞƚƌŝĐĚĞƐĐƌŝƉƚŝŽŶĂŶĚ ƚŚĞĞƋƵĂƚŝŽŶĨŽƌĂĐŽŶŝĐƐĞĐƚŝŽŶ͘

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Derive the equation of a parabola given a focus and directrix.

DŽĚƵůĞϮ: Vectors and Matrices 7

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ ĨŽƌDĂƚŚĞŵĂƚŝĐĂůWƌĂĐƚŝĐĞ

ZĞĂƐŽŶĂďƐƚƌĂĐƚůǇĂŶĚƋƵĂŶƚŝƚĂƚŝǀĞůǇ͘Students recognize matrices and justify the transformations that they represent. Students use ͵ൈ͵ matrices to solve systems of equations and continue to calculate the determinant of matrices. Students also represent complex numbers as vectors and determine magnitude and direction. Students reason to determine the effect of scalar multiplication and the result of a zero vector.

DŽĚĞůǁŝƚŚŵĂƚŚĞŵĂƚŝĐƐ͘Students initially study matrix multiplication as a tool for modeling networks and create a model of a bus route. Later, students look at matrix transformations and their role in developing video games and create their own video game. The focus of the mathematics in the computer animation is such that students come to see rotating and translating as dependent on matrix operations and the addition vectors.

hƐĞĂƉƉƌŽƉƌŝĂƚĞƚŽŽůƐƐƚƌĂƚĞŐŝĐĂůůǇ͘As students study ͵ൈ͵ matrices, they begin to view matrices as a tool that can solve problems including networks, payoffs, velocity, and force. Students use calculators and computer software to solve systems of three equations and three unknowns using matrices. Computer software is also used to help students visualize three-dimensional changes on a two-dimensional screen and in the creation of their video games. dĞƌŵŝŶŽůŽŐǇ

EĞǁŽƌZĞĐĞŶƚůǇ/ŶƚƌŽĚƵĐĞĚdĞƌŵƐ ƌŐƵŵĞŶƚ (The argument of the complex number ݖ is the radian (or degree) measure of the counterclockwise rotation of the complex plane about the origin that maps the initial ray (i.e., the ray corresponding to the positive real axis) to the ray from the origin through the complex number ݖ (.in the complex plane. The argument of ݖ is denoted ሺݖሻ ŽƵŶĚsĞĐƚŽƌ(A bound vector is a directed line segment (an arrow). For example, the directed line segment ܣܤሬሬሬሬሬԦ is a bound vector whose initial point (or tail) is ܣ and terminal point (or tip) is ܤ. Bound vectors are bound to a particular location in space. A bound vector ܣܤሬሬሬሬሬԦ has a magnitude given by the length of segment ܣܤ and direction given by the ray ܣܤሬሬሬሬሬԦ . Many times, only the magnitude and direction of a bound vector matters, not its position in space. In that case, we consider any translation of that bound vector to represent the same free vector. à A bound vector or a free vector (see below) is often referred to as just a vector throughout this module. The context in which the word vector is being used determines if the vector is free or bound.)

8 DŽĚƵůĞϮ: Vectors and Matrices

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

ŽŵƉůĞǆEƵŵďĞƌ (A complex number is a number that can be represented by a point in the complex plane. A complex number can be expressed in two forms: The rectangular form of a complex number ݖ is ܽ൅ܾ݅ where ݖ corresponds to the point .1 ሺܽǡ ܾሻ in the complex plane, and ݅ is the imaginary unit. The number ܽ is called the real part of ܽ൅ܾ݅, and the number ܾ is called the imaginary part of ܽ൅ܾ݅. Note that both the real and imaginary parts of a complex number are themselves real numbers. ൌȁݖȁ andݎ ሺߠሻ ൅݅ሺߠሻሻ where ሺݎ For ݖ്Ͳ, the polar form of a complex number ݖ is .2 (.ߠൌሺݖሻ, and ݅ is the imaginary unit ŽŵƉůĞǆWůĂŶĞ (The complex plane is a Cartesian plane equipped with addition and multiplication operators defined on ordered pairs by à ĚĚŝƚŝŽŶ: ሺܽǡ ܾሻ ൅ ሺܿǡ ݀ሻ ൌሺܽ൅ܿǡܾ൅݀ሻ When expressed in rectangular form, if ݖൌܽ൅ܾ݅ and ݓൌܿ൅݀݅, then .ݖ൅ݓൌሺܽ൅ܿሻ ൅ ሺܾ൅݀ሻ݅ ሺܿǡ ݀ሻ ൌ ሺܽܿ െ ܾ݀ǡ ܽ݀ ൅ ܾܿሻ ڄ à DƵůƚŝƉůŝĐĂƚŝŽŶ: ሺܽǡ ܾሻ When expressed in rectangular form, if ݖൌܽ൅ܾ݅ and ݓൌܿ൅݀݅, then ݓൌሺܽܿ െ ܾ݀ሻ ൅ሺܽ݀൅ܾܿሻ݅. The horizontal axis corresponding to points of the formڄݖ ሺݔǡ Ͳሻ is called the real axis, and a vertical axis corresponding to points of the form ሺͲǡ ݕሻ is called the imaginary axis.) ŽŶũƵŐĂƚĞ (The conjugate of a complex number of the form ܽ൅ܾ݅ is ܽെܾ݅. The conjugate of ݖ is (.denoted ݖ ܾܽ ĞƚĞƌŵŝŶĂŶƚŽf ૛ൈ૛ DĂƚƌŝǆ(The determinant of the ʹൈʹ matrix ቂ ቃ is the number computed ܿ݀ ܾܽ by evaluating ܽ݀ െ ܾܿ and is denoted by ቀቂ ቃቁ.) ܿ݀ ܽଵଵ ܽଵଶ ܽଵଷ ĞƚĞƌŵŝŶĂŶƚŽf ૜ൈ૜ DĂƚƌŝǆ(The determinant of the ͵ൈ͵ matrix ൥ܽଶଵ ܽଶଶ ܽଶଷ൩ is the number ܽଷଵ ܽଷଶ ܽଷଷ computed by evaluating the expression ܽଶଶ ܽଶଷ ܽଶଵ ܽଶଷ ܽଶଵ ܽଶଶ ܽଵଵ ቀቂ ቃቁ െ ܽଵଶ ቀቂ ቃቁ ൅ ܽଵଷ ቀቂ ቃቁ ܽଷଶ ܽଷଷ ܽଷଵ ܽଷଷ ܽଷଵ ܽଷଶ

ܽଵଵ ܽଵଶ ܽଵଷ and is denoted by ൭൥ܽଶଵ ܽଶଶ ܽଶଷ൩൱.) ܽଷଵ ܽଷଶ ܽଷଷ ŝƌĞĐƚĞĚ'ƌĂƉŚ (A directed graph is an ordered pair ܦൌሺܸǡܧሻ with à ܸ a set whose elements are called vertices or nodes, and à ܧ a set of ordered pairs of vertices, called arcs or directed edges.) ŝƌĞĐƚĞĚ^ĞŐŵĞŶƚ (A directed segment ܣܤሬሬሬሬሬԦ is the line segment ܣܤ together with a direction given by connecting an initial point ܣ to a terminal point ܤ.)

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A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

&ƌĞĞsĞĐƚŽƌ (A free vector is the equivalence class of all directed line segments (arrows) that are equivalent to each other by translation. For example, scientists often use free vectors to describe physical quantities that have magnitude and direction only, freely placing an arrow with the given magnitude and direction anywhere in a diagram where it is needed. For any directed line segment in the equivalence class defining a free vector, the directed line segment is said to be a representation of the free vector or is said to represent the free vector. à A bound vector (see above) or a free vector is often referred to as just a vector throughout this module. The context in which the word vector is being used determines if the vector is free or bound.) /ĚĞŶƚŝƚǇDĂƚƌŝǆ (The ݊ൈ݊ identity matrix is the matrix whose entry in row ݅ and column ݅ for ͳ൑݅൑݊ is ͳ, and whose entries in row ݅ and column ݆ for ͳ൑݅, ݆൑݊ and ്݆݅ are all zero. The identity matrix is denoted by ܫ.) /ŵĂŐŝŶĂƌǇǆŝƐ (See complex plane.) /ŵĂŐŝŶĂƌǇEƵŵďĞƌ (An imaginary number is a complex number that can be expressed in the form ܾ݅ where ܾ is a real number.) /ŵĂŐŝŶĂƌǇWĂƌƚ (See complex number.) /ŵĂŐŝŶĂƌǇhŶŝƚ (The imaginary unit, denoted by ݅, is the number corresponding to the point ሺͲǡ ͳሻ in the complex plane.) /ŶĐŝĚĞŶĐĞDĂƚƌŝǆ (The incidence matrix of a network diagram is the ݊ൈ݊ matrix such that the entry in row ݅ and column ݆ is the number of edges that start at node ݅ and end at node ݆.) /ŶǀĞƌƐĞDĂƚƌŝǆ (An ݊ൈ݊ matrix ܣ is invertible if there exists an ݊ൈ݊ matrix ܤ so that ܣܤ ൌ ܤܣ ൌ ܫ, where ܫ is the ݊ൈ݊ identity matrix. The matrix ܤ, when it exists, is unique and is called the inverse of ܣ and is denoted by ܣିଵ.) >ŝŶĞĂƌ&ƵŶĐƚŝŽŶ (A function ݂ǣ Թ ՜ Թ is called a linear function if it is a polynomial function of degree one, that is, a function with real number domain and range that can be put into the form ሺݔሻ ൌ݉ݔ൅ܾ for real numbers ݉ and ܾ. A linear function of the form ݂ሺݔሻ ൌ݉ݔ൅ܾ is a linear݂ transformation only if ܾൌͲ.) >ŝŶĞĂƌdƌĂŶƐĨŽƌŵĂƚŝŽŶ (A function ܮǣ Թ௡ ՜Թ௡ for a positive integer ݊ is a linear transformation if the following two properties hold Թ௡, and א ܡ ǡܠ ሻ for allܡሺܮሻ ൅ܠሺܮሻ ൌܡ൅ܠሺܮ à ,Թא݇ Թ௡ andאܠ ሻ for allܠሺܮڄሻ ൌ݇ܠሺ݇ܮ à (.is a point in Թ௡ ܠ Թ௡ means thatאܠ where >ŝŶĞĂƌdƌĂŶƐĨŽƌŵĂƚŝŽŶ/ŶĚƵĐĞĚďǇDĂƚƌŝǆ࡭ (Given a ʹൈʹ matrix ܣ, the linear transformation ݔ ݔ ቂݕቃ. Given aڄܣቀቂݕቃቁൌܮ given by the formula ܮ is the linear transformation ܣ induced by matrix ͵ൈ͵ matrix ܣ, the linear transformation induced by matrix ܣ ݅s the linear transformation ܮ given by ݔ ݔ (.ቈݕ቉ڄܣቆቈݕ቉ቇ ൌ ܮ the formula ݖ ݖ

10 DŽĚƵůĞϮ: Vectors and Matrices

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

DĂƚƌŝǆ(An ݉ൈ݊ matrix is an ordered list of ݊݉ real numbers, ܽଵଵǡܽଵଶǡǥǡܽଵ௡ǡܽଶଵǡܽଶଶǡǥǡܽଶ௡ǡǥǡܽ௠ଵǡܽ௠ଶǡǥǡܽ௠௡, organized in a rectangular array of ݉ rows ଵ௡ܽڮ ଵଵ ܽଵଶܽ ଶ௡ܽڮ ଶଵ ܽଶଶܽ (.݆ ൪. The number ܽ௜௝ is called the entry in row ݅ and columnڭڰڭڭ and ݊ columns: ൦ ௠௡ܽڮ ௠ଵ ܽ௠ଶܽ

DĂƚƌŝǆŝĨĨĞƌĞŶĐĞ(Let ܣ be an ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܽ௜௝, and let ܤ be an ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܾ௜௝. Then the matrix difference ܣെܤ is the ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܽ௜௝ െܾ௜௝.) DĂƚƌŝǆWƌŽĚƵĐƚ (Let ܣ be an ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܽ௜௝, and let ܤ be an ݊ൈ݌ matrix whose entry in row ݅ and column ݆ is ܾ௜௝. Then the matrix product ܣܤ is the ݉ൈ݌ (.൅ܽ௜௡ܾ௡௝ڮmatrix whose entry in row ݅ and column ݆ is ܽ௜ଵܾଵ௝ ൅ܽ௜ଶܾଶ௝ ൅ DĂƚƌŝǆ^ĐĂůĂƌDƵůƚŝƉůŝĐĂƚŝŽŶ (Let ݇ be a real number, and let ܣ be an ݉ൈ݊ matrix whose entry in ݅ is the ݉ൈ݊ matrix whose entry in row ܣڄ݇ row ݅ and column ݆ is ܽ௜௝. Then the scalar product (.௜௝ܽڄ݇ and column ݆ is DĂƚƌŝǆ^Ƶŵ (Let ܣ be an ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܽ௜௝, and let ܤ be an ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܾ௜௝. Then the matrix sum ܣ൅ܤ is the ݉ൈ݊ matrix whose entry in row ݅ and column ݆ is ܽ௜௝ ൅ܾ௜௝.) DŽĚƵůƵƐ (The modulus of a complex number ݖ, denoted ȁݖȁ, is the distance from the origin to the (.point corresponding to ݖ in the complex plane. If ݖൌܽ൅ܾ݅, then ȁݖȁ ൌ ξܽଶ ൅ܾଶ EĞƚǁŽƌŬŝĂŐƌĂŵ (A network diagram is a graphical representation of a directed graph where the ݊ vertices are drawn as circles with each circle labeled by a number ͳ through ݊, and the directed edges are drawn as segments or arcs with an arrow pointing from the tail vertex to the head vertex.) ,ሬሬሬሬሬԦ , the opposite vectorܤܣ KƉƉŽƐŝƚĞsĞĐƚŽƌ (For a vector ݒԦ represented by the directed line segment ݒଵ ݒ ሬሬሬሬሬԦ ଶ ௡ ൪ in Թ , then ڭ ൌ൦ܞ If .ܣܤ is the vector represented by the directed line segment ,ܞdenoted െ ݒ௡ െݒଵ െݒଶ (.൪ ڭ ൌ ൦ ܞെ െݒ௡ ሺߠሻ ൅݅ሺߠሻሻ ሺݎ WŽůĂƌ&ŽƌŵŽĨĂŽŵƉůĞǆEƵŵďĞƌ (The polar form of a complex number ݖ is (.ൌȁݖȁ and ߠൌሺݖሻݎ where ݒଵ ௡ ݒଶ ൪ or ڭ denoted by ൦ ,ܞ WŽƐŝƚŝŽŶsĞĐƚŽƌ(For a point ܲሺݒଵǡݒଶǡǥǡݒ௡ሻ in Թ , the position vector ݒ௡ that is represented by the directed line segment ܱܲሬሬሬሬሬԦ from the origin ܞ is a free vector ,ۄݒଵǡݒଶǡǥǡݒ௡ۃ th (.ܞ ሺͲǡͲǡͲǡ ǥ ǡͲሻ to the point ܲ. The real number ݒ௜ is called the ݅ component of the vectorܱ

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A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

ZĞĂůŽŽƌĚŝŶĂƚĞ^ƉĂĐĞ (For a positive integer ݊, the ݊-dimensional real coordinate space, denoted Թ௡, is the set of all ݊-tuple of real numbers equipped with a distance function ݀ that satisfies ଶ ଶ ଶ ൅ሺݕ௡ െݔ௡ሻڮሾሺݔଵǡݔଶǡǥǡݔ௡ሻǡ ሺݕଵǡݕଶǡǥǡݕ௡ሻሿ ൌ ඥሺݕଵ െݔଵሻ ൅ ሺݕଶ െݔଵሻ ൅݀ for any two points in the space. One-dimensional real coordinate space is called a number line, and the two-dimensional real coordinate space is called the Cartesian plane.) ZĞĐƚĂŶŐƵůĂƌ&Žƌŵof a ŽŵƉůĞǆEƵŵďĞƌ (The rectangular form of a complex number ݖ is ܽ൅ܾ݅ where ݖ corresponds to the point ሺܽǡ ܾሻ in the complex plane, and ݅ is the imaginary unit. The number ܽ is called the real part of ܽ൅ܾ݅, and the number ܾ is called the imaginary part of ܽ൅ܾ݅.) dƌĂŶƐůĂƚŝŽŶďy a sĞĐƚŽƌŝŶZĞĂůŽŽƌĚŝŶĂƚĞ^ƉĂĐĞ(A translation by a vector ܞ in Թ௡ is the translation ݒଵ ௡ ௡ ௡ ݒଶ ൪ in ڭ ൌ൦ܞ in Թ . If ܠ for all ܞ൅ܠհܠ ǣԹ ՜Թ given by the map that takesܞܶ transformation ݒ௡ ݔଵ ݔଵ ൅ݒଵ ௡ ݔଶ ݔଶ ൅ݒଶ ௡ (. in Թ ܠ ൪ for all ڭ ൪൲ ൌ ൦ ڭ ൮൦ ܞܶ Թ , then ݔ௡ ݔ௡ ൅ݒ௡ sĞĐƚŽƌ (A vector is described as either a bound or free vector depending on the context. We refer to both bound and free vectors as vectors throughout this module.) sĞĐƚŽƌĚĚŝƚŝŽŶ(For vectors ܞ and ܟ in Թ௡, the sum ܞ൅ܟ is the vector whose ݅th component is the ݒଵ ݓଵ th ݒଶ ݓଶ ௡ ൪ in Թ , then ڭ ൌ൦ܟ ൪ and ڭ ൌ൦ܞ for ͳ൑݅൑݊. If ܟ and ܞ sum of the ݅ components of ݒ௡ ݓ௡ ݒଵ ൅ݓଵ ݒଶ ൅ݓଶ (.൪ ڭ ൌ൦ܟ൅ܞ ݒ௡ ൅ݓ௡ sĞĐƚŽƌDĂŐŶŝƚƵĚĞ(The magnitude or length of a vector ܞ, denoted ԡܞԡ, is the length of any directed ݒଵ ݒଶ ௡ ଶ ଶ ଶ ൅ݒ௡ , whichڮԡ ൌ ඥݒଵ ൅ݒଶ ൅ܞ൪ in Թ , then ԡ ڭ ൌ൦ܞ line segment that represents the vector. If ݒ௡ (.is the distance from the origin to the associated point ܲሺݒଵǡݒଶǡǥǡݒ௡ሻ sĞĐƚŽƌZĞƉƌĞƐĞŶƚĂƚŝŽŶof a ŽŵƉůĞǆEƵŵďĞƌ (The vector representation of a complex number ݖ is associated to the point ݖ in the complex plane. If ݖൌܽ൅ܾ݅ for two real ܢ the position vector ܽ numbers ܽ and ܾ, then ܢൌቂ ቃ.) ܾ

12 DŽĚƵůĞϮ: Vectors and Matrices

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

is the ܞڄ݇ in Թ௡ and a real number ݇, the scalar product ܞ sĞĐƚŽƌ^ĐĂůĂƌDƵůƚŝƉůŝĐĂƚŝŽŶ (For a vector vector whose ݅th component is the product of ݇ and the ݅th component of ܞ for ͳ൑݅൑݊. If ݇ is a ݒଵ ݇ݒଵ ݒ ݇ݒ the ,ܟ and ܞ ൌ൦ ଶ൪.)sĞĐƚŽƌ^ƵďƚƌĂĐƚŝŽŶ (For vectorsܞڄ݇ ൌ൦ ଶ൪ in Թ௡, thenܞ real number and ڭ ڭ ݒ௡ ݇ݒ௡ ݒଵ ݒଶ ൪ and ڭ ൌ൦ܞ ሻ. Ifܟ൅ሺെܞൌܟെܞ ,that is ;ܟ and the opposite of ܞ is the sum of ܟെܞ difference ݒ௡ ݓଵ ݒଵ െݓଵ ݓଶ ௡ ݒଶ െݓଶ ൪.)ĞƌŽDĂƚƌŝǆ (The ݉ൈ݊ zero matrix is the ݉ൈ݊ matrix ڭ ൌ൦ܟെܞ ൪ in Թ , then ڭ ൌ൦ܟ ݓ௡ ݒ௡ െݓ௡ ͲͲ in which all entries are equal to zero. For example, the ʹൈʹ zero matrix is ቂ ቃ, and the ͵ൈ͵ ͲͲ ͲͲͲ zero matrix is ൥ͲͲͲ൩.) ͲͲͲ ĞƌŽsĞĐƚŽƌ (The zero vector in Թ௡ is the vector in which each component is equal to zero. For Ͳ Ͳ example, the zero vector in Թଶ is ቂ ቃ, and the zero vector in Թଷ is ൥Ͳ൩.) Ͳ Ͳ

&ĂŵŝůŝĂƌ dĞƌŵƐĂŶĚ^ǇŵďŽůƐϮ Dilation Rectangular Form RotationTranslation

^ƵŐŐĞƐƚĞĚdŽŽůƐĂŶĚZĞƉƌĞƐĞŶƚĂƚŝŽŶƐ Graphing Calculator Wolfram Alpha Software Geometer’s Sketchpad Software ALICE 3.1 Geogebra Software

2These are terms and symbols students have seen previously.

DŽĚƵůĞϮ: Vectors and Matrices 13

A STORY OF FUNCTIONS DŽĚƵůĞKǀĞƌǀŝĞǁ DϮ PRECALCULUS AND ADVANCED TOPICS

ƐƐĞƐƐŵĞŶƚ^ƵŵŵĂƌǇ

ƐƐĞƐƐŵĞŶƚdǇƉĞ ĚŵŝŶŝƐƚĞƌĞĚ Format

Mid-Module After Topic B Constructed response with rubric Assessment Task

End-of-Module After Topic E Constructed response with rubric Assessment Task

14 DŽĚƵůĞϮ: Vectors and Matrices

^dKZzK&&hEd/KE^

DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ

PRECALCULUS AND ADVANCED TOPICS ͻDKh> 2

Topic A EĞƚǁŽƌŬƐĂŶĚDĂƚƌŝĐĞƐ

&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

(+) Add, subtract, and multiply matrices of appropriate dimensions. /ŶƐƚƌƵĐƚŝŽŶĂůĂǇƐ͗ 3 >ĞƐƐŽŶϭ͗ Introduction to Networks (E)1 >ĞƐƐŽŶϮ͗ Networks and Matrix Arithmetic (E) >ĞƐƐŽŶϯ͗ Matrix Arithmetic in Its Own Right (E)

In Topic A, students are introduced to networks, which are a second application of matrices, and use public transportation routes to study the usefulness of matrices. In Lesson 1, students discover the value of matrices in counting routes. Students see that the arithmetic and properties of matrices are the same regardless of the application. Lesson 2 builds on the work of networks as students study a network of subway lines between four cities and a social network. This work allows students to further explore multiplication by a scalar, and matrix addition and subtraction. In Lesson 3, students continue their study of public transportation networks. Students revisit matrix addition, subtraction, and multiplication as well as multiplication by a scalar as they work with square and rectangular matrices. They begin to see that matrix multiplication is not commutative. In this topic, students make sense of problems involving transportation networks and matrices. Students are asked to relate and explain the connection between real-world situations, such as networks, and their matrix representations. Matrices are used as a tool to organize and represent transportation and social network systems and to show links between these systems with precise and careful calculations.

1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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A STORY OF FUNCTIONS Lesson 1 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 1: Introduction to Networks

Student Outcomes Students use matrices to represent and manipulate data from network diagrams.

Lesson Notes In Module 1, students used matrix multiplication to perform a linear transformation in the plane or in space. In this module, they use matrices to model new phenomena, starting with networks and graphs, and develop arithmetic operations on matrices in this context. Students extend their understanding of the usefulness of matrices—from representing functions to representing connections between people and places—as they represent and solve a wide variety of problems using matrices. These lessons emphasize understanding matrix arithmetic in authentic situations and then performing appropriate calculations to solve problems. Technology may be incorporated to perform arithmetic operations on large matrices. These lessons address representing and manipulating data using matrices, understanding multiplication by a scalar, and performing arithmetic operations on matrices of appropriate dimensions.

NETWORK DIAGRAM: A network diagram is a graph that represents vertices with circles and connects the vertices by segments, edges, or arcs. A directed graph, or digraph, is a network diagram in which the edges have arrows indicating the direction being traversed. Note: The term arc is defined differently here than it was in Geometry. Here it means any curve that connects one vertex to another vertex (or a vertex to itself). Also, the vertices are sometimes represented by circles in which the number of the node is written. The directed edges can intersect each other. The figure at left is a specific type of network diagram that has arrows indicating direction. This is the more specific network diagram, which can also be called a directed graph. The figure at right is a network diagram.

In these lessons, a network is any system that can be described by a network diagram that represents the connections between the parts of the system. Matrices are tools that allow us to organize information from these networks and to perform calculations.

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Classwork Opening (5 minutes) Scaffolding: Open this lesson by asking students to share what they think of when they hear the word Use a word wall to support network. This activates their prior knowledge about everyday usage of this word, which English language learners should then transition to the mathematical meaning of a network. Give students a minute in your classroom when to think silently, and then have them turn and discuss their thoughts with a partner. Ask you introduce new one or two students to share their responses with the entire class. vocabulary. The word network can be What do you think of when you hear the word network? posted on your word wall à I think of social networks, a computer network, a television network, or after you complete the networking with professionals in a particular career field or group. opening. For our purposes, a network is a system of interrelated objects (such as people Create a Frayer diagram or places) that we can represent using a network diagram, as shown below. for networks (see Precalculus and Advanced Topics, Module 1, Lesson 5 Exploratory Challenges 1–3 (15 minutes) for an example). Introduce the network diagram representing bus routes shown in the student materials. Have students read silently and then summarize with a partner the information on the student materials. Ask them to point out the four cities and the bus routes connecting them. Use the terminology of vertices for the cities and edges for the routes. Organize students into small groups, and have them respond to Exercises 1–7. These exercises provide a fairly simple context to explore the number of bus lines that connect four cities. However, students are being shown an entirely new representation of information so they have to make sense and persevere when working on this with their group members.

Exploratory Challenge 1

A network diagram depicts interrelated objects by using circles to represent the objects and directed edges drawn as segments or arcs between related objects with arrows to denote direction. The network diagram below shows the bus routes that run between four cities, forming a network. The arrows indicate the direction the buses travel.

Figure 1

a. How many ways can you travel from City 1 to City 4? Explain how you know.

There are three ways to travel from City 1 to City 4. According to the arrows, you can travel from City 1 to City 2 to City 4, from City 1 to City 3 to City 4, or from City 1 to City 3 to City 2 to City 4.

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b. What about these bus routes doesn’t make sense?

It is not possible to leave City 4. The direction of the arrows show that there are no bus routes that lead from City 4 to any other city in this network. It is not possible to travel to City 1. The direction of the arrows show that there are no bus routes that lead to City 1.

c. It turns out there was an error in printing the first route map. An updated network diagram showing the bus routes that connect the four cities is shown below in Figure 2. Arrows on both ends of an edge indicate that buses travel in both directions.

Figure 2

How many ways can you reasonably travel from City 4 to City 1 using the route map in Figure 2? Explain how you know.

There is only one reasonable way. You must go from City 4 to City 2 to City 3 to City 1. The arrows indicate that there is only one route to City 1, which comes from City 3. However, it is possible to travel from City 4 to City 2 to City 3 as many times as desired before traveling from City 4 to City 2 to City 1.

Exploratory Challenge 2

A rival bus company offers more routes connecting these four cities as shown in the network diagram in Figure 3.

Figure 3

a. What might the loop at City 1 represent?

This loop could represent a tour bus that takes visitors around City 1 but does not leave the city limits.

b. How many ways can you travel from City 1 to City 4 if you want to stop in City 2 and make no other stops?

There are three bus routes from City 1 to City 2 and two bus routes from City 2 to City 4, so there are ૟ possible ways to travel from City 1 to City 4.

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c. How many possible ways are there to travel from City 1 to City 4 without repeating a city?

City 1 to City 4 with no stops: no routes

City 1 to City 4 with a stop in City 2: ૟ routes

City 1 to City 4 with a stop in City 3: ૚ route

City 1 to City 4 via City 3 then City 2: ૛ routes

City 1 to City 4 via City 2 then City 3: ૟ routes

Total ways: ૙൅૟൅૚൅૛൅૟ൌ૚૞ possible ways to travel from City 1 to City 4 without visiting a city more than once.

Pause here to debrief student work so far. Have different groups present their findings, and use the discussion questions that follow to make sure students begin to understand that, as networks become more complicated, an organized numeric representation may be beneficial. The next discussion focuses on using a table as a tool to organize information. As a transportation network grows, these diagrams become more complicated, and keeping track of all of the information can be challenging. People that work with complicated networks use computers to manage and manipulate this information. Provide students with an example of a more complex network diagram by displaying a visual representation of the New York City subway system, an airline flight map, or show a computer networking diagram. What challenges did you encounter as you tried to answer Exercises 1–6? à We had to keep track of the different routes in an organized manner to make sure our answers were accurate. How might we represent the possible routes in a more organized manner? à We could make a list or a table. Organizing information in a table can make things easier, especially if computers are used to count and keep track of routes and schedules. What if we wanted to represent a count of ALL the routes that connect ALL the cities? How could we best organize that information? à We need a grid or table to show connections from city to city. Have students return to their groups to complete this exercise to conclude the Exploratory Challenge 3 using the table below.

Exploratory Challenge 3

Let’s consider a “direct route” to be a route from one city to another without going through any other city. Organize the number of direct routes from each city into the table shown below. The first row showing the direct routes between City 1 and the other cities is complete for you.

Destination Cities 1 2 3 4 1 ૚ ૜ ૚ ૙ 2 ૛ ૙ ૛ ૛ 3 ૛ ૚ ૙ ૚

Cities of Origin 4 ૙ ૛ ૚ ૙

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Discussion (5 minutes) After students have had a few minutes to complete this exercise, copy the table onto a white board, and lead this next discussion to connect the tabular representation of the network to a matrix. Circulate to check for understanding. In particular, ensure that students understand that a direct route means a route from one city to another without going through any other city. If we agree that the rows represent the cities of origin and the columns represent the destination cities, then we do not really need the additional labels. At this point, erase or display the same table with the row and column labels removed. The new table is shown below. 1 2 3 4 1 ͳ ͵ ͳ Ͳ 2 ʹ Ͳ ʹ ʹ 3 ʹ ͳ Ͳ ͳ 4 Ͳ ʹ ͳ Ͳ Next, if we agree to list the cities in numerical order from top to bottom and left to right, then we don’t need the individual row or column labels. At this point, erase the city number labels or display the table shown below. ͳ ͵ ͳ Ͳ ʹ Ͳ ʹ ʹ ʹ ͳ Ͳ ͳ Ͳ ʹ ͳ Ͳ Does this array of numbers written like this look familiar to you? What about if we erase the table grid lines? à It looks like a matrix. Now erase the lines and just show the numbers. ͳ ͵ ͳ Ͳ ʹ Ͳ ʹ ʹ ʹ ͳ Ͳ ͳ Ͳ ʹ ͳ Ͳ In the last module, we used matrices to represent transformations, but they are Scaffolding: useful in a wide variety of mathematical situations. What do you recall about For struggling students, matrices from Module 1? emphasize that the entry in à A matrix is an array of numbers organized into ݉ rows and ݊ columns. row ݅ and column ݆ of the A matrix containing ݉ rows and ݊ columns has dimensions ݉ൈ݊. The matrix counts the number of

entry in the first row and first column is referred to as ܽଵǡଵ. In general, direct routes from city ݅ to city th th the entry in the ݅ row and ݆ column would be denoted ܽ௜ǡ௝. ݆. Illustrate with a concrete example.

Exercises (12 minutes) These exercises can be done as a whole class or within small groups, depending on how much students recall about matrix terminology from the previous module.

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Exercises

1. Use the network diagram in Figure 3 to represent the number of direct routes between the four cities in a matrix ࡾ. ૚૜૚૙ ૛૙૛૛ ࡾൌ൦ ൪ ૛૚૙૚ ૙૛૚૙

2. What is the value of ࢘૛ǡ૜? What does it represent in this situation? The value is ૛. It is the number of direct routes from City 2 to City 3.

3. What is the value of ࢘૛ǡ૜ ή࢘૜ǡ૚, and what does it represent in this situation? The value is ૝. It represents the number of one-stop routes between City 2 and City 1 that pass through City 3.

4. Write an expression for the total number of one-stop routes from City 4 and City 1, and determine the number of routes stopping in one city.

࢘૝ǡ૛ ή࢘૛ǡ૚ ൅࢘૝ǡ૜ ή࢘૜ǡ૚ ൌ૛ή૛൅૚ή૛ൌ૟

5. Do you notice any patterns in the expression for the total number of one-stop routes from City 4 and City 1?

The middle indices in each expression are the same in each term and represent the cities where a stop was made.

6. Create a network diagram for the matrices shown below. Each matrix represents the number of transportation routes that connect four cities. The rows are the cities you travel from, and the columns are the cities you travel to. ૙૚૙૚ ૚૙૚૚ a. ࡾൌ൦ ൪ ૚૚૙૙ ૚૚૙૙

૙૚૙૙ ૙૚૛૚ b. ࡾൌ൦ ൪ ૚૙૙૛ ૙૚૛૙

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This last exercise introduces a new type of network diagram. This exercise helps students understand that networks can be used to represent a wide variety of situations where tracking connections between a number of entities is important. If time is running short, this exercise could be used as an additional Problem Set exercise.

Here is a type of network diagram called an arc diagram.

Suppose the points represent eleven students in your mathematics class, numbered 1 through 11. Suppose the arcs above and below the line of vertices 1–11 represent people who are friends on a social network.

7. Complete the matrix that shows which students are friends with each other on this social network. The first row has been completed for you. ͲͲͳͳͳͲͳͳͳͲͳ ې̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ۍ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ۑ ێ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ۑ ێ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ۑ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ێ ے̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ ̴̴ۏ

ͲͲͳͳͳͲͳͳͳͲͳ ېͲͲͲͲͲͲͳͲͲͲͲۍ ۑͳͲͲͲͳͳͳͲͲͲͲێ ۑ ێ ͳͲͲͲͲͲͲͲͲͲͲ ۑ ێ ۑͳͲͳͲͲͲͳͲͳͳͳێ ۑͲͲͳͲͲͲͲͲͲͲͲێ ۑͳͳͳͲͳͲͲͲͳͳͲێ ۑͳͲͲͲͲͲͲͲͲͲͲێ ۑͳͲͲͲͳͲͳͲͲͲͳێ ۑͲͲͲͲͳͲͳͲͲͲͲێ ےͳͲͲͲͳͲͲͲͳͲͲۏ

a. Number 1 is not friends with number 10. How many ways could number 1 get a message to number 10 by only going through one other friend?

There are ૛ ways. Number 1 to number 5 and then number 5 to number 10 or number 1 to number 7 and numbers 7 to number 10.

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b. Who has the most friends in this network? Explain how you know.

Number 1 is friends with ૠ people, and that’s more than anyone else, so number 1 has the most friends in this network.

c. Is everyone in this network connected at least as a friend of a friend? Explain how you know.

No. Number 2 is not connected to number 4 as a friend of a friend because number 2 is only friends with number 7, and number 7 and number 4 are not friends.

d. What is entry ࢇ૛ǡ૜? Explain its meaning in this context. The entry is ૙. Number 2 is not friends with number 3.

Closing (3 minutes) Ask students to summarize their work from this lesson with a partner or on their own in writing. How can you find the total number of possible routes between two locations in a network? à Multiply the number of routes from each city stopping at common cities and add the products. How does a matrix help you to organize and represent information in a network? à A matrix allows for succinct numerical representations which are advantageous when network diagrams are complex and contain many nodes and edges. How does a network diagram help you to organize and represent information? à It shows connections between points to help you see the data before organizing it.

Lesson Summary

Students organize data and use matrices to represent data in an organized way.

A network diagram is a graphical representation of a directed graph where the ࢔ vertices are drawn as circles with each circle labeled by a number ૚ through ࢔, and the directed edges are drawn as segments or arcs with an arrow pointing from the tail vertex to the head vertex.

A matrix is an array of numbers organized into ࢓ rows and ࢔ columns. A matrix containing ࢓ rows and ࢔ columns

has dimensions ࢓ൈ࢔. The entry in the first row and first column is referred to as ࢇ૚ǡ૚. In general, the entry in the th th ࢏ row and ࢐ column would be denoted ࢇ࢏ǡ࢐.

Exit Ticket (5 minutes)

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A STORY OF FUNCTIONS Lesson 1 M2 PRECALCULUS AND ADVANCED TOPICS

Name Date

Lesson 1: Introduction to Networks

Exit Ticket

The following directed graph shows the major roads that connect four cities.

1 2

3 4

1. Create a matrix ܥ that shows the direct routes connecting the four cities.

2. Use the matrix to determine how many ways are there to travel from City 1 to City 4 with one stop in City 2.

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3. What is the meaning of ܿଶǡଷ?

4. Write an expression that represents the total number of ways to travel between City 2 and City 3 without passing through the same city twice. (You can travel through another city on the way from City 2 to City 3.)

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Exit Ticket Sample Solutions

The following directed graph shows the major roads that connect four cities.

1. Create a matrix ࡯ that shows the direct routes connecting the four cities. ૙૜૛૙ ૚૙ ࡯ൌ቎ ૚૛቏ ૚ ૚ ૙૛ ૙ ૛ ૛૙

2. Use the matrix to determine how many ways are there to travel from City 1 to City 4 with one stop in City 2.

૛ൌ૟ڄࢉ૛ǡ૝ ൌ૜ڄ ࢉ૚ǡ૛

3. What is the meaning of ࢉ૛ǡ૜?

Since ࢉ૛ǡ૜ ൌ૚, it means that there is just one road that goes to City 3 from City 2 without going through the other cities.

We will count the routes by the number of stops in intermediate cities.

Direct routes between City 2 and City 3: ࢉ૛ǡ૜ ൌ૚

૛ൌ૛ڄࢉ૚ǡ૜ ൌ૚ڄ One-stop routes through City 1: ࢉ૛ǡ૚

૛ൌ૝ڄࢉ૝ǡ૜ ൌ૛ڄ One-stop routes through City 4: ࢉ૛ǡ૝

૛ൌ૙ڄ૙ڄࢉ૝ǡ૜ ൌ૚ڄ ࢉ૚ǡ૝ڄ Two-stop routes through City 1 and then City 4: ࢉ૛ǡ૚

૛ൌ૙ڄ૙ڄࢉ૚ǡ૜ ൌ૛ڄ ࢉ૝ǡ૚ڄ Two-stop routes through City 4 and then City 1: ࢉ૛ǡ૝

ࢉ૚ǡ૜ ൌ૚൅૛൅૝൅૙൅૙ൌૠڄ ࢉ૝ǡ૚ڄ ࢉ૝ǡ૜ ൅ࢉ૛ǡ૝ڄ ࢉ૚ǡ૝ڄ ࢉ૝ǡ૜ ൅ࢉ૛ǡ૚ڄ ࢉ૚ǡ૜ ൅ࢉ૛ǡ૝ڄ ࢉ૛ǡ૜ ൅ࢉ૛ǡ૚ So there are ૠ ways to travel between City 2 and City 3 without passing through the same city twice.

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Problem Set Sample Solutions

1. Consider the railroad map between Cities 1, 2, and 3, as shown. a. Create a matrix ࡾto represent the railroad map between Cities 1, 2, and 3. ૙૛૚ ࡾൌ൥૚૙૙൩ ૛૜૙

b. How many different ways can you travel from City 1 to City 3 without passing through the same city twice?

૚൅૚ൌ૚ڄ૙൅૙ڄ࢘૚ǡ૛ ή࢘૛ǡ૜ ൅࢘૚ǡ૚ ή࢘૚ǡ૜ ൅࢘૚ǡ૜ ൌ૛

c. How many different ways can you travel from City 2 to City 3 without passing through the same city twice?

૚ൌ૚ڄ࢘૛ǡ૜ ൅࢘૛ǡ૚ ή࢘૚ǡ૜ ൌ૙൅૚

d. How many different ways can you travel from City 1 to City 2 with exactly one connecting stop?

૜ൌ૜ڄ࢘૚ǡ૜ ή࢘૜ǡ૛ ൌ૚

e. Why is this not a reasonable network diagram for a railroad?

More trains arrive in City 2 than leave, and more trains leave City 3 than arrive.

2. Consider the subway map between stations 1, 2, and 3, as shown. a. Create a matrix ࡿto represent the subway map between stations 1, 2, and 3. ૙૛૚ ࡿൌ൥૚૙૛൩ ૛૚૙

b. How many different ways can you travel from station 1 to station 3 without passing through the same station twice?

࢙૚ǡ૜ ൅࢙૚ǡ૛ ή࢙૛ǡ૜ ൌ૚൅૛ή૛ൌ૞

c. How many different ways can you travel directly from station 1 to station 3 with no stops?

࢙૚ǡ૜ ൌ૚

d. How many different ways can you travel from station 1 to station 3 with exactly one stop?

૛ൌ૝ڄ࢙૚ǡ૛ ή࢙૛ǡ૜ ൌ૛

e. How many different ways can you travel from station 1 to station 3 with exactly two stops? Allow for stops at repeated stations.

࢙૚ǡ૜ ൌ૛ή૚ή૚൅૚ή૚ή૛൅૚ή૛ή૚ൌ૟ڄ ࢙૜ǡ૚ڄ ࢙૛ǡ૜ ൅࢙૚ǡ૜ڄ ࢙૜ǡ૛ڄ ࢙૚ǡ૜ ൅࢙૚ǡ૜ڄ ࢙૛ǡ૚ڄ ࢙૚ǡ૛

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3. Suppose the matrix ࡾ represents a railroad map between cities 1, 2, 3, 4, and 5. ૙૚૛૚૚ ې૛૙૚૚૙ۍ ۑ ێ ۑ૚૚૙૛૛ێࡾൌ ۑ૚૚૙૙૛ێ ے૚૚૜૙૙ۏ a. How many different ways can you travel from City 1 to City 3 with exactly one connection?

࢘૞ǡ૜ ൌ૚ή૚൅૚ή૙൅૚ή૜ൌ૝ڄ ࢘૝ǡ૜ ൅࢘૚ǡ૞ڄ ࢘૛ǡ૜ ൅࢘૚ǡ૝ڄ ࢘૚ǡ૛

b. How many different ways can you travel from City 1 to City 5 with exactly one connection?

࢘૝ǡ૞ ൌ૚ή૙൅૛ή૛൅૚ή૛ൌ૟ڄ ࢘૜ǡ૞ ൅࢘૚ǡ૝ڄ ࢘૛ǡ૞ ൅࢘૚ǡ૜ڄ ࢘૚ǡ૛

c. How many different ways can you travel from City 2 to City 5 with exactly one connection?

࢘૝ǡ૞ ൌ૛ή૚൅૚ή૛൅૚ή૛ൌ૟ڄ ࢘૜ǡ૞ ൅࢘૛ǡ૝ڄ ࢘૚ǡ૞ ൅࢘૛ǡ૜ڄ ࢘૛ǡ૚

૙૛૚ 4. Let ࡮ൌ൥૚૚૛൩ represent the bus routes between ૜ cities. ૛૚૚ a. Draw an example of a network diagram represented by this matrix.

b. Calculate the matrix ࡮૛ ൌ࡮࡮. ૝૜૞ ࡮૛ ൌ ൥૞૞૞൩ ૜૟૞

c. How many routes are there between City 1 and City 2 with one stop in between?

૚ൌ૜ڄ૚൅૚ڄ૚൅૛ڄ࢈૚ǡ૚࢈૚ǡ૛ ൅࢈૚ǡ૛࢈૛ǡ૛ ൅࢈૚ǡ૜࢈૜ǡ૛ ൌ૙

d. How many routes are there between City 2 and City 2 with one stop in between?

૚ൌ૞ڄ૚൅૛ڄ૛൅૚ڄ࢈૛ǡ૚࢈૚ǡ૛ ൅࢈૛ǡ૛࢈૛ǡ૛ ൅࢈૛ǡ૜࢈૜ǡ૛ ൌ૚

e. How many routes are there between City 3 and City 2 with one stop in between?

૚ൌ૟ڄ૚൅૚ڄ૛൅૚ڄ࢈૜ǡ૚࢈૚ǡ૛ ൅࢈૜ǡ૛࢈૛ǡ૛ ൅࢈૜ǡ૜࢈૜ǡ૛ ൌ૛

f. What is the relationship between your answers to parts (b)–(e)? Formulate a conjecture.

The numbers ૜, ૞, and ૟ appear in the second column of matrix ࡮૛. It seems that the entry in row ࢏ and column ࢐ of matrix ࡮૛ is the number of ways to get from city ࢏ to city ࢐ with one stop.

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5. Consider the airline flight routes between Cities 1, 2, 3, and 4, as shown.

a. Create a matrix ࡲto represent the flight map between Cities 1, 2, 3, and 4. ૙ ૛૚૛ ૚ ૙૚૛ ࡲൌ቎ ቏ ૚ ૚૙૚ ૛ ૛૚૙

b. How many different routes can you take from City 1 to City 4 with no stops?

ࢌ૚ǡ૝ ൌ૛

c. How many different routes can you take from City 1 to City 4 with exactly one stop?

૚ൌ૞ڄ૛൅૚ڄࢌ૚ǡ૛ࢌ૛ǡ૝ ൅ࢌ૚ǡ૜ࢌ૜ǡ૝ ൌ૛

d. How many different routes can you take from City 3 to City 4 with exactly one stop?

ࢌ૜ǡ૚ࢌ૚ǡ૝ ൅ࢌ૜ǡ૛ࢌ૛ǡ૝ ൌ૚ή૛൅૚ή૛ൌ૝

e. How many different routes can you take from City 1 to City 4 with exactly two stops? Allow for routes that include repeated cities.

ࢌ૚ǡ૛ࢌ૛ǡ૚ࢌ૚ǡ૝ ൅ࢌ૚ǡ૛ࢌ૛ǡ૜ࢌ૜ǡ૝ ൅ࢌ૚ǡ૜ࢌ૜ǡ૚ࢌ૚ǡ૝ ൅ࢌ૚ǡ૜ࢌ૜ǡ૛ࢌ૛ǡ૝ ൅ࢌ૚ǡ૝ࢌ૝ǡ૚ࢌ૚ǡ૝ ൅ࢌ૚ǡ૝ࢌ૝ǡ૛ࢌ૛ǡ૝ ൅ࢌ૚ǡ૝ࢌ૝ǡ૜ࢌ૜ǡ૝ ൌ ૚ሻൌ૛ૡڄ૚ڄ૛ሻ ൅ሺ૛ڄ૛ڄ૛ሻ ൅ ሺ૛ڄ૛ڄ૛ሻ ൅ ሺ૛ڄ૚ڄ૛ሻ൅ሺ૚ڄ૚ڄ૚ሻ൅ሺ૚ڄ૚ڄ૛ሻ൅ሺ૛ڄ૚ڄሺ૛

f. How many different routes can you take from City 2 to City 4 with exactly two stops? Allow for routes that include repeated cities.

ࢌ૛ǡ૚ࢌ૚ǡ૛ࢌ૛ǡ૝ ൅ࢌ૛ǡ૚ࢌ૚ǡ૜ࢌ૜ǡ૝ ൅ࢌ૛ǡ૜ࢌ૜ǡ૚ࢌ૚ǡ૝ ൅ࢌ૛ǡ૜ࢌ૜ǡ૛ࢌ૛ǡ૝ ൅ࢌ૛ǡ૝ࢌ૝ǡ૚ࢌ૚ǡ૝ ൅ࢌ૛ǡ૝ࢌ૝ǡ૛ࢌ૛ǡ૝ ൅ࢌ૛ǡ૝ࢌ૝ǡ૜ࢌ૜ǡ૝ ൌ

૚ሻ ൌ૛ૠڄ૚ڄ૛ሻ ൅ ሺ૛ڄ૛ڄ૛൯ ൅ ሺ૛ڄሶ ૛ڄሺ૚ή૛ή૛ሻ ൅ ሺ૚ή૚ή૚ሻ ൅ ሺ૚ή૚ή૛ሻ ൅ ሺ૚ή૚ή૛ሻ ൅ ൫૛

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6. Consider the following directed graph representing the number of ways Trenton can get dressed in the morning (only visible options are shown):

a. What reasons could there be for there to be three choices for shirts after “traveling” to shorts but only two after traveling to pants?

It could be that Trenton has shirts that only make sense to wear with one or the other. For instance, maybe he does not want to wear a button-up shirt with a pair of shorts.

b. What could the order of the vertices mean in this situation?

The order of the vertices is probably the order Trenton gets dressed in.

c. Write a matrix ࡭ representing this directed graph. ૙ ૛ ૜ ૙ ૙ ې૙ ૙ ૙ ૛ ૙ۍ ۑ ێ ۑ૙ ૙ ૙ ૜ ૙ێ࡭ൌ ۑ૙ ૙ ૙ ૙ ૚ێ ے૙ ૙ ૙ ૙ ૙ۏ

d. Delete any rows of zeros in matrix ࡭, and write the new matrix as matrix ࡮. Does deleting this row change the meaning of any of the entries of ࡮? If you had deleted the first column, would the meaning of the entries change? Explain. ૙ ૛ ૜ ૙ ૙ ࡮ൌ቎૙ ૙ ૙ ૛ ૙቏ ૙ ૙ ૙ ૜ ૙ ૙ ૙ ૙ ૙ ૚

Deleting the row did not change the meaning of any of the other entries. Each entry ࢈࢏ǡ࢐ still says how to get from article of clothing ࢏ to article of clothing ࢐. If we had deleted the first column, then each entry ࢈࢏ǡ࢐ would represent how to get from article of clothing ࢏െ૚ to article of clothing ࢐.

?࢈૝ǡ૞. What does this product representڄ ࢈૛ǡ૝ڄ e. Calculate ࢈૚ǡ૛ ૚ൌ૝ڄ૛ڄ૛ The product represents the number of outfits that Trenton can wear assuming he wears pants instead of shorts.

f. How many different outfits can Trenton wear assuming he always wears a watch?

࢈૝ǡ૞ ൌ૝൅ૢൌ૚૜ڄ ࢈૜ǡ૝ڄ ࢈૝ǡ૞ ൅࢈૚ǡ૜ڄ ࢈૛ǡ૝ڄ ࢈૚ǡ૛

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7. Recall the network representing bus routes used at the start of the lesson:

Faced with competition from rival companies, you have been tasked with considering the option of building a toll road going directly from City 1 to City 4. Once built, the road will provide income in the form of tolls and also enable the implementation of a nonstop bus route to and from City 1 and City 4.

Analysts have provided you with the following information (values are in millions of dollars):

Start-up costs Projected minimum Projected maximum (expressed as profit) profit per year profit per year Road െ̈́૟૜ ̈́૟૞ ̈́૚૙૙ New bus route െ̈́૞ ̈́૙Ǥ ૠ૞ ̈́૚Ǥ ૛૞

a. Express this information in a matrix ࡼ. െ૟૜ ૟૞ ૚૙૙ ቂ ቃ െ૞ ૙Ǥ ૠ૞ ૚Ǥ ૛૞

b. What are the dimensions of the matrix?

૛ൈ૜

c. Evaluate ࢖૚ǡ૚ ൅࢖૚ǡ૛. What does this sum represent? െ૟૜ ൅ ૟૞ ൌ ૛ ̈́૛ǡ ૙૙૙ is the worst-case profit of the road after one year.

?࢖૚ǡ૛ ൌ૙ for ࢚. What does the solution representڄd. Solve ࢖૚ǡ૚ ൅࢚ െ૟૜ ൅ ૟૞࢚ ൌ ૙ ૟૞࢚ ൌ ૟૜ ૟૜ ࢚ൌ ൎ ૙Ǥ ૢ૟ૢ૛ ૟૞

It will take about ૚ year for the road to break even if we assume the worst-case profit.

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?࢖૚ǡ૜ ൌ૙ for ࢚. What does the solution representڄe. Solve ࢖૚ǡ૚ ൅࢚ െ૟૜ ൅ ૚૙૙࢚ ൌ ૙ ૚૙૙࢚ ൌ ૟૜ ࢚ൌ૙Ǥ૟૜

It will take about ૠǤ ૞ months for the road to break even if we assume the best-case profit every year.

f. Summarize your results to parts (d) and (e).

It will take between ૠǤ ૞ months and ૚ year for the road to break even.

g. Evaluate ࢖૚ǡ૚ ൅࢖૛ǡ૚. What does this sum represent? െ૟૜ ൅ െ૞ ൌ െ૟ૡ The total cost of the new road and new bus route is ̈́૟ૡǡ ૙૙૙ǡ ૙૙૙.

h. Solve ࢖૚ǡ૚ ൅࢖૛ǡ૚ ൅࢚൫࢖૚ǡ૛ ൅࢖૛ǡ૛൯ ൌ૙ for ࢚. What does the solution represent? െ૟ૡ ൅ ૟૞Ǥ ૠ૞࢚ ൌ ૙ ૟૞Ǥ ૠ૞࢚ ൌ ૟ૡ ૟ૡ ࢚ൌ ൎ ૚Ǥ ૙૜૝ ૟૞Ǥ ૠ૞

It will take about ૚ year for the road and new bus route to break even assuming the worst-case scenario for profit.

i. Make your recommendation. Should the company invest in building the toll road or not? If they build the road, should they also put in a new bus route? Explain your answer.

Answers will vary but should include the length of time it will take for the company to be profitable after the initial investment. Other factors can include considering the positive press for the company from building and maintaining a nonstop route between the cities as well as how this would affect the other routes of their buses.

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A STORY OF FUNCTIONS Lesson 2 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 2: Networks and Matrix Arithmetic

Student Outcomes Students use matrices to represent data based on transportation networks. Students multiply a matrix by a scalar, add and subtract matrices of appropriate dimensions, and interpret the meaning of this arithmetic in terms of transportation networks.

Lesson Notes This lesson builds on the work in the previous lesson in which students modeled transportation networks with matrices. The primary example used in Lesson 1 was a set of bus routes that connect four cities. This situation is used to help students discover and define multiplication by a scalar and matrix addition and subtraction. This lesson helps students understand the meaning of this matrix arithmetic. Matrix multiplication and the properties of matrix arithmetic is explored further in Lesson 3. Throughout this lesson, students make sense of transportation network diagrams and matrices, reason about contextual and abstract situations, and use matrices as tools to represent network diagrams with care and precision.

Classwork Scaffolding: Opening Exercise (5 minutes) Teachers can offer a simplified task for the Opening Exercise Have students turn and talk to a partner about the following questions to activate prior by displaying the network knowledge about different types of transportation networks and to remind them of diagram and the matrix yesterday’s scenario. representation side by side and In yesterday’s lesson, you looked at bus routes and roads that connected four asking: cities. What other types of transportation might connect cities? Explain why ܾଶǡଷ ൌͳ. à Other types of transportation could include trains, airplanes, boats, Explain why ݏସǡସ ൌͲ. walking, or biking routes. Use this exercise to check for student understanding about how to create a matrix from a network diagram. Students should work independently on this exercise and then confirm their answers with a partner. If time is a factor, try having half of the class create the matrix for the subway line and the other half for the bus line. Ask one or two students to share their answers with the class. Take time to review the meaning of the arrows and to make sure students are distinguishing between bus routes (solid) and subway routes (dashed).

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Opening Exercise

Suppose a subway line also connects the four cities. Here is the subway and bus line network. The bus routes connecting the cities are represented by solid lines, and the subway routes are represented by dashed arcs.

Write a matrix ࡮ to represent the bus routes and a matrix ࡿ to represent the subway lines Scaffolding: connecting the four cities. Teachers can offer a simplified ૚૜૚૙ ૙૚૚૚ diagram if students are ૛૙૛૛ ૚૙૚૛ ࡮ൌ൦ ൪ and ࡿൌ൦ ൪ struggling with the one ૛૚૙૚ ૚૛૙૚ ૙૛૚૙ ૚૚૛૙ presented.

Exploratory Challenge/Exercises 1–6 (15 minutes): Matrix Arithmetic Organize students into small groups, and invite them to work through the exercises in this Exploratory Challenge. Students create new matrices and interpret them in terms of the situation. They consider the product of a scalar and a matrix and the sum of two matrices. At the end of this challenge, lead a discussion debriefing the results, and then give students time to work with group members to start populating a graphic organizer for these matrix arithmetic operations.

Exploratory Challenge/Exercises 1–6: Matrix Arithmetic

Use the network diagram from the Opening Exercise and your answers to help you complete this challenge with your group.

1. Suppose the number of bus routes between each city were doubled. a. What would the new bus route matrix be? ૛૟૛૙ ૝૙૝૝ ൦ ൪ ૝૛૙૛ ૙૝૛૙

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b. Mathematicians call this matrix ૛࡮. Why do you think they call it that?

Each entry is multiplied by ૛, so the values are twice what they were before.

2. What would be the meaning of ૚૙࡮ in this situation?

It would mean each entry in matrix ࡮ is multiplied by ૚૙, so the city now has ૚૙ times as many bus routes as before connecting the cities.

3. Write the matrix ૚૙࡮. ૚૙ ૜૙ ૚૙ ૙ ૛૙ ૙ ૛૙ ૛૙ ૚૙࡮ ൌ ൦ ൪ ૛૙ ૚૙ ૙ ૚૙ ૙૛૙૚૙૙

4. Ignore whether or not a line connecting cities represents a bus or subway route. a. Create one matrix that represents all the routes between the cities in this transportation network. ૚૝૛૚ ૜૙૜૝ ൦ ൪ ૜૜૙૛ ૚૜૜૙

b. Why would it be appropriate to call this matrix ࡮൅ࡿ? Explain your reasoning.

The total number of routes regardless of transportation mode is found by counting the total routes. We can also find that by adding the number of bus and subway lines between each pair of cities.

5. What would be the meaning of ૝࡮ ൅ ૛ࡿ in this situation?

It would mean the total routes connecting the cities if the number of bus lines were increased by a factor of ૝ and the number of subway lines were doubled.

6. Write the matrix ૝࡮ ൅ ૛ࡿ. Show work and explain how you found your answer.

First multiply each entry in ࡮ by ૝, and then multiply each entry in ࡿ by ૛. ૚૜૚૙ ૝૚૛૝૙ ૛૙૛૛ ૡ૙ૡૡ ૝࡮ ൌ ૝ ൦ ൪ൌ൦ ൪ ૛૚૙૚ ૡ૝૙૝ ૙૛૚૙ ૙ૡ૝૙ ૙૚૚૚ ૙૛૛૛ ૚૙૚૛ ૛૙૛૝ ૛ࡿ ൌ ૛ ൦ ൪ൌ൦ ൪ ૚૛૙૚ ૛૝૙૛ ૚૚૛૙ ૛૛૝૙ Finally, add these two new matrices together by adding corresponding entries. ૝૚૛૝૙ ૙૛૛૛ ૝൅૙ ૚૛൅૛ ૝൅૛ ૙൅૛ ૝૚૝૟ ૛ ૡ૙ૡૡ ૛૙૛૝ ૡ൅૛ ૙൅૙ ૡ൅૛ ૡ൅૝ ૚૙ ૙ ૚૙ ૚૛ ૝࡮ ൅ ૛ࡿ ൌ ൦ ൪൅൦ ൪ൌ൦ ൪ൌ൦ ൪ ૡ૝૙૝ ૛૝૙૛ ૡ൅૛ ૝൅૝ ૙൅૙ ૝൅૛ ૚૙ ૡ ૙ ૟ ૙ૡ૝૙ ૛૛૝૙ ૙൅૛ ૡ൅૛ ૝൅૝ ૙൅૙ ૛૚૙ૡ ૙

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To debrief these exercises, have groups present their solutions to the class. As they present, encourage them to explain how they created the new matrices and what they mean in this situation. Assign each group in your class to present a portion of these exercises. For example, have one or two groups present their solutions to Exercise 3, have another group present Exercises 4 and 5, and then have one or two groups present Exercise 6.

Discussion (10 minutes) Introduce matrix arithmetic vocabulary, and discuss how matrix subtraction might be defined. How would you describe the process used to create the matrix ͳͲܤ in Exercise 4? à To create this matrix, multiply each element that represents the number of bus lines by ͳͲ. This operation is called scalar multiplication. Describe in words how to create the matrix ݇ܣ where ݇ is a real number and ܣ is a matrix. à You would multiply each element in matrix ܣ by the real number ݇. Scaffolding: How would you describe the process you used to create the matrix ܤ൅ܵ in If students are having trouble Exercise 5? describing the abstract à To create this matrix, you would count (or add) the corresponding entries operations (e.g., ݇ܣ), provide for bus and subway routes to find the total between each pair of cities them with concrete examples and then record these in a new matrix. like the ones shown below. This operation is called matrix addition. Describe in words how to create a ʹʹ If ܣൌቂ ቃ, describe matrix ܣ൅ܤ where ܣ and ܤ are matrices with equal dimensions. ͵ͳ

à Add each entry in matrix ܣ to its corresponding entry in matrix ܤ. how to create ʹܣ, ͵ܣ, and Why would ܣ and ܤ need to have the same dimensions in order to find their ݇ܣ. ʹʹ sum? If ܣൌቂ ቃ and ͵ͳ à Because we have to add corresponding elements, the matrices ܣ and ܤ ʹͳ must have the same dimensions or the operation does not make sense. ܤൌቂ ቃ, describe how ͷͲ

How would we use addition to represent the difference between ͷ and ͵? to create ܣ൅ܤ and െܤ à You add the opposite of the second number. For example, and ܣെܤ. ͷെ͵ൌͷ൅ሺെ͵ሻ ൌʹ. Model using technology How could we create the opposite of a matrix? (graphing calculators or à Multiply it by the scalar െͳ. software) so students have a means to check their work or How could you produce a matrix ܣെܤ if ܣ and ܤ are matrices with equal to help them to accurately dimensions? calculate when working with à You would simply subtract each element in matrix ܤ from its larger matrices. corresponding element in matrix ܣ. More formally, you would compute the sum ܣ൅ሺെͳሻܤ.

Exercise 7 (5 minutes) Now give students time to work with their groups to complete the first two rows of the graphic organizer in Exercise 7. If you would like, you can provide students with sample matrices to work from in the example column.

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Exercise 7

7. Complete this graphic organizer.

Matrix Operations Graphic Organizer

Operation Symbols Describe How to Calculate Example Using ૜ൈ૜ Matrices

Multiply each element of ૚૚૛ ૛૛૝ Scalar ࢑࡭ matrix ࡭ by the real ૛ ൥૛૛૝൩ ൌ ൥૝૝ૡ൩ Multiplication number ࢑. ૜૙૙ ૟૙૙ Add corresponding elements in each row and ૚૚૛ ૛૛૝ ૜૜ ૟ The Sum of column of ࡭ and ࡮. ࡭൅࡮ ൥૛૛૝൩ ൅ ൥૝૝ૡ൩ ൌ ൥૟૟૚૛൩ Two Matrices Matrices ࡭ and ࡮ must ૜૙૙ ૟૙૙ ૢ૙ ૙ have the same dimensions. ૚૚૛ ૛૛૝ െ૚ െ૚ െ૛ Subtract corresponding ൥૛૛૝൩ െ ൥૝૝ૡ൩ ൌ ൥െ૛ െ૛ െ૝൩ ૜૙૙ ૟૙૙ െ૜ ૙ ૙ The ࡭െ࡮ elements in each row and Difference of column of ࡭ and ࡮. The Or Two Matrices ൌ࡭൅ሺെ૚ሻ࡮ matrices must have the ૚૚૛ ૛૛૝ െ૚ െ૚ െ૛ same dimensions. ൥૛૛૝൩ ൅ሺെ૚ሻ൥૝૝ૡ൩ ൌ ൥െ૛ െ૛ െ૝൩ ૜૙૙ ૟૙૙ െ૜ ૙ ૙

Closing (5 minutes) Have students review their entries in Exercise 7 with a partner, and then ask one or two students to share their responses with the entire class. Take a minute to clarify any questions students have about the notation used in the Lesson Summary shown below. Be sure to emphasize that the processes of matrix addition, subtraction, and scalar multiplication apply to matrices that are not square, even though the examples in this section used only square matrices. Tell students they can add or subtract any two matrices that have the same dimensions, and they can multiply any matrix by a real number.

Lesson Summary

MATRIX SCALAR MULTIPLICATION: Let ࢑ be a real number, and let ࡭ be an ࢓ൈ࢔ matrix whose entry in row ࢏ and

.ࢇ࢏ǡ࢐ڄ࡭ is the ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢑ڄcolumn ࢐ is ࢇ࢏ǡ࢐. Then the scalar product ࢑

MATRIX SUM: Let ࡭ be an ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢇ࢏ǡ࢐, and let ࡮ be an ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢈࢏ǡ࢐. Then the matrix sum ࡭൅࡮ is the ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢇ࢏ǡ࢐ ൅࢈࢏ǡ࢐.

MATRIX DIFFERENCE: Let ࡭ be an ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢇ࢏ǡ࢐, and let ࡮ be an ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢈࢏ǡ࢐. Then the matrix difference ࡭െ࡮ is the ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢇ࢏ǡ࢐ െ࢈࢏ǡ࢐.

Exit Ticket (5 minutes)

Lesson 2: Networks and Matrix Arithmetic 37

A STORY OF FUNCTIONS Lesson 2 M2 PRECALCULUS AND ADVANCED TOPICS

Name Date

Lesson 2: Networks and Matrix Arithmetic

Exit Ticket

The diagram below represents a network of highways and railways between three cities. Highways are represented by black lines, and railways are represented by red lines.

1. Create matrix ܣ that represents the number of major highways connecting the three cities and matrix ܤ that represents the number of railways connecting the three cities.

2. Calculate and interpret the meaning of each matrix in this situation. a. ܣ൅ܤ

b. ͵ܤ

3. Find ܣെܤ. Does the matrix ܣെܤ have any meaning in this situation? Explain your reasoning.

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Exit Ticket Sample Solutions

The diagram below represents a network of highways and railways between three cities. Highways are represented by black lines, and railways are represented by red lines.

1. Create matrix ࡭ that represents the number of major highways connecting the three cities and matrix ࡮ that represents the number of railways connecting the three cities. ૙૚૛ ࡭ൌ൥૛૙૛൩ ૜૚૙ ૙૙૛ ࡮ൌ൥૛૙૚൩ ૚૚૙

2. Calculate and interpret the meaning of each matrix in this situation a. ࡭൅࡮ ૙૚૝ ࡭൅࡮ൌ൥૝૙૜൩ The resulting matrix represents the numbers of ways that a person can travel between ૝૛૙ the three cities by taking any of the major highways or railways.

b. ૜࡮ ૙૙૟ ૜࡮ ൌ ൥૟૙૜൩ The resulting matrix means the number of trains between the three cities has tripled. ૜૜૙

3. Find ࡭െ࡮. Does the matrix ࡭െ࡮ have any meaning in this situation? Explain your reasoning. ૙૚૙ ࡭െ࡮ൌ൥૙૙૚൩ The entries in matrix ࡭െ࡮ represent how many more routes there are by highway than there ૛૙૙ are by train between two cities. For example, the ૛ in the third row and first column indicates that there are two more ways to get from city 3 to city 1 by highway than there are by rail.

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A STORY OF FUNCTIONS Lesson 2 M2 PRECALCULUS AND ADVANCED TOPICS

Problem Set Sample Solutions

1. For the matrices given below, perform each of the following calculations, or explain why the calculation is not possible. ૚૛ ૛૚ ࡭ൌቂ ቃ ࡮ൌቂ ቃ ૙૚ െ૚ ૝

૞૛ૢ ૚૟ ૙ ࡯ൌ൥ ૟૚૜൩ ࡰൌ൥૜૙ ૛൩ െ૚ ૚ ૙ ૚૜െ૛

a. ࡭൅࡮ ૜૜ ቂ ቃ െ૚ ૞

b. ૛࡭ െ ࡮ ૙૜ ቂ ቃ ૚െ૛

c. ࡭൅࡯ Matrices ࡭ and ࡯ cannot be added together because they do not have the same dimensions.

d. Ȃ૛࡯ െ૚૙ െ૝ െ૚ૡ ൥െ૚૛ െ૛ െ૟ ൩ ૛െ૛૙

e. ૝ࡰ െ ૛࡯ െ૟ ૛૙ െ૚ૡ ൥ ૙െ૛૛൩ ૟૚૙െૡ

f. ૜࡮ െ ૜࡮ ૙૙ ቂ ቃ ૙૙

g. ૞࡮ െ ࡯ Matrix ࡯ cannot be subtracted from ૞࡮ because they do not have the same dimensions.

h. ࡮െ૜࡭ െ૚ െ૞ ቂ ቃ െ૚ ૚

i. ࡯ ൅ ૚૙ࡰ ૚૞ ૟૛ ૢ ൥૜૟ ૚ ૛૜ ൩ ૢ ૜૚ െ૛૙

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૚ j. ࡯൅ࡰ ૛ ૠ ૢ ې ૠ ۍ ۑ ૛ ૛ێ ૚ ૠ ۑ ૟ێ ۑ ૛ ૛ ێ ۑ ૚ ૠێ െ૛ ے ૛ ૛ۏ

૚ k. ࡮ ૝ ૚ ૚ ൦ ૛ ૝൪ ૚ െ ૚ ૝

l. ૜ࡰ െ ૝࡭ Matrix ૝࡭ cannot be subtracted from ૜ࡰ because they do not have the same dimensions.

૚ ૛ m. ࡮െ ࡭ ૜ ૜ ૙െ૚ ൥ ૚ ૛ ൩ െ ૜ ૜

2. For the matrices given below, perform each of the following calculations, or explain why the calculation is not possible. ૚૛૚ ૛૚ ࡭ൌቂ ቃ ૜૙૛ ࡮ൌ൥૜૟൩ ૚૙ ૚െ૛૜ ૛െ૚ ࡯ൌቂ ቃ ૚૚૝ ࡰൌ൥െ૚ ૙ ൩ ૝૚

a. ࡭൅૛࡮ Matrices ࡭ and ૛࡮ cannot be added together because they do not have the same dimensions.

b. ૛࡭ െ ࡯ ૚૟െ૚ ቂ ቃ ૞െ૚૙

c. ࡭൅࡯ ૛૙૝ ቂ ቃ ૝૚૟

d. െ૛࡯ െ૛ ૝ െ૟ ቂ ቃ െ૛ െ૛ െૡ

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A STORY OF FUNCTIONS Lesson 2 M2 PRECALCULUS AND ADVANCED TOPICS

e. ૝ࡰ െ ૛࡯ Matrix ૛࡯ cannot be subtracted from ૝ࡰ because the matrices do not have the same dimensions.

f. ૜ࡰ െ ૜ࡰ ૙૙ ൥૙૙൩ ૙૙

g. ૞࡮ െ ࡰ ૡ૟ ൥૚૟ ૜૙൩ ૚െ૚

h. ࡯െ૜࡭ െ૛ െૡ ૙ ቂ ቃ െૡ ૚ െ૛

i. ࡮ ൅ ૚૙ࡰ ૛૛ െૢ ൥െૠ ૟ ൩ ૝૚ ૚૙

૚ j. ࡯൅࡭ ૛ ૜ ૞ ૚ ൦૛ ૛൪ ૠ ૚ ૝ ૛ ૛

૚ k. ࡮ ૝ ૚ ૚ ې ۍ ۑ૛ ૝ێ ૜ ૜ ۑ ێ ۑ૝ ૛ێ ۑ ૚ێ ૙ ے ૝ۏ

l. ૜࡭ ൅ ૜࡮ Matrices ૜࡭ and ૜࡮ cannot be added together because they do not have the same dimension.

૚ ૛ m. ࡮െ ࡰ ૜ ૜ ૛ ې െ ૚ۍ ۑ ૜ ێ ૞ ۑ ૛ ێ ۑ ૜ ێ ۑૠ ૛ ێ െ െ ے૜ ૜ ۏ

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3. Let ૛ ૜ ࡭ൌ൥ ૜൩ െ૚ ૞ and ૚ ૜ ࡮ൌ൥૛ ૛൩ ૝૚

a. Let ࡯ൌ૟࡭൅૟࡮. Find matrix ࡯. ૛૚ ૚૜ ࡯ൌቂ ቃ ૚ૡ ૜૟

b. Let ࡰൌ૟ሺ࡭൅࡮ሻ. Find matrix ࡰ. ૛૚ ૚૜ ࡰൌቂ ቃ ૚ૡ ૜૟

c. What is the relationship between matrices ࡯ and ࡰ? Why do you think that is?

The matrices are the same. Multiplying by a scalar appears to be distributive with matrices.

૜૛ െ૛ ૜ 4. Let ࡭ൌ൥െ૚ ૞ ൩ and ࢄ be a ૜ൈ૛ matrix. If ࡭൅ࢄൌ൥ ૝૚൩, then find ࢄ. ૜െ૝ ૚െ૞ െ૞ ૚ ࢄൌ൥ ૞െ૝൩ െ૛ െ૚

૚૜૛ ૛૚૜ 5. Let ࡭ൌ൥૜૚૛൩ and ࡮ൌ൥૛૛૚൩ represent the bus routes of two companies between three cities. ૝૜૛ ૚૜૚

a. Let ࡯ൌ࡭൅࡮. Find matrix ࡯. Explain what the resulting matrix and entry ࢉ૚ǡ૜ mean in this context. ૜૝૞ ࡯ൌ൥૞૜૜൩. Entry ࢉ૚ǡ૜ ൌ૞ means that there are ૞ ways to get from city 1 to city 3 using either bus ૞૟૜ company.

b. Let ࡰൌ࡮൅࡭. Find matrix ࡰ. Explain what the resulting matrix and entry ࢊ૚ǡ૜ mean in this context. ૜૝૞ ࡰൌ൥૞૜૜൩. Entry ࢊ૚ǡ૜ ൌ૞ means that there are ૞ ways to get from city 1 to city 3 using either bus ૞૟૜ company.

c. What is the relationship between matrices ࡯ and ࡰ? Why do you think that is?

Matrices ࡯ and ࡰ are equal. When we add two matrices, we add the real numbers in the corresponding spots. When two real numbers are added, the order doesn’t matter, so it makes sense that it doesn’t matter the order in which we add matrices.

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A STORY OF FUNCTIONS Lesson 2 M2 PRECALCULUS AND ADVANCED TOPICS

6. Suppose that April’s Pet Supply has three stores in cities 1, 2, and 3. Ben’s Pet Mart has two stores in cities 1 and 2. Each shop sells the same type of dog crates in sizes 1 (small), 2 (medium), 3 (large), and 4 (extra large). April’s and Ben’s inventory in each city is stored in the tables below.

April’s Pet Supply Ben’s Pet Mart City 1 City 2 City 3 City 1 City 2 Size 1 ૜ ૞ ૚ Size 1 ૛ ૜ Size 2 ૝ ૛ ૢ Size 2 ૙ ૛ Size 3 ૚ ૝ ૛ Size 3 ૝ ૚ Size 4 ૙ ૙ ૚ Size 4 ૙ ૙

a. Create a matrix ࡭ so that ࢇ࢏ǡ࢐ represents the number of crates of size ࢏ available in April’s store in City ࢐. ૜૞૚ ૝૛ૢ ࡭ൌ൦ ൪ ૚૝૛ ૙૙૚

૛૜૙ ૙૛૙ b. Explain how the matrix ࡮ൌ൦ ൪ can represent the dog crate inventory at Ben’s Pet Mart. ૝૚૙ ૙૙૙ The first two columns represent the inventory in Ben’s stores in cities 1 and 2. Since there is no store in city 3, Ben has no inventory of dog crates in this city, which is represented by the zeros in the third column of matrix ࡮.

c. Suppose that April and Ben merge their inventories. Find a matrix that represents their combined inventory in each of the three cities. ૞ૡ૚ ૝૝ૢ Matrix ࡭൅࡮ൌ൦ ൪ represents the combined inventory in each of the three cities. For example, since ૞૞૛ ૙૙૚ the entry in row 4, column 1 is ૙, there are no extra large dog crates in either store in city 1. And, since the entry in row 1, column 2 is ૡ, there are a combined ૡ small crates between the stores in city 2.

7. Jackie has two businesses she is considering buying and a business plan that could work for both. Consider the tables below, and answer the questions following.

Horus’s One-Stop Warehouse Supply Re’s 24-Hour Distributions If Business If Business If Business Stays If Business Stays Improves as Improves as the Same the Same Projected Projected Expand to Multiple Expand to Multiple െ̈́ૠ૞ǡ ૙૙૙ǡ ૙૙૙ ̈́૝૞ǡ ૙૙૙ǡ ૙૙૙ െ̈́ૢૢǡ ૙૙૙ǡ ૙૙૙ ̈́૟૛ǡ ૞૙૙ǡ ૙૙૙ States States Invest in Drone Invest in Drone െ̈́૜૜ǡ ૙૙૙ǡ ૙૙૙ ̈́૜૙ǡ ૙૙૙ǡ ૙૙૙ െ̈́૝ૢǡ ૙૙૙ǡ ૙૙૙ ̈́૛ૢǡ ૙૙૙ǡ ૙૙૙ Delivery Delivery Close and Sell Out ̈́૛૙ǡ ૙૙૙ǡ ૙૙૙ ̈́૛૙ǡ ૙૙૙ǡ ૙૙૙ Close and Sell Out ̈́૜૞ǡ ૙૙૙ǡ ૙૙૙ ̈́૜૞ǡ ૙૙૙ǡ ૙૙૙

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a. Create matrices ࡴ and ࡾ representing the values in the tables above such that the rows represent the different options and the columns represent the different outcomes of each option. െૠ૞ ૝૞ ࡴൌ൥െ૜૜ ૜૙൩ ૛૙ ૛૙ െૢૢ ૟૛Ǥ ૞ ࡾൌ൥െ૝ૢ ૛ૢ ൩ ૜૞ ૜૞

b. Calculate ࡾെࡴ. What does ࡾെࡴ represent? െૢૢ ૟૛Ǥ ૞ െૠ૞ ૝૞ െ૛૝ ૚ૠǤ ૞ ൥െ૝ૢ ૛ૢ ൩ െ ൥െ૜૜ ૜૙൩ ൌ ൥െ૚૟ െ૚ ൩ ૜૞ ૜૞ ૛૙ ૛૙ ૚૞ ૚૞ ࡾെࡴ represents how much more money Jackie makes buying the second business instead of the first in each situation. Note that negative values mean that Jackie loses more money with those choices.

c. Calculate ࡴ൅ࡾ. What does ࡴ൅ࡾ represent? െૠ૞ ૝૞ െૢૢ ૟૛Ǥ ૞ െ૚ૠ૝ ૚૙ૠǤ ૞ ൥െ૜૜ ૜૙൩ ൅ ൥െ૝ૢ ૛ૢ ൩ ൌ ൥ െૡ૛ ૞ૢ ൩ ૛૙ ૛૙ ૜૞ ૜૞ ૞૞ ૞૞ ࡴ൅ࡾ represents what could happen if Jackie buys both businesses.

d. Jackie estimates that the economy could cause fluctuations in her numbers by as much as ૞Ψ both ways. Find matrices to represent the best and worst case scenarios for Jackie. െૠ૚Ǥ ૛૞ ૝૛Ǥ ૠ૞ ૙Ǥ ૢ૞ࡴ ൌ ൥െ૜૚Ǥ ૛૞ ૛ૡǤ ૞ ൩ ૚ૢ ૚ૢ െૠૡǤ ૠ૞ ૝ૠǤ ૛૞ ૚Ǥ ૙૞ࡴ ൌ ൥െ૜૝Ǥ ૟૞ ૜૚Ǥ ૞ ൩ ૛૚ ૛૚ െૢ૝Ǥ ૙૞ ૞ૢǤ ૜ૠ૞ ૙Ǥ ૢ૞ࡾ ൌ ൥െ૝૟Ǥ ૞૞ ૛ૠǤ ૞૞ ൩ ૜૜Ǥ ૛૞ ૜૜Ǥ ૛૞ െ૚૙૜Ǥ ૢ૞ ૟૞Ǥ ૟૛૞ ૚Ǥ ૙૞ࡾ ൌ ൥ െ૞૚Ǥ ૝૞ ૜૙Ǥ ૝૞ ൩ ૜૟Ǥ ૠ૞ ૜૟Ǥ ૠ૞

e. Which business should Jackie buy? Which of the three options should she choose? Explain your choices.

Answers may vary depending on how students feel about risk and income. There are two poor choices. If Jackie is planning to sell out the business she buys, then Re’s business results in a guaranteed increase of ̈́૚૞ million. Also, if Jackie is planning to invest in drones, then she should buy Horus’s business since it is both less risky and results in a greater return.

Lesson 2: Networks and Matrix Arithmetic 45

A STORY OF FUNCTIONS Lesson 3 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 3: Matrix Arithmetic in Its Own Right

Student Outcomes Students use matrices to represent data based on transportation networks. Students multiply matrices of appropriate dimensions and interpret the meaning of this arithmetic in terms of transportation networks.

Lesson Notes This lesson builds on the work in the previous lesson by considering a transportation network with two different types of public transportation. This situation is used to help students discover and define multiplication of two matrices. Students understand the meaning of this matrix arithmetic. They begin to discover that matrix multiplication is not commutative. Matrix arithmetic and the properties of matrix arithmetic are explored further in this lesson as well. Throughout this lesson, students make sense of transportation network diagrams and matrices, reasoning about contextual and abstract situations, and using matrices as tools to represent network diagrams with care and precision.

Classwork Opening Exercise (5 minutes) Students begin to explore the concept of matrix multiplication by calculating the total number of ways to travel from City 2 to City 1 if the first leg of the trip is by bus and the second leg is by subway. This activity helps students understand how to determine the number of available routes that use two modes of transportation in a particular order. The table helps students understand how to calculate the elements in the product of two matrices and to understand the meaning of that product. What do the rows in this table represent? à Each row is the number of ways to travel from City 2 to City 1 via one of the four cities. Why are the total ways to travel from City 2 to City 1 through City 1 equal to Ͳ? à The exercise asks us to find the routes from City 2 to City 1 with the first leg being by bus and the second leg being by subway. There is one way to travel from City 2 to City 1 via bus, but there are no ways to travel from City 1 to City 1 by subway. That would require the diagram to have a dashed loop from City 1 to City 1. Therefore, there are no routes from City 2 to City 1 that use a bus for the first leg of the trip and the subway for the second leg. How did you complete the table below? à We counted the routes between each city and then multiplied the number of bus routes by the number of subway routes. Finally, we had to add all the numbers in the last column to determine the total possible routes.

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How does this table ensure that you counted all the possible routes? à We counted all the bus routes from City 2 to another city (including itself), and then we counted all the subway routes from one of the four cities to City 1. We considered all possible paths.

Opening Exercise

The subway and bus line network connecting four cities that we used in Lesson 2 is shown below. The bus routes connecting the cities are represented by solid lines, and the subway routes are represented by dashed lines.

Scaffolding: Challenge advanced students to create a matrix to represent this network. Struggling students can use the diagram below that contains fewer bus routes and subway lines.

Suppose we want to travel from City 2 to City 1, first by bus and then by subway, with no more than one connecting stop.

a. Complete the chart below showing the number of ways to travel from City 2 to City 1 using first a bus and then the subway. The first row has been completed for you.

First Leg (BUS) Second Leg (SUBWAY) Total Ways to Travel

City 2 to City 1: ૛ City 1 to City 1: ૙ ૛ ή ૙ ൌ ૙

City 2 to City 2: ૙ City 2 to City 1: ૚ ૙ή૚ൌ૙

City 2 to City 3: ૛ City 3 to City 1: ૚ ૛ή૚ൌ૛

City 2 to City 4: ૛ City 4 to City 1: ૚ ૛ή૚ൌ૛

b. How many ways are there to travel from City 2 to City 1, first on a bus and then on a subway? How do you know?

The total number of ways is the sum of the numbers in the last column. There are ૝ ways.

Lesson 3: Matrix Arithmetic in Its Own Right 47

A STORY OF FUNCTIONS Lesson 3 M2 PRECALCULUS AND ADVANCED TOPICS

Exploratory Challenge/Exercises 1–12 (20 minutes): The Meaning of Matrix Multiplication This challenge continues to introduce the notion of matrix multiplication in a meaningful situation. Students consider travel routes if we use two different means of transportation in a particular order and have at most one connecting city for all pairs of cities.

Exploratory Challenge/Exercises 1–12: The Meaning of Matrix Multiplication

Suppose we want to travel between all cities, traveling first by bus and then by subway, with no more than one connecting stop.

1. Use a chart like the one in the Opening Exercise to help you determine the total number of ways to travel from City 1 to City 4 using first a bus and then the subway.

First Leg (BUS) Second Leg (SUBWAY) Total Ways to Travel

City 1 to City 1: ૚ City 1 to City 4: ૚ ૚ή૚ൌ૚

City 1 to City 2: ૜ City 2 to City 4: ૛ ૜ή૛ൌ૟

City 1 to City 3: ૚ City 3 to City 4: ૚ ૚ή૚ൌ૚

City 1 to City 4: ૙ City 4 to City 4: ૙ ૙ή૙ൌ૙

Summing the entries in the last column shows that there are 8 different ways to travel from City 1 to City 4 using first a bus and then the subway.

2. Suppose we create a new matrix ࡼ to show the number of ways to travel between the cities, first by bus and then by subway, with no more than one connecting stop. Record your answers to Opening Exercise, part (b) and Exercise 1 in this matrix in the appropriate row and column location. We do not yet have enough information to complete the entire matrix. Explain how you decided where to record these numbers in the matrix shown below. ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ࡼൌ቎̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴቏ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴

̴̴̴ ̴̴̴ ̴̴̴ ૡ ૝ ̴̴̴ ̴̴̴ ̴̴̴ ࡼൌ൦ ൪ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ ̴̴̴ If the rows represent the starting city and the columns represent the destination city, then the ways to travel from

City 2 to City 1 would be ࢖૛ǡ૚, and the ways to travel from City 1 to City 4 would be ࢖૚ǡ૝.

At this point, circulate around the room to check for understanding by making sure that groups are recording the correct entries in the proper locations in the matrix. Debrief as a whole class if a majority of students are struggling. Consider reviewing matrix notation, especially since students are going to use it in upcoming exercises. For example, if ܲ is a th th matrix, then ݌௜ǡ௝ is the element in the ݅ row and ݆ column of the matrix. In Exercise 5, some groups may continue to use a table, while some may simply create a list. Make sure that students are recording a written explanation of how they calculated their answers. If students can describe the process of multiplying and summing verbally or in writing, then they are able to calculate the product of two matrices without difficulty.

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3. What is the total number of ways to travel from City 3 to City 2 first by bus and then by subway with no more than one connecting stop? Explain how you got your answer and where you would record it in matrix ࡼ above.

Find the ways to travel by bus from City 3 to each of the other cities, including City 3 itself. Then for each city, find the ways to travel by subway from that city to City 2. Multiply the number of ways from City 3 to the connecting city by the number of ways from the connecting city to City 2. Finally, add the result of the number of ways through each connecting city.

City 3 to City 2 via City 1: ૛ή૚ൌ૛

City 3 to City 2 via City 2: ૚ή૙ൌ૙

City 3 to City 2 via City 3: ૙ή૛ൌ૙

City 3 to City 2 via City 4: ૚ή૚ൌ૚

There are ૛൅૙൅૙൅૚ൌ૜ ways to travel from City 3 to City 2, using first a bus and then the subway. This would be entered in the 3rd row and 2nd column of matrix ࡼ.

Recall matrix ࡮, which shows the number of bus lines connecting the cities in this transportation network, and matrix ࡿ, which represents the number of subway lines connecting the cities in this transportation network. ૚૜૚૙ ૙૚૚૚ ૛૙૛૛ ૚૙૚૛ ࡮ൌ൦ ൪ and ࡿൌ൦ ൪ ૛૚૙૚ ૚૛૙૚ ૙૛૚૙ ૚૚૛૙

4. What does the product ࢈૚ǡ૛࢙૛ǡ૝ represent in this situation? What is the value of this product?

The product ࢈૚ǡ૛࢙૛ǡ૝ represents the number of ways to travel from City 1 to City 4 via City 2, first by bus and then by subway.

࢈૚ǡ૛࢙૛ǡ૝ ൌ૜ή૛ൌ૟

5. What does ࢈૚ǡ૝࢙૝ǡ૝ represent in this situation? What is the value of this product? Does this make sense?

The product ࢈૚ǡ૝࢙૝ǡ૝ represents the number of ways to travel from City 1 to City 4 via City 4, first by bus and then by subway.

࢈૚ǡ૝࢙૝ǡ૝ ൌ૙ή૙ൌ૙ Since there is no bus route from City 1 to City 4 and no subway route from City 4 to City 4, it is impossible to travel from City 1 to City 4 using first a bus and then a subway. Thus, there are no possible routes.

6. Calculate the value of the expression ࢈૚ǡ૚࢙૚ǡ૝ ൅࢈૚ǡ૛࢙૛ǡ૝ ൅࢈૚ǡ૜࢙૜ǡ૝ ൅࢈૚ǡ૝࢙૝ǡ૝. What is the meaning of this expression in this situation?

࢈૚ǡ૚࢙૚ǡ૝ ൅࢈૚ǡ૛࢙૛ǡ૝ ൅࢈૚ǡ૜࢙૜ǡ૝ ൅࢈૚ǡ૝࢙૝ǡ૝ ൌ૚ή૚൅૜ή૛൅૚ή૚൅૙ή૙ൌૡ Thus, there are ૡ total ways to travel from City 1 to City 4 via any one of the cities, first by bus and then by subway.

7. Circle the first row of ࡮ and the second column of ࡿ. How are these entries related to the expression above and your work in Exercise 1?

We multiplied the first element in the row by the first element in the column, the second by the second, third by third and so on, and then we found the sum of those products. This is the same as the sum of the entries in the last column of the table in Exercise 1.

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8. Write an expression that represents the total number of ways you can travel between City 2 and City 1,first by bus and then by subway, with no more than one connecting stop. What is the value of this expression? What is the meaning of the result?

࢈૛ǡ૚࢙૚ǡ૚ ൅࢈૛ǡ૛࢙૛ǡ૚ ൅࢈૛ǡ૜࢙૜ǡ૚ ൅࢈૛ǡ૝࢙૝ǡ૚ ൌ૛ή૙൅૙ή૚൅૛ή૚൅૛ή૚ൌ૝ Thus, there are૝ total ways to travel from City 2 to City 1 via any one of the cities, first by bus and then by subway.

9. Write an expression that represents the total number of ways you can travel between City 4 and City 1, first by bus and then by subway, with no more than one connecting stop. What is the value of this expression? What is the meaning of the result?

࢈૝ǡ૚࢙૚ǡ૚ ൅࢈૝ǡ૛࢙૛ǡ૚ ൅࢈૝ǡ૜࢙૜ǡ૚ ൅࢈૝ǡ૝࢙૝ǡ૚ ൌ૙ή૙൅૛ή૚൅૚ή૚൅૙ή૚ൌ૜ Thus, there are ૜ total ways to travel from City 4 to City 1 via any one of the cities, first by bus and then by subway.

At this point, stop and debrief. Have several students share their results with the whole class. Have different groups present different portions of the Exploratory Challenge on the board or on a document camera. Use the questions that follow to focus the discussion. Have students answer them with a partner or in their small groups before asking for volunteers to share the answer with the entire class. What does each element of matrix ܲ represent? th th à Entry ݌௜ǡ௝ represents the number of ways to travel from the ݅ city to the ݆ city, changing from a bus to the subway one time. What patterns do you notice in the expression in Exercise 6 and the expressions you wrote for Exercises 8 and 9? à In the indices, there is a pattern where the first number is the starting city and the last number is the ending city, and the middle indices are the same. Complete this sentence: To calculate the element of matrix ܲ in the 2nd row and 4th column, you would … à Multiply each element in the 2nd row of ܤby each corresponding element in the 4th column of ܵ, and then add the resulting products. Complete the sentence: The element of matrix ܲ in the 4th row and 2nd column represents the number of ways to travel … à From City 4 to City 2 first via bus and then changing to a subway in the connecting city. Describe how to calculate any element in matrix ܲ. à You multiply each element in the ݅th row of the first matrix by the corresponding element in the ݆th column of the second matrix, and then add the resulting products. The last two exercises have students construct the complete matrix. Allow them time to work through the process. There are many opportunities for careless errors, so taking the time to be accurate and precise is essential.

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10. Complete matrix ࡼ that represents the routes connecting the four cities if you travel first by bus and then by subway. ૝૜૝ૡ ૝ૡ૟૝ ࡼൌ൦ ൪ ૛૜૞૝ ૜૛૛૞

11. Construct a matrix ࡹ that represents the routes connecting the four cities if you travel first by subway and then by bus. ૝૜૜૜ ૜ૡ૜૚ ࡹൌ൦ ൪ ૞૞૟૝ ૠ૞૜૝

12. Should these two matrices be the same? Explain your reasoning.

This transportation network has different numbers of routes connecting cities in each direction. Traveling from City 1 to City 4 does not have the same number of options as traveling from City 4 to City 1. Therefore, the order in which you select your method of travel makes a difference, which is why the matrices ࡹ and ࡼ are not equal.

Discussion (5 minutes) Use this time to finish debriefing the exercises in the Exploratory Challenge. Define and describe the product of two matrices more formally, and give students an opportunity to take notes on the process. During this Exploratory Challenge, you have been calculating what mathematicians define to be the product of two matrices. Does this matrix arithmetic operation seem to relate back to real number multiplication in any way? à We are working with an array and doing some multiplying when working with real number multiplication. We do not really find products and then add the results. Our examples used square ͶൈͶ matrices, but matrix multiplication can also be applied to matrices of ͳʹ ʹͳͲ different dimensions. Consider matrices ܣൌቂ ቃ and ܤൌቂ ቃ. Would it make sense to compute ͵ͳ ͳെʹͳ ?Why or why not ?ܤڄܣ ͳڄʹͲ൅ڄሺെʹሻ ͳڄʹͳ൅ڄͳ ͳڄʹ൅ʹڄͳ ൌ൤ ൨, soܤڄܣ and we would get ,ܤڄܣ à We could compute ͳڄͲ൅ͳڄ͵ ሺെʹሻڄͳ൅ͳڄ͵ ͳڄ൅ͳʹڄ͵ Ͷെ͵ʹ .ൌቂ ቃܤڄܣ ͹ͳͳ ǫܣڄܤ What about the product because there are a different number of ܣڄܤ à Our multiplication process fails if we try to compute entries in the columns of ܤ than in the rows of ܣ. You described how to calculate each element of the product matrix in words. Using symbols, if ܲൌܣήܤ, then and ݉ is the ܣ ൅ܽ௜ǡ௡ܾ௠ǡ௝, where ݊ is the number of columns inڮelement ݌௜ǡ௝ ൌܽ௜ǡଵܾଵǡ௝ ൅ܽ௜ǡଶܾଶǡ௝ ൅ܽ௜ǡଷܾଷǡ௝ ൅ number of rows in ܤ. What must be true about the dimensions of two matrices if we wish to find their product? à This implies that you can only multiply two matrices if the number of columns in the first matrix is equal to the number of rows in the second matrix.

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Have students record this information in their notes to show examples of the product of two matrices. Have them color code a few entries as shown below to highlight the process. ܤήܵൌܲ ͳ͵ͳͲ Ͳͳͳͳ Ͷ͵Ͷͺ ʹͲʹʹ ͳͲͳʹ Ͷͺ͸Ͷ ൦ ൪ή൦ ൪ൌ൦ ൪ ʹͳͲͳ ͳʹͲͳ ʹ͵ͷͶ ͲʹͳͲ ͳͳʹͲ ͵ʹʹͷ

݌ଵǡଶ ൌͳήͳ൅͵ήͲ൅ͳήʹ൅Ͳήͳൌ͵

ܵήܤൌܯ Ͳͳͳͳ ͳ͵ͳͲ Ͷ͵͵͵ ͳͲͳʹ ʹͲʹʹ ͵ͺ͵ͳ ൦ ൪ή൦ ൪ൌ൦ ൪ ͳʹͲͳ ʹͳͲͳ ͷͷ͸Ͷ ͳͳʹͲ ͲʹͳͲ ͹ͷ͵Ͷ

When we multiply two matrices together, such as an ݉ൈ݊ matrix by an ݊ൈ݌ matrix, what is the size of the resulting matrix? à The resulting matrix has size ݉ൈ݌.

Size of final product

ሺ݊ൈ݌ሻڄ ሺ݉ൈ݊ሻ

Inner numbers match

In Exercises 10 and 11, you showed that matrix ܲ and matrix ܯ were not the same. In real number arithmetic, the operation of multiplication is commutative. What does that mean? à It means that ܽήܾൌܾήܽ when ܽ and ܾ are real numbers. Is matrix multiplication commutative? Explain your reasoning. à The results of Exercises 10–12 and the meaning of the product matrix in this situation implies that even if the two matrices being multiplied are square matrices (i.e., equal number of rows and columns), matrix multiplication is not commutative.

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Exercises 13–16 (5 minutes) These exercises talk about arithmetic operations with matrices. Students are beginning to think abstractly about the properties of matrices and how they compare to properties of real numbers. ͵ൈ͵ matrices are used in the example, and square matrices are used in these exercises, but advanced students could be asked to show that these properties hold regardless of the size of the matrices, as long as added matrices are of equal dimensions and multiplied matrices have appropriate dimensions.

Exercises 13–16 ૛૜െ૚ 13. Let ࡭ൌ൥૙૚ ૛൩ ૚૙ ૙ a. Construct a matrix ࢆ such that ࡭൅ࢆൌ࡭. Explain how you got your answer. ૙૙૙ ࢆൌ൥૙૙૙൩ ૙૙૙ We would need to add ૙ to each element of ࡭ to get the same matrix, so each element of ࢆ must be equal to ૙. Also, ࢆ must have the same dimensions as ࡭ so we can add corresponding elements.

b. Explain why ࢑ήࢆൌࢆ for any real number ࢑.

Any number multiplied by ૙ is equal to ૙. Scalar multiplication multiplies each element of ࢆ by ࢑.

c. The real number ૙ has the properties that ࢇ൅૙ൌ૙ and ࢇή૙ൌ૙ for all real numbers ࢇ. Why would mathematicians call ࢆ a zero matrix?

The matrix ࢆ has the same properties in matrix addition as the real number ૙ has with real number addition.

14. Suppose each city had a trolley car that ran a route between tourist destinations. The blue loops represent the trolley car routes. Remember that straight lines indicate bus routes, and dotted lines indicate subway routes.

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a. Explain why the matrix ࡵ shown below would represent the number of routes connecting cities by trolley car in this transportation network. ૚૙૙૙ ૙૚૙૙ ࡵൌ൦ ൪ ૙૙૚૙ ૙૙૙૚ It shows that you can travel on the trolley from City 1 to City 1, City 2 to City 2, City 3 to City 3, and City 4 to City 4 but not between cities.

b. Recall that ࡮ is the bus route matrix. Show that ࡵή࡮ൌ࡮. Explain why this makes sense in terms of the transportation network.

The entries in matrix ࡵή࡮ represent the number of ways to travel from city to city, first on a trolley and then on the bus. Since there is only one way in each city to travel by trolley and there are no ways to travel by trolley between cities, the number of routes is the same as the number of bus routes.

c. The real number ૚ has the property that ૚ήࢇൌࢇ for all real numbers ࢇ, and we call ૚ the multiplicative identity. Why would mathematicians call ࡵ an identity matrix?

Matrixࡵ has the same property under matrix multiplication that the real number ૚ does for real number multiplication, so it makes sense to call ࡵ an identity matrix.

d. What would be the form of a ૛ൈ૛ identity matrix? What about a ૜ൈ૜ identity matrix? ૚૙૙ ૚૙ A ૛ൈ૛ identity matrix would be ࡵൌቂ ቃ, and a ૜ൈ૜ identity matrix would be ࡵൌ൥૙૚૙൩. ૙૚ ૙૙૚

15. In this lesson you learned that the commutative property does not hold for matrix multiplication. This exercise asks you to consider other properties of real numbers applied to matrix arithmetic. a. Is matrix addition associative? That is, does ሺ࡭൅࡮ሻ ൅࡯ൌ࡭൅ሺ࡮൅࡯ሻ for matrices ࡭, ࡮, and ࡯ that have the same dimensions? Explain your reasoning.

Yes. Because all three matrices have the same dimensions, we add corresponding entries. The entries are real numbers. Since the associative property holds for adding real numbers, it would make sense to hold for the addition of matrices.

b. Is matrix multiplication associative? That is, does ሺ࡭ή࡮ሻ ή࡯ൌ࡭ήሺ࡮ή࡯ሻ for matrices ࡭, ࡮, and ࡯ for which the multiplication is defined? Explain your reasoning.

Yes. As long as the dimensions of the matrices are such that the multiplications are all defined, computing the products requires that we add and multiply real numbers. Since these operations are associative for real numbers, multiplication is associative for matrices. It would be like finding the number of routes if we had three modes of transportation and changed modes twice. We can count those totals in different groupings as long as we maintain the order of the transportation modes (e.g., bus to subway to train).

c. Is matrix addition commutative? That is, does ࡭൅࡮ൌ࡮൅࡭ for matrices ࡭ and ࡮ with the same dimensions?

Yes, because addition of the individual elements is commutative.

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16. Complete the graphic organizer to summarize your understanding of the product of two matrices.

Operation Symbols Describe How to Calculate Example Using ૜ൈ૜ Matrices To find the element in the ࢏th row and ࢐th column of the product matrix, multiply corresponding elements from Matrix the ࢏th row of the first matrix by ૛૙ ૛૚૚૛ ૡ૛ ૝ ࡭ή࡮ ൥૛૛െ૚൩൥૛૛૝൩ ൌ ൥ ૜૟૚૛൩ Multiplication the ࢐th column of the second ૚െ૜૙ ૜૙૙ െ૞ െ૞ െ૚૙ matrix, and then add the results. Repeat this process for each element in the product matrix.

Closing (5 minutes) Have students review their graphic organizer entries in Exercise 16 with a partner, and then ask one or two students to share their responses with the entire class. Take a minute to clarify any questions students have about the notation used in the Lesson Summary shown below.

Lesson Summary

MATRIX PRODUCT: Let ࡭ be an ࢓ൈ࢔ matrix whose entry in row ࢏ and column ࢐ is ࢇ࢏ǡ࢐, and let ࡮ be an ࢔ൈ࢖ matrix whose entry in row ࢏ and column ࢐ is ࢈࢏ǡ࢐. Then the matrix product ࡭࡮ is the ࢓ൈ࢖ matrix whose entry in row ࢏ and .൅ࢇ࢏ǡ࢔࢈࢔ǡ࢐ڮcolumn ࢐ is ࢇ࢏ǡ૚࢈૚ǡ࢐ ൅ࢇ࢏ǡ૛࢈૛ǡ࢐ ൅

IDENTITY MATRIX: The ࢔ൈ࢔ identity matrix is the matrix whose entry in row ࢏ and column ࢏ for ૚൑࢏൑࢔ is ૚, and whose entries in row ࢏ and column ࢐ for ૚൑࢏, ࢐൑࢔ and ࢏്࢐ are all zero. The identity matrix ૚૙૙ ૚૙ is denoted by ࡵ. The ૛ൈ૛ identity matrix is ቂ ቃ, and the ૜ൈ૜ identity matrix is ൥૙૚૙൩. If the size of the ૙૚ ૙૙૚ identity matrix is not explicitly stated, then the size is implied by context.

ZERO MATRIX: The ࢓ൈ࢔ zero matrix is the ࢓ൈ࢔ matrix in which all entries are equal to zero. ૙૙૙ ૙૙ For example, the ૛ൈ૛ zero matrix is ቂ ቃ, and the ૜ൈ૜ zero matrix is ൥૙૙૙൩. If the size of the zero matrix ૙૙ ૙૙૙ is not explicitly stated, then the size is implied by context.

Exit Ticket (5 minutes)

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Name Date

Lesson 3: Matrix Arithmetic in Its Own Right

Exit Ticket

Matrix ܣ represents the number of major highways connecting three cities. Matrix ܤ represents the number of railways connecting the same three cities. Ͳ͵Ͳ Ͳͳͳ ܣൌ൥ʹͲʹ൩ and ܤൌ൥ͳͲͳ൩ ͳͳʹ ͳͳͲ

Draw a network diagram for the transportation network of highways and railways between these cities. Use solid lines for highways and dotted lines for railways.

1. Calculate and interpret the meaning of each matrix in this situation. ܤڄܣ .a

ܣڄܤ .b

?ܣڄܤ്ܤڄܣ In this situation, why does it make sense that .2

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Exit Ticket Sample Solutions

Matrix ࡭ represents the number of major highways connecting three cities. Matrix ࡮ represents the number of railways connecting the same three cities. ૙૜૙ ૙૚૚ ࡭ൌ൥૛૙૛൩ and ࡮ൌ൥૚૙૚൩ ૚૚૛ ૚૚૙ 1. Draw a network diagram for the transportation network of highways and railways between these cities. Use solid lines for highways and dotted lines for railways.

2. Calculate and interpret the meaning of each matrix in this situation. ࡮ڄa. ࡭ ૜૙૜ ࡭࡮ ൌ ൥૛૝૛൩ It indicates the number of ways to get around between three cities by taking highways ૜૜૛

first and then railways second.

࡭ڄb. ࡮ ૜૚૝

࡮࡭ ൌ ൥૚૝૛൩ It indicates the number of ways to get around between three cities by taking railways first ૛૜૛ and then highways second.

?࡭ڄ࡮്࡮ڄIn this situation, why does it make sense that ࡭ .3

It makes sense because both transportation networks have a different number of routes connecting cities in each direction. Not every route goes both directions. For example, there are a different number of possible routes from City 1 to City 2 than from City 2 to City 1.

Problem Set Sample Solutions

૚૜ ૚૛ ڄLet ࡭ൌቂ ቃ and ࡮ൌቂ ቃ represent the bus routes of two companies between ૛ cities. Find the product ࡭ .1 ૛૙ ૝૜ .࡮ in the context of this scenarioڄ࡮, and explain the meaning of the entry in row 1, column 2 of ࡭ ૚૜ ૚૛ ૚૜ ૚૚ ቂ ቃൌቂ ቃ. The entry in row 1, column 2 to ૚૚, which means that there areڄ࡮ൌቂ ቃڄThe product is ࡭ ૛૙ ૝૜ ૛૝ ૚૚ possible routes from the first city to the second city, taking first a bus from Company A and then a bus from Company B.

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૚૜૛ ૛૚૜ 2. Let ࡭ൌ൥૜૚૛൩ and ࡮ൌ൥૛૛૚൩ represent the bus routes of two companies between three cities. ૝૜૛ ૚૜૚

.࡮. Find matrix ࡯, and explain the meaning of entry ࢉ૚ǡ૜ڄa. Let ࡯ൌ࡭ ૚૙ ૚૜ ૡ The product is ࡯ൌ൥૚૙ ૚૚ ૚૛൩, and ࢉ૚ǡ૜ ൌૡ means that there are ૡ different ways to travel to City 3 ૚૟ ૚૟ ૚ૠ from City 1 by taking a bus from Company A and then a bus from Company B.

b. Nina wants to travel from City 3 to City 1 and back home to City 3 by taking a direct bus from Company A on the way to City 1 and a bus from Company B on the way back home to City 3. How many different ways are there for her to make this trip?

Nina wants to travel from City 3 to City 1 and back to City 3, ࢇ૜ǡ૚ ή࢈૚ǡ૜ ൌ૝ή૜ൌ૚૛ means that she has ૚૛ ways to make the trip.

c. Oliver wants to travel from City 2 to City 3 by taking first a bus from Company A and then taking a bus from Company B. How many different ways can he do this? ૚૙ ૚૜ ૡ ࡮ൌ൥૚૙ ૚૚ ૚૛൩, and Oliver wants to travel from City 2 to City 3, he has ࢉ૛ǡ૜ ൌ૚૛ ways toڄSince ࡯ൌ࡭ ૚૟ ૚૟ ૚ૠ make the trip.

d. How many routes can Oliver choose from if he travels from City 2 to City 3 by first taking a bus from Company B and then taking a bus from Company A? ૚ૠ ૚૟ ૚૛ ࡭ൌ൥૚૛ ૚૚ ૚૙൩, and the entry in row 2 , column 3 is ૚૙, which means that there are ૚૙ ways toڄSince ࡮ ૚૝ ૢ ૚૙ get from City 2 to City 3 by taking a bus from Company B first and then a bus from Company A.

3. Recall the bus and trolley matrices from the lesson: ૚૜૚૙ ૚૙૙૙ ૛૙૛૛ ૙૚૙૙ ࡮ൌ൦ ൪ and ࡵൌ൦ ൪ ૛૚૙૚ ૙૙૚૙ ૙૛૚૙ ૙૙૙૚

a. Explain why it makes sense that ࡮ࡵ ൌ ࡵ࡮ in the context of the problem.

Since there is only one way to take a trolley in each city, no matter whether you take the trolley before you take the bus or if you take the bus first, the result is the same number of ways as there are bus routes.

b. Multiply out ࡮ࡵ and ࡵ࡮ to show ࡮ࡵ ൌ ࡵ࡮. ૚૜૚૙ ૚૜૚૙ ૛૙૛૛ ૛૙૛૛ ࡮ࡵ ൌ ൦ ൪ǡࡵ࡮ൌ൦ ൪ ૛૚૙૚ ૛૚૙૚ ૙૛૚૙ ૙૛૚૙

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c. Consider the multiplication that you did in part (b). What about the arrangement of the entries in the identity matrix causes ࡮ࡵ ൌ ࡮?

When you multiply ࡮ࡵ, you are multiplying the rows of ࡮ by the columns of ࡵ; since every entry in the columns of ࡵ but the row that you are currently multiplying by is ૙, you only get a single value of ࡮ to carry over, and it is carried over in the same position. A similar thing happens with ࡵ࡮, but this time it is because the rows of the identity are zero everywhere except at the position you want to carry over.

4. Consider the matrices ૚ ૜૚െ ૚૙ ࡭ൌ൦ ૛൪and࡮ ൌ ൥ ൩ ૛ ૙૚ ૛ ૝ ૙૙ ૜

a. Multiply ࡭࡮ and ࡮࡭ or explain why you cannot. ૚ ې ૜ ૚ െۍ ૜૚ ۑ૛ ێ ࡭࡮ ൌ ൥ ૛൩ ǡ࡮࡭ൌ ૛ ۑ ૛ ૝ێ ૛ ۑ ૜ ێ ૜ ے૙૙ ૙ۏ

b. Would you consider ࡮ to be an identity matrix for ࡭? Why or why not?

No. ࡭࡮ has all of the same entries as ࡭ for those entries that ࡭࡮ has, but ࡭ possesses an additional column that ࡭࡮ does not. ࡮࡭ possesses all of the same entries as ࡭ for those that exist in ࡭, but it also possesses a row of zeros that did not exist in ࡭; all three matrices have different dimensions.

૚૙૙ ૚૙ c. Would you consider ࡵ ൌቂ ቃ or ࡵ ൌ ൥૙૚૙൩ an identity matrix for ࡭? Why or why not? ૛ ૙૚ ૜ ૙૙૚

Answers may vary. It is true that ࡵ૛࡭ൌ࡭ and ࡭ࡵ૜ ൌ࡭, but neither of these can commute with ࡭ based on their dimensions. We can say that ࡵ૛ is an identity for ࡭ on the left, and ࡵ૜ is an identity for ࡭ on the right.

5. We’ve shown that matrix multiplication is generally not commutative, meaning that as a general rule for two .ࡲ in each of the following examplesڄࡳൌࡳڄ࡭. Explain why ࡲڄ࡮്࡮ڄmatrices ࡭ and ࡮, ࡭ ૚૜ ૛૟ a. ࡲൌቂ ቃ ǡࡳ ൌ ቂ ቃ ૛૙ ૝૙ ૚૝ ૟ ૚૝ ૟ ࡲ ൌ ቂ ቃ. Because ࡳൌ૛ࡲ, we have ڄ ࡳൌቂ ቃandࡳڄWe see that ࡲ ૝૚૛ ૝૚૛ .ࡲڄࡲൌࡳڄ ࡲሻ ൌ ሺ૛ࡲሻڄሺ૛ࡲሻ ൌ૛ሺࡲڄࡳൌࡲڄࡲ

૚૜૛ ૙૙૙ b. ࡲൌ൥૜૚૛൩ ǡࡳ ൌ ൥૙૙૙൩ ૝૜૛ ૙૙૙ ૙૙૙ ૙૙૙ ࡲ ൌ ൥૙૙૙൩. The matrix ࡳ is the zero matrix, and any matrix ڄ ࡳൌ൥૙૙૙൩andࡳڄWe see that ࡲ ૙૙૙ ૙૙૙

multiplied by a zero matrix will result in the zero matrix.

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A STORY OF FUNCTIONS Lesson 3 M2 PRECALCULUS AND ADVANCED TOPICS

૚૜૛ ૚૙૙ c. ࡲൌ൥૜૚૛൩ ǡࡳ ൌ ൥૙૚૙൩ ૝૜૛ ૙૙૚ ૚૜૛ ૚૜૛ ࡲ ൌ ൥૜૚૛൩. The matrix ࡳ is an identity matrix. Any square ڄ ࡳൌ൥૜૚૛൩andࡳڄWe see that ࡲ ૝૜૛ ૝૜૛

matrix multiplied by an identity matrix will result in the original matrix.

૚૜૛ ૜૙૙ d. ࡲൌ൥૜૚૛൩ ǡࡳ ൌ ൥૙૜૙൩ ૝૜૛ ૙૙૜ ૜ૢ૟ ૜ૢ૟ ࡲൌ൥ ૢ૜૟൩. Because ࡳൌ૜ࡵ, we haveڄࡳൌ൥ ૢ૜૟൩and ࡳڄWe see that ࡲ ૚૛ ૢ ૟ ૚૛ ૢ ૟ .ࡲڄࡲൌࡳڄ ࡲሻ ൌ ሺ૜ࡵሻڄࡵሻ ൌ૜ሺࡵڄሺ૜ࡵሻ ൌ૜ሺࡲڄࡳൌࡲڄࡲ

6. Let ࡵ࢔ be the ࢔ൈ࢔ identity matrix. For the matrices given below, perform each of the following calculations or explain why the calculation is not possible: ૚ ૜ ૢെ૚૛ ࡭ൌ൦૛ ൪࡮ൌቂቃ ૛ െ૜ ૝ ૚ ૛ ૜ ૜૚૜ ૚ ૛ ξ૛ െ૛ ࡯ൌ൥૚૙૚൩ ࡰൌ൥ ૛൩ ૜૚૜ ૜૛ ૚૙

a. ࡭࡮ ૢ ૛૜ െ ૝ ൦ ૛ ૛ ൪ ૛ ૚૝ ૚૟ ૜ ૜

b. ࡮࡭ These matrices have incompatible dimensions multiplied this way.

c. ࡭࡯ ࡭ has ૛ columns, and ࡯ has ૜ rows, which means we cannot multiply them.

d. ࡭࡮࡯ ૚ ૚૙ െ ૚૙ ൦ ૛ ൪ ૚ૡૡ ૟૛ ૚ૡૡ ૜ ૜ ૜

e. ࡭࡮࡯ࡰ Matrix ࡭࡮࡯ is a ૛ൈ૜ matrix, and matrix ࡰ is a ૛ൈ૝ matrix, which means we cannot multiply them.

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f. ࡭ࡰ

ξ૛ ૚ ې ૚૙ ൅૟ ૛ۍ ۑ ૛ ૝ ێ ۑ ૝ ૚૙ ێ ૟૛ξ૛ ൅ െ ૚ ے ૜ ૜ ۏ

g. ࡭૛ ૛૞ ૠ ൦ ૝ ૛ ൪ ૠ ૞ૡ ૜ ૢ

h. ࡯૛ ૚ૢ ૟ ૚ૢ ൥ ૟૛૟൩ ૚ૢ ૟ ૚ૢ

i. ࡮࡯૛ ૛૙૜ ૟૝ ૛૙૜ ቂ ቃ െ૚૝ െ૝ െ૚૝

j. ࡭࡮࡯ ൅ ࡭ࡰ ࡭࡮࡯ and ࡭ࡰ have different dimensions, so they cannot be added.

k. ࡭࡮ࡵ૛

࡮ is ૛ൈ૜ and ࡵ૛ is ૛ൈ૛, so they cannot be multiplied in this order.

l. ࡭ࡵ૛࡮ This is the same as ૢ ૛૜ െ ૝ ࡭࡮ ൌ ൦ ૛ ૛ ൪ ૛ ૚૝ ૚૟ ૜ ૜

m. ࡯ࡵ૜࡮

࡯ࡵ૜ ൌ࡯, so this is ࡯࡮, but these have incompatible dimensions and cannot be multiplied.

n. ࡵ૛࡮࡯ This is the same as ૜૛ ૚૚ ૜૛ ࡮࡯ ൌ ቂ ቃ െ૛ െ૛ െ૛

o. ૛࡭ ൅ ࡮ ૛࡭ and ࡮ have different dimensions, so they cannot be added.

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p. ࡮ሺࡵ૜ ൅࡯ሻ ૝૚૜ ࡵ૜ ൅࡯ൌ൥૚૚૚൩ ૜૚૝ ૝૚ ૚૙ ૜૝ ࡮ሺࡵ ൅࡯ሻ ൌቂ ቃ ૜ െ૞ ૛ െ૚

q. ࡮൅࡮࡯ ૝૚ ૚૙ ૜૝ ቂ ቃ െ૞ ૛ െ૚

r. ૝ࡰࡵ૝

൤ ૡ૝ξ૛ െૡ ૛൨ ૚૛ ૡ ૝૙

7. Let ࡲ be an ࢓ൈ࢔ matrix. Then what do you know about the dimensions of matrix ࡳ in the problems below if each expression has a value? a. ࡲ൅ࡳ ࡳ must have the exact same dimensions as ࡲ in order to add them together. That is, the dimensions of ࡳ are ࢓ൈ࢔.

b. ࡲࡳ Here we know that the rows of ࡳ must be the same as the columns of ࡲ; that is, ࡳ has ࢔ rows.

c. ࡳࡲ We know that ࡳ has ࢓ columns.

d. ࡲࡴࡳ for some matrix ࡴ.

We know nothing about the dimensions of matrix ࡳ based on the dimensions of ࡲ.

8. Consider an ࢓ൈ࢔ matrix ࡭ such that ࢓്࢔. Explain why you cannot evaluate ࡭૛.

The only way to evaluate ࡭૛ is to multiply ࡭࡭, which implies that ࡭ has the same number of rows as it does columns. Since ࢓്࢔, then ࡭૛ does not exist.

૙૚૛ ૙૛૚ ૙૛૚ 9. Let ࡭ൌ൥૛૙૚൩ ǡ࡮ ൌ ൥૚૙૚൩ ǡ࡯ൌ൥૙૙૚൩ represent the routes of three airlines ࡭, ࡮, and ࡯ between ૚૛૙ ૛૛૙ ૚૛૙

three cities. a. Zane wants to fly from City 1 to City 3 by taking Airline ࡭ first and then Airline ࡮ second. How many different ways are there for him to travel? ૞૝૚ .࡮ൌ൥૛૟૛൩, and the entry in row 1, column 3 is ૚, there is only ૚ way for Zane to travelڄSince ࡭ ૛૛૜

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b. Zane did not like Airline ࡭ after the trip to City 3, so on the way home, Zane decides to fly Airline ࡯ first and then Airline ࡮ second. How many different ways are there for him to travel? ૝૛૛ .࡮ ൌ ൥૛૛૙൩, and the entry in row 3, column 1 is ૛, there are ૛ ways for Zane to travel ڄ Since࡯ ૛૛૜

૙ ૚૚૛ ૚ ૙૛૚ 10. Let ࡭ൌ቎ ቏represent airline flights of one airline between ૝ cities. ૛ ૚૙૛ ૚ ૛૚૙ ࡭. Calculate ࡭૛. What do the entries in matrix ࡭૛ڄa. We use the notation ࡭૛ to represent the product ࡭ represent? ૞ ૞૝૜ We see that ࡭૛ ൌ൦૞ ૞૛૟൪. The entry in row ࢏, column ࢐ of matrix ࡭૛ is the number of ways to get ૜ ૟૟૞ ૝ ૛૞૟ from

City ࢏ to City ࢐ with one stop in between at one of the other cities.

b. Jade wants to fly from City 1 to City 4 with exactly one stop. How many different ways are there for her to travel?

૛ Since ሺ࡭ ሻ૚ǡ૝ ൌ૜, Jade can choose between ૜ different ways to travel.

c. Now Jade wants to fly from City 1 to City 4 with exactly two stops. How many different ways are there for her to choose? ૚૟ ૚૞ ૚ૡ ૛૜ ૚૞ .࡭ൌ൦ ૚ૢ ૛૚ ૚ૢ൪, and ሺ࡭૜ሻ ൌ૛૜, there are ૛૜ different ways for Jade to travelڄ Since ࡭૜ ൌ࡭૛ ૛૜ ૚ૢ ૛૙ ૛૝ ૚ǡ૝ ૚ૡ ૛૚ ૚૝ ૛૙

Lesson 3: Matrix Arithmetic in Its Own Right 63

^dKZzK&&hEd/KE^

DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ

PRECALCULUS AND ADVANCED TOPICS ͻDKh>2

Topic B >ŝŶĞĂƌdƌĂŶƐĨŽƌŵĂƚŝŽŶƐŽĨWůĂŶĞƐĂŶĚ^ƉĂĐĞ

&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

(+) Add, subtract, and multiply matrices of appropriate dimensions.

(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. /ŶƐƚƌƵĐƚŝŽŶĂůĂǇƐ͗ 10 >ĞƐƐŽŶ4: Linear Transformations Review (S)1 >ĞƐƐŽŶϱ: Coordinates of Points in Space (P) >ĞƐƐŽŶϲ: Linear Transformations as Matrices (E) >ĞƐƐŽŶϳ͗ Linear Transformations Applied to Cubes (E) >ĞƐƐŽŶƐ ϴ–ϵ͗ Composition of Linear Transformations (E, E) >ĞƐƐŽŶϭϬ͗ Matrix Multiplication Is Not Commutative (P) >ĞƐƐŽŶϭϭ͗ Matrix Addition Is Commutative (P) >ĞƐƐŽŶϭϮ͗ Matrix Multiplication Is Distributive and Associative (P) >ĞƐƐŽŶϭϯ͗ Using Matrix Operations for Encryption (M)

Topic B explores the usefulness of matrices with dimensions higher than ʹൈʹ. The concept of linear transformations from Module 1 is extended to linear transformations in three- (and higher-) dimensional space. In Lessons 4–6, students use what they know about linear transformations performed on real and complex numbers in two dimensions and extend that to three-dimensional space. They verify the conditions

1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

64 dŽƉŝĐ: Linear Transformations of Planes and Space

©201ϴ Great Minds ®. eureka-math.org A STORY OF FUNCTIONS dŽƉŝĐ DϮ PRECALCULUS AND ADVANCED TOPICS of linearity in two-dimensional space and make conjectures about linear transformations in three- dimensional space. Students add matrices and perform scalar multiplication, exploring the geometric interpretations for these operations in three dimensions. In Lesson 7, students examine the geometric effects of linear transformations in Թଷ induced by various ͵ൈ͵ matrices on the unit cube. Students explore these transformations and discover the connections between a ͵ൈ͵ matrix and the geometric effect of the transformation produced by the matrix. The materials support the use of geometry software, such as the freely available GeoGebra, but software is not required. Students extend their knowledge of the multiplicative inverse and that it exists precisely when the determinant of the matrix is nonzero from the area of a unit square in two dimensions to the volume of the unit cube in three dimensions.

In Lesson 8, students explore a sequence of transformations in two dimensions, and this is extended in Lesson 9 to three dimensions. Students see a sequence of transformations as represented by multiplication of several matrices and relate this to a composition. In Lessons 8–9, students practice scalar and matrix multiplication extensively, setting the stage for properties of matrices studied in Lessons 10 and 11. In Lesson 10, students discover that matrix multiplication is not commutative and verify this finding algebraically for ʹൈʹ and ͵ൈ͵ matrices. In Lesson 11, students translate points by matrix addition and see that, while matrix multiplication is not commutative, matrix addition is commutative. They also write points in two and three dimensions as single column matrices (vectors) and multiply matrices by vectors.

The study of matrices continues in Lesson 12 as students discover that matrix multiplication is associative and distributive. In Lesson 13, students recap their understanding of matrix operations—matrix product, matrix sum, and scalar multiplication—and properties of matrices by using matrices and matrix operations to discover encrypted codes. The geometric and arithmetic roles of the zero matrix and the identity matrix are explored in Lessons 12 and 13. Students understand that the zero matrix is similar to the role of Ͳ in the real number system, and the identity matrix is similar to ͳ. Throughout Topic B, students study matrix operations in two- and three-dimensional space and relate these abstract representations to the transformations they represent . Students have opportunities to use computer programs as tools for examining and understanding the geometric effects of transformations produced by matrices on the unit circle.

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A STORY OF FUNCTIONS Lesson 4 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 4: Linear Transformations Review

Student Outcome Students review their understanding of linear transformations on real and complex numbers as well as linear transformations in two-dimensional space. They use their understanding to form conjectures about linear transformations in three-dimensional space.

Lesson Notes In this lesson, students respond to questions to demonstrate what they understand about linear transformations performed on real and complex numbers, as well as those performed in two-dimensional space and three-dimensional space. They describe special cases of linear transformations for real and complex numbers. They also verify the conditions for linear transformations in two-dimensional space and make conjectures about how to define and represent linear transformations in three-dimensional space.

Classwork Discussion (5 minutes): Linear Transformations Թ՜Թ During this Discussion, call on a variety of students to answer the questions as they are posed. Recall that when we write ܮǣ Թ ՜ Թ, we mean that ܮ is a function that takes real numbers as inputs and produces real numbers as outputs.

Let’s recall the conditions a function ܮǣ Թ ՜ Թ must meet to be called a linear Scaffolding: transformation. What are they? Show the following :ሺݕሻ. examplesܮሺݔሻ ൅ܮሺݔ൅ݕሻ ൌܮ à First, we need .ሺݔሻܮሺܽήݔሻ ൌܽήܮ à Second, we need ሺݔሻ ൌʹݔ݂ ?Good. And what exactly are ܽ, ݔ, and ݕ in this case ሺݔሻ ൌʹݔ൅ͳ݃ à Each of these symbols represents a real number. ͳ Okay. Now give three examples of formulas that you know represent linear ݄ሺݔሻ ൌ ݔ ʹ transformations from Թ to Թ. ͳ ሺݔሻ ൌ ݔെͳ݆ ʹ ሺݔሻ ൌʹήݔܮ à ሺݔሻ ൌ͵ήݔ Which are linearܮ à ሺݔሻ ൌͶήݔ transformations? Explainܮ à Good. So we see that a linear transformation ܮǣ Թ ՜ Թ must have the form your answer. ሺݔሻ ൌܽήݔ for some real number ܽ. Now let’s explore the geometric effect Advanced students can beܮ that linear transformations have on the real number line. The value of ܽ could challenged to explore the be positive, negative, or zero, and the size of ܽ could be big or small. How are questions about the these cases different geometrically? Let’s explore together. geometric implications of different values of ܽ without any additional cueing.

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?ሺݔሻ ൌ͵ήݔܮ Consider the interval ሾെʹǡ ʹሿ. What is the image of the interval under the rule à We have ܮሺെʹሻ ൌ͵ήെʹൌെ͸, and ܮሺʹሻ ൌ͵ήʹൌ͸. It looks as though the image of ሾെʹǡ ʹሿ is ሾെ͸ǡ ͸ሿ.

When we use a scale factor of ͵, the interval expands to ͵ times its original size. Now let’s try a role reversal: Can you think of a mapping that would take the interval at the bottom of the figure and map it to the one at the top? Explain your answer. ͳ ሺݔሻ ൌቀ ቁήݔ. This makes sense because we are contracting the interval so that itܮ à The inverse map is ͵ ଵ ends up being of its original size. ଷ .ሺݔሻ ൌܽήݔ expands an interval, and in other cases it contracts the intervalܮ In some cases, the transformation Can you think of a mapping that leaves the size of the interval unchanged? .ሺݔሻ ൌͳήݔ does not change the size of the intervalܮ à ሺݔሻ ൌͳήݔ as the identity mapping? Try to make sense of thisܮ Can you see why mathematicians refer to phrase. à Under the identity mapping, each point on the number line is mapped to a location that is identical to its original location. Now let’s consider some cases where ܽ is a negative number. What is the image of ሾെʹǡ ʹሿ under the rule ?ሺݔሻ ൌെͳήݔܮ

à In this case, each point on the number line is reflected across the origin. The result is the same interval we started with, but each point has been taken to the opposite side of the number Ͳ. ͳ .ሺݔሻ ൌቀെ ቁήݔܮ ሺݔሻ ൌെʹήݔ orܮ Describe what happens after the application of ʹ à First, each interval gets reflected across the origin, and then a dilation gets applied. In the first case, each interval dilates to twice its original size, and in the second case, each interval dilates to half its original size. ?ሺݔሻ ൌͲήݔܮ ,Now let’s consider the only remaining case. What happens when we apply the zero map ,ሺݔሻ ൌͲήݔൌͲ for every real number ݔ, the entire number line gets mapped to a single pointܮ à Since namely, Ͳ.

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Perhaps we should call this kind of transformation a collapse, since what used to be a line has now collapsed to a single point. Write a quick summary of this discussion in your notebook, and then share what you wrote with a partner. ሺݔሻ ൌܽήݔ for someܮ ǣ Թ ՜ Թ has the formܮ à Every linear transformation Scaffolding: real number ܽ. Each of these transformations is essentially a dilation with scale factor ܽ. Special cases include ܽൌͲ, which causes the whole Students could create a graphic number line to collapse to the origin; ܽൌͳ, which leaves the whole organizer listing the matrix in number line unchanged; and ܽൌെͳ, which causes each point on the one column and the number line to undergo a reflection through the origin. transformation represented in the next column.

Exercises 1–2 (3 minutes) Students should complete the exercises in pairs. One pair should be asked to share each Scaffolding: solution. The presenting pair could represent the mapping geometrically on the board to aid students who are strong visual learners. Provide a specific interval that students can use when solving Exercise 1 (e.g., have students Exercises explain the effect of the 1. Describe the geometric effect of each mapping. transformations on the interval a. ࡸሺ࢞ሻ ൌૢή࢞ ሾെͳǡͳሿ). Dilates the interval by a factor of ૢ

૚ ࡸሺ࢞ሻ ൌെ ή࢞ b. ૛ ૚ Reflects the interval over the origin and then applies a dilation with a scale factor of ૛

2. Write the formula for the mappings described. a. A dilation that expands each interval to ૞ times its original size

ࡸሺ࢞ሻ ൌ૞ή࢞

b. A collapse of the interval to the number ૙

ࡸሺ࢞ሻ ൌ૙ή࢞

Discussion (10 minutes): Linear Transformations ԧ՜ԧ In addition to transformations from Թ to Թ, in the previous module we also studied transformations that take complex numbers as inputs and produce complex numbers as outputs, and we used the symbol ܮǣԧ ՜ ԧ to denote this. As we did for functions from Թ to Թ, let’s recall the conditions a function ܮǣ ԧ ՜ ԧ must meet to be called a linear transformation. What are they? .ሺݔሻܮሺܽήݔሻ ൌܽήܮ ሺݕሻ andܮሺݔሻ ൅ܮሺݔ൅ݕሻ ൌܮ à Just as before, we need ?What exactly do ܽ, ݔ, and ݕ represent in this case à Each of these symbols represents a complex number.

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Give three examples of formulas that you know represent linear transformations from ԧ to ԧ. ሺݔሻ ൌʹήݔܮ à ሺݔሻ ൌ݅ήݔܮ à ሺݔሻ ൌሺ͵൅ͷ݅ሻήݔܮ à .ܽ ሺݔሻ ൌܽήݔ for some complex numberܮ ǣ ԧ ՜ ԧ must have the formܮ So we see that a linear transformation As we did above with functions from Թ to Թ, let’s explore the geometric effect of some special linear transformations from ԧ to ԧ. What do you remember about the result of multiplying ܽ൅ܾ݅ by ܿ൅݀݅? à We know that ሺܽ൅ܾ݅ሻ ή ሺܿ൅݀݅ሻ ൌܽܿെܾ݀൅ሺܽ݀ ൅ ܾܿሻ ή݅. Recall that we visualize complex numbers as points in the complex plane. What do you remember about the geometric effect of multiplying one complex number by another? à Multiplication by a complex number induces a rotation and a dilation. Now let’s get back to the subject of linear transformations. Discuss the following questions with another student. Analyze each question both algebraically and geometrically. Be prepared to share your analysis with the whole class in a few minutes. Is there a linear transformation ܮǣ ԧ ՜ ԧ that leaves each point in the complex plane unchanged? That is, is there an identity map from ԧ՜ԧ? :à Yes. If we take ܽൌͳ൅Ͳ݅ and ݔൌܿ൅݀݅, we get this ሺݔሻ ൌܽݔൌሺͳ൅Ͳ݅ሻሺܿ൅݀݅ሻൌͳήܿെͲή݀൅ሺͳή݀൅Ͳήܿሻ݅ൌܿ൅݀݅ൌݔܮ à Geometrically, multiplying by the complex number ͳ൅Ͳ݅ is the same thing as multiplying by the real number ͳ, so it makes sense that this action leaves points unchanged. Describe the set of linear transformations ܮǣ ԧ ՜ ԧ that induce a pure rotation of points in the complex plane.

à We know that multiplication by a complex number ܽ induces a rotation through the argument of ܽ, as well as a dilation by the modulus of ܽ. So if we want a pure rotation, then we need to have ȁܽȁ ൌͳ. ሺݔሻ ൌܽݔ will induce a pureܮ Thus, if ܽ is any point on the unit circle surrounding the origin, then rotation of the complex plane.

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Describe the set of linear transformations ܮǣ ԧ ՜ ԧ that induce a pure dilation of points in the complex plane. ሺݔሻ ൌܽݔ induces a pure dilation, then the argument of ܽ must be Ͳ. This means that ܽ is, in fact, aܮ à If real number.

Is there a linear transformation ܮǣ ԧ ՜ ԧ that collapses the entire complex plane into a single point? ሺݔሻ ൌܽݔൌሺͲ൅Ͳ݅ሻݔൌͲ൅Ͳ݅ for every ݔ. Thus, everyܮ à Yes. If we choose ܽൌͲ൅Ͳ݅, then we get point in the complex plane gets mapped to the origin.

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Is there a linear transformation ܮǣ ԧ ՜ ԧ that induces a reflection across the real axis? In particular, where would such a transformation take ͷ൅ʹ݅? Where would it take ͳ൅Ͳ݅? Where would it take Ͳ൅ͳ݅? à Since ሺͷǡʹሻ maps to ሺͷǡ െʹሻ, the image of ͷ൅ʹ݅ must be ͷെʹ݅. à Since ሺͳǡͲሻ maps to ሺͳǡͲሻ, the image of ͳ൅Ͳ݅ must be ͳ൅Ͳ݅. à Since ሺͲǡͳሻ maps to ሺͲǡ െͳሻ, the image of Ͳ൅ͳ݅ must be Ͳെͳ݅.

ሺݔሻ ൌܽݔ induces a reflection across the real axis, then what wouldܮ If there is a complex number ܽ such that its argument have to be? Analyze your work above carefully. à Since points on the real axis map to themselves, it would appear that the argument of ܽ would have to be Ͳ. In other words, there must be no rotational component. à On the other hand, points on the imaginary axis get rotated through ͳͺͲι. But this is irreconcilable with the statement above. Thus, it appears that there is no complex number ܽ that meets the necessary requirements, so we conclude that reflection across the real axis is not a linear transformation from ԧ to ԧ.

Discussion (7 minutes): Linear Transformations Թ૛ ՜Թ૛ Recall that we can visualize complex numbers as points in the complex plane. Let’s briefly review some arithmetic with complex numbers.

?Let ݖଵ ൌ͵൅Ͷ݅, and let ݖଶ ൌͷെʹ݅. What is ݖଵ ൅ݖଶ

݅ʹà ݖଵ ൅ݖଶ ൌ ሺ͵൅ͷሻ ൅ ሺͶെʹሻ݅ൌͺ൅ ͵ Thinking of these two complex numbers as points in the coordinate plane, we can write ݖ ൌቀ ቁ and ଵ Ͷ ͷ ͵ ͷ ͺ ݖ ൌቀ ቁ. Thus, we could just as well write ݖ ൅ݖ ൌቀ ቁ൅ቀ ቁൌቀ ቁ. In fact, we can even abandon the ଶ െʹ ଵ ଶ Ͷ െʹ ʹ

context of complex numbers and let addition of ordered pairs take on a life of its own. ͻ ͵ With this in mind, what is ቀ ቁ൅ቀ ቁ? െͳ ͸ ͻ ͵ ͻ൅͵ ͳʹ à ቀ ቁ൅ቀ ቁൌቀ ቁൌቀ ቁ െͳ ͸ െͳ ൅ ͸ ͷ ?Now let’s review scalar multiplication. With ݖൌ͵൅ͷ݅, what is ͳͲݖ à ͳͲݖ ൌ ͳͲሺ͵൅ͷ݅ሻ ൌ͵Ͳ൅ͷͲ݅

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͵ ͵ ͵Ͳ Thinking of ݖ as a point in the coordinate plane, we have ݖൌቀ ቁ, and ͳͲݖ ൌ ͳͲ ቀ ቁൌቀ ቁ. Once again, we ͷ ͷ ͷͲ can set complex numbers to the side and let scalar multiplication of ordered pairs be an operation in its own right. Ͷ With this in mind, what is ͷቀ ቁ? െʹ Ͷ ͷήͶ ʹͲ à ͷቀ ቁൌቀ ቁൌቀ ቁ െʹ ͷήെʹ െͳͲ We saw in Module 1 that multiplication by a complex number is a linear transformation that can be modeled ݔ ͵െͶ ݔ .ቀ ቁൌቀ ቁቀ ቁܮ using matrices. For instance, to model the product ሺ͵ ൅ Ͷ݅ሻ ή ሺݔ ൅ ݕ݅ሻ, we can write ݕ Ͷ͵ ݕ Describe the geometric effect of this transformation on points in the plane. ݔ ଶ ଶ à Each point ቀݕቁ is dilated by a factor of ξ͵ ൅Ͷ ൌͷ and rotated through an angle that is given by Ͷ ቀ ቁ. ͵ ܽെܾ When a matrix is used to model complex-number multiplication, it always has the form ቀ ቁ, and we ܾܽ already know that these matrices represent linear transformations. Now let’s consider a more general matrix ݔ െ͵ ʹ ݔ transformation, ܣቀ ቁൌቀ ቁቀ ቁ. Does this also represent a linear transformation? Let’s begin by ݕ ͳͶݕ checking the addition requirement.

ݔଵ ݔଶ ݔଵ ൅ݔଶ െ͵ ʹ ݔଵ ൅ݔଶ െ͵ሺݔ ൅ݔ ሻ ൅ʹሺݕ ൅ݕ ሻ à ܣቆቀ ቁ൅ቀ ቁቇ ൌ ܣ ቀ ቁൌቀ ቁቀ ቁൌ൬ ଵ ଶ ଵ ଶ ൰ ݕଵ ݕଶ ݕଵ ൅ݕଶ ͳͶݕଵ ൅ݕଶ ሺݔଵ ൅ݔଶሻ ൅Ͷሺݕଵ ൅ݕଶሻ

ݔ ݔ െ͵ݔ െ͵ݔ ൅ʹݕ ൅ʹݕ ܣቆቀ ଵቁ൅ቀ ଶቁቇ ൌ ൬ ଵ ଶ ଵ ଶ൰ ݕଵ ݕଶ ݔଵ ൅ݔଶ ൅Ͷݕଵ ൅Ͷݕଶ

ݔଵ െ͵ ʹ ݔଵ െ͵ݔ ൅ʹݕ à ܣቀ ቁൌቀ ቁቀ ቁൌ൬ ଵ ଵ൰ ݕଵ ͳͶݕଵ ݔଵ ൅Ͷݕଵ

ݔଶ െ͵ ʹ ݔଶ െ͵ݔ ൅ʹݕ ܣቀ ቁൌቀ ቁቀ ቁൌ൬ ଶ ଶ൰ ݕଶ ͳͶݕଶ ݔଶ ൅Ͷݕଶ ݔ ݔ െ͵ݔ ൅ʹݕ െ͵ݔ ൅ʹݕ െ͵ݔ ൅ʹݕ െ͵ݔ ൅ʹݕ ܣቀ ଵቁ൅ܣቀ ଶቁൌ൬ ଵ ଵ൰൅൬ ଶ ଶ൰ൌ൬ ଵ ଵ ଶ ଶ൰ ݕଵ ݕଶ ݔଵ ൅Ͷݕଵ ݔଶ ൅Ͷݕଶ ݔଵ ൅Ͷݕଵ ൅ݔଶ ൅Ͷݕଶ ݔ ݔ ݔ ݔ à When we compare ܣቆቀ ଵቁ൅ቀ ଶቁቇ with ܣቀ ଵቁ൅ܣቀ ଶቁ, we see that the results are the same. ݕଵ ݕଶ ݕଵ ݕଶ

Now let’s check the scalar multiplication requirement. ݔ ݇ݔ െ͵ ʹ ݇ݔ െ͵ ή ݇ݔ ൅ ʹ ή ݇ݕ à ܣቆ݇ቀ ቁቇ ൌ ܣ ൬ ൰ൌቀ ቁ൬ ൰ൌ൬ ൰ ݕ ݇ݕ ͳͶ݇ݕ ͳή݇ݔ൅Ͷή݇ݕ

ݔ െ͵ ʹ ݔ െ͵ήݔ൅ʹήݕ െ͵ ή ݇ݔ ൅ ʹ ή ݇ݕ à ݇ήܣቀ ቁൌ݇ήቀ ቁቀ ቁൌ݇ή൬ ൰ൌ൬ ൰ ݕ ͳͶݕ ͳήݔ൅Ͷήݕ ͳή݇ݔ൅Ͷή݇ݕ ݔ ݔ .ቀݕቁ, we see that they are the sameܣቆ݇ቀݕቁቇ with ݇ήܣ à When we compare

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ݔ െ͵ ʹ ݔ So we see that the matrix mapping ܣቀ ቁൌቀ ቁቀ ቁ is indeed a linear transformation. In fact, we could ݕ ͳͶݕ

ݔ ܽܿ ݔ use the same technique to prove in general that ܣቀ ቁൌቀ ቁቀ ቁ is a linear transformation also. ݕ ܾ݀ ݕ ݔ ݕ :So if ݔൌቀ ଵቁ and ݕൌቀ ଵቁ, then we can summarize the linearity requirements as follows ݔଶ ݕଶ ݕܣݔ൅ܣሺݔ൅ݕሻ ൌܣ .1 ݔܣ ሺ݇ ή ݔሻ ൌ ݇ ήܣ .2 This is much cleaner and easier to handle! ݔ ܽܿ ݔ ܿܽ Suppose that the determinant of ቀ ቁ is zero. What can we say about the mapping ቀ ቁ฽ቀ ቁቀ ቁ in ݕ ܾ݀ ݕ ܾ݀ this case? à In Module 1, we learned that the determinant of a matrix represents the area of the image of the unit square. If this area is Ͳ, then the transformation represents a collapse of some kind. This implies that the transformation is not invertible.

Discussion (5 minutes): Linear Transformations Թ૜ ՜Թ૜ ݔ We saw that points in Թଶ can be represented as ordered pairs ቀ ଵቁ. How do you suppose we might represent ݔଶ points in three-dimensional space? ݔଵ ଷ .à Points in Թ can be represented as ordered triples ൭ݔଶ൱ ݔଷ How would you guess to define addition and scalar multiplication of points in a three-dimensional setting?

ݔଵ ݕଵ ݔଵ ൅ݕଵ à ൭ݔଶ൱൅൭ݕଶ൱ൌ൭ݔଶ ൅ݕଶ൱ ݔଷ ݕଷ ݔଷ ൅ݕଷ

ݔଵ ݇ήݔଵ à ݇ή൭ݔଶ൱ൌ൭݇ήݔଶ൱ ݔଷ ݇ήݔଷ What might it mean for a function ܮǣ Թଷ ՜Թଷ to be a linear transformation? ሺݕሻ andܮሺݔሻ ൅ܮሺݔ൅ݕሻ ൌܮ à If ݔ and ݕ are points in Թଷ and ݇ is any real number, then we should have .ሺݔሻܮሺ݇ήݔሻ ൌ݇ήܮ Any ideas about how to represent such transformations? à Perhaps a ͵ൈ͵ matrix will come into play! One last point to ponder: We’ve explored the geometric effects of linear transformations in one and two dimensions. Give some thought to what effects linear transformations might have in a three-dimensional setting. We’ll take up this issue further in the next lesson!

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Closing (10 minutes) Students should complete a graphic organizer that summarizes their understanding of linear transformations. They can work for a few minutes independently and then compare their results with a partner. During the last few minutes, volunteers can share the information they included in their graphic organizers. They can share this information aloud or display it on the board so that students can make revisions to their organizers. A suggested format for the organizer is shown.

ࡸǣԹ ՜ Թ ࡸǣԧ ՜ ԧ ࡸǣԹ૛ ՜Թ૛ ࡸǣԹ૜ ՜Թ૜

Conditions for ࡸ

General form of ࡸ

What ࡸ represents

Note: Students form conjectures about linear transformations in three-dimensional space, which is discussed further in later lessons.

Exit Ticket (5 minutes)

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Name Date

Lesson 4: Linear Transformations Review

Exit Ticket

1. In Module 1, we learned about linear transformations for any real number functions. What are the conditions of a linear transformation? If a real number function is a linear transformation, what is its form? What are the two characteristics of the function?

2. Describe the geometric effect of each mapping: ሺݔሻ ൌ͵ݔܮ .a

ሺݖሻ ൌሺξʹ ൅ ξʹ݅ሻ ή ݖܮ .b

Ͳെͳ ݔ ሺݖሻ ൌቀ ቁቀ ቁ, where ݖis a complex numberܮ .c ͳͲ ݕ

Ͳ ݔʹ ሺݖሻ ൌቀ ቁቀ ቁ, where ݖis a complex numberܮ .d Ͳʹ ݕ

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Exit Ticket Sample Solutions

ࡸሺ࢞൅࢟ሻ ൌࡸሺ࢞ሻ ൅ࡸሺ࢟ሻ, ࡸሺ࢑࢞ሻ ൌ࢑ࡸሺ࢞ሻ, where ࢞, ࢟, and ࢑ are real numbers

It is in the form of ࡸሺ࢞ሻ ൌ࢓࢞, and its graph is a straight line going through the origin. It is an odd function.

2. Describe the geometric effect of each mapping: a. ࡸሺ࢞ሻ ൌ૜࢞ Dilate the interval by a factor of ૜.

b. ࡸሺࢠሻ ൌሺξ૛ ൅ ξ૛࢏ሻ ή ࢠ ࣊ Rotate radians counterclockwise about the origin and dilate the interval by a factor of 2. ૝

૙െ૚ ࢞ c. ࡸሺࢠሻ ൌቀ ቁቀ ቁ, where ࢠis a complex number ૚૙ ࢟ ࣊ Rotate radians counterclockwise about the origin. ૛

૛૙ ࢞ d. ࡸሺࢠሻ ൌቀ ቁቀ ቁ, where ࢠis a complex number ૙૛ ࢟ Dilate ࢠ by a factor of ૛.

Problem Set Sample Solutions

1. Suppose you have a linear transformation ࡸǣ Թ ՜ Թ, where ࡸሺ૜ሻ ൌ૟, ࡸሺ૞ሻ ൌ૚૙. a. Use the addition property to find ࡸሺ૟ሻ, ࡸሺૡሻ, ࡸሺ૚૙ሻ, and ࡸሺ૚૜ሻ.

ࡸሺ૟ሻ ൌࡸሺ૜൅૜ሻ ൌࡸሺ૜ሻ ൅ࡸሺ૜ሻ ൌ૟൅૟ൌ૚૛ ࡸሺૡሻ ൌࡸሺ૜൅૞ሻ ൌࡸሺ૜ሻ ൅ࡸሺ૞ሻ ൌ૟൅૚૙ൌ૚૟ ࡸሺ૚૙ሻ ൌࡸሺ૞ሻ ൅ࡸሺ૞ሻ ൌ૚૙൅૚૙ൌ૛૙ ࡸሺ૚૜ሻ ൌࡸሺ૚૙ ൅ ૜ሻ ൌࡸሺ૚૙ሻ ൅ࡸሺ૜ሻ ൌ૛૙൅૟ൌ૛૟

b. Use the multiplication property to find ࡸሺ૚૞ሻ, ࡸሺ૚ૡሻ, andࡸሺ૜૙ሻ.

ࡸሺ૚૞ሻ ൌࡸሺ૜ή૞ሻ ൌ૜ήࡸሺ૞ሻ ൌ૜ή૚૙ൌ૜૙ ࡸሺ૚ૡሻ ൌࡸሺ૜ή૟ሻ ൌ૜ήࡸሺ૟ሻ ൌ૜ή૚૛ൌ૜૟ ࡸሺ૜૙ሻ ൌࡸሺ૞ή૟ሻ ൌ૞ήࡸሺ૟ሻ ൌ૞ή૚૛ൌ૟૙

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c. Find ࡸሺെ૜ሻ, ࡸሺെૡሻ, and ࡸሺെ૚૞ሻ.

ࡸሺെ૜ሻ ൌࡸሺെ૚ ή ૜ሻ ൌെ૚ήࡸሺ૜ሻ ൌെ૟ ࡸሺെૡሻ ൌࡸሺെ૚ ή ૡሻ ൌെ૚ήࡸሺૡሻ ൌ െ૚૟ ࡸሺെ૚૞ሻ ൌࡸሺെ૜ ή ૞ሻ ൌെ૜ήࡸሺ૞ሻ ൌ െ૜ ή ૚૙ ൌ െ૜૙

d. Find the formula for ࡸሺ࢞ሻ.

Given ࡸሺ࢞ሻ is a linear transformation; therefore, it must have a form of ࡸሺ࢞ሻ ൌ࢓࢞, where ࢓ is real number.

Given ࡸሺ૜ሻ ൌ૟; therefore, ૜࢓ ൌ ૟, ࢓ൌ૛. ࡸሺ࢞ሻ ൌ૛࢞.

e. Draw the graph of the function ࡸሺ࢞ሻ.

2. A linear transformation ࡸǣԹ ՜ Թ must have the form of ࡸሺ࢞ሻ ൌࢇ࢞ for some real number ࢇ. Consider the interval ሾെ૞ǡ ૛ሿ. Describe the geometric effect of the following, and find the new interval. a. ࡸሺ࢞ሻ ൌ૞࢞ It dilates the interval by a scale factor of ૞; ሾെ૛૞ǡ ૚૙ሿ

b. ࡸሺ࢞ሻ ൌ െ૛࢞ It reflects the interval over the origin and then dilates it with a scale factor of ૛; ሾെ૝ǡ ૚૙ሿ

3. A linear transformation ࡸǣԹ ՜ Թ must have the form of ࡸሺ࢞ሻ ൌࢇ࢞ for some real number ࢇ. Consider the interval ሾെ૛ǡ ૟ሿ. Write the formula for the mapping described, and find the new interval. a. A reflection over the origin

ࡸሺ࢞ሻ ൌെ࢞,ሾെ૟ǡ ૛ሿ

b. A dilation with a scale of ξ૛

ࡸሺ࢞ሻ ൌ ξ૛ ή࢞,ൣെ૛ξ૛ǡ૟ξ૛൧

૚ c. A reflection over the origin and a dilation with a scale of ૛ ૚ ࡸሺ࢞ሻ ൌെ ࢞ ሾെ૜ǡ ૚ሿ ૛ ,

d. A collapse of the interval to the number ૙

ࡸሺ࢞ሻ ൌ૙࢞,ሾ૙ǡ ૙ሿ

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4. In Module 1, we used ૛ൈ૛ matrices to do transformations on a square, such as a pure rotation, a pure reflection, a pure dilation, and a rotation with a dilation. Now use those matrices to do transformations on this complex number: ࢠൌ૛൅࢏. For each transformation below, graph your answers. a. A pure dilation with a factor of ૛

૛૙ ૛૙ ૛ ૝ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ ૙૛ ૙૛ ૚ ૛

࣊ b. A pure radians counterclockwise rotation about the origin ૛

૙െ૚ ૙െ૚ ૛ െ૚ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ ૚૙ ૚૙ ૚ ૛

c. A pure ࣊ radians counterclockwise rotation about the origin

െ૚ ૙ െ૚ ૙ ૛ െ૛ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ ૙െ૚ ૙െ૚૚ െ૚

d. A pure reflection about the real axis

૚૙ ૚૙ ૛ ૛ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ ૙െ૚ ૙െ૚ ૚ െ૚

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e. A pure reflection about the imaginary axis

െ૚ ૙ െ૚ ૙ ૛ െ૛ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ ૙૚ ૙૚૚ ૚

f. A pure reflection about the line ࢟ൌ࢞

૙૚ ૙૚ ૛ ૚ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ ૚૙ ૚૙ ૚ ૛

g. A pure reflection about the line ࢟ൌെ࢞

૙െ૚ ૙െ૚૛ െ૚ ቀ ቁ,ࡸሺࢠሻ ൌቀ ቁቀ ቁൌቀ ቁ െ૚ ૙ െ૚ ૙ ૚ െ૛

૙െ૚ ࣊ 5. Wesley noticed that multiplying the matrix ቀ ቁ by a complex number ࢠ produces a pure radians ૚૙ ૛

૛૙ counterclockwise rotation, and multiplying by ቀ ቁ produces a pure dilation with a factor of ૛. So, he thinks he ૙૛

૛െ૚ ࣊ can add these two matrices, which will produce ቀ ቁ and will rotate ࢠ by radians counterclockwise and dilate ૚૛ ૛

ࢠ with a factor of ૛. Is he correct? Explain your reason. ࢇെ࢈ ࢞ No, he is not correct. For the general transformation of complex numbers, the form is ࡸሺࢆሻ ൌቀ ቁቀ ቁ. ࢈ࢇ ࢟ ࢇെ࢈ ࢈ By multiplying the matrix ቀ ቁ to ࢠ, it rotates ࢠ an angle ܉ܚ܋ܜ܉ܖ ቀ ቁ and dilates ࢠ with a factor of ξࢇ૛ ൅࢈૛. ࢈ࢇ ࢇ ૚ ࣊ ܉ܚ܋ܜ܉ܖ ቀ ቁ ൌ ૛૟Ǥ ૞૟૞ι, and ඥሺ૚ሻ૛ ൅ሺ૛ሻ૛ ൌ ξ૞, which is not radians or a dilation of a factor of ૛. ૛ ૛

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૚ ૚ ૚ 6. In Module 1, we learned that there is not any real number that will satisfy ൌ ൅ , which is the addtition ࢇା࢈ ࢇ ࢈ property of linear transformation. However, we discussed that some fixed complex numbers might work. Can you find two pairs of complex numbers that will work? Show your work. ૚ ૚ ૚ Given ൌ ൅ ,ࢇǡ ࢈ ് ૙ ࢇା࢈ ࢇ ࢈ ૚ ࢈൅ࢇ ൌ ࢇ൅࢈ ࢇ࢈ ࢇ࢈ ൌ ሺࢇ൅࢈ሻ૛ ࢇ࢈ ൌ ࢇ૛ ൅૛ࢇ࢈൅࢈૛ ࢇ૛ ൅ࢇ࢈൅࢈૛ ൌ૙ ૚ ૜ ࢇ૛ ൅ࢇ࢈൅ ࢈૛ ൌെ ࢈૛ ૝ ૝ ૚ ૛ ૜ ൬ࢇ ൅ ࢈൰ ൌെ ࢈૛ ૛ ૝ ૚ ξ૜ ࢇ൅ ࢈ൌേ ࢈ή࢏ ૛ ૛ െ૚ േ ξ૜࢏ ࢇൌ ࢈ ૛

For example, ࢈ൌ૛ and ࢇൌെ૚൅ξ૜ ή࢏ or ࢇൌെ૚െξ૜ ή࢏. If ࢈ൌ૝ and ࢇൌെ૛൅૛ξ૜ ή࢏ or ࢇൌെ૛െ૛ξ૜ ή࢏. Answers may vary.

7. Suppose ࡸis a complex-number function that satisfies the dream conditions: ࡸሺࢠ൅࢝ሻ ൌࡸሺࢠሻ ൅ࡸሺ࢝ሻ and ࡸሺ࢑ࢠሻ ൌ࢑ࡸሺࢠሻ for all complex numbers ࢠ, ࢝, and ࢑. Show ࡸሺࢠሻ ൌ࢓ࢠ for a fixed complex number ࢓, the only type of complex-number function that satisfies these conditions.

Let ࢠൌࢇ൅࢈࢏ǡ࢝ൌࢉ൅ࢊ࢏, for addition property:

ࡸሺࢠ൅࢝ሻ ൌࡸሺࢠሻ ൅ࡸሺ࢝ሻ ൌ࢓ሺࢇ൅࢈࢏ሻ ൅࢓ሺࢉ൅ࢊ࢏ሻ ൌ࢓൫ሺࢇ൅ࢉሻ ൅ ሺ࢈൅ࢊሻ࢏൯

ࡸሺࢠ൅࢝ሻ ൌࡸሺࢇ൅࢈࢏൅ࢉ൅ࢊ࢏ሻ ൌࡸ൫ሺࢇ൅ࢉሻ ൅ ሺ࢈൅ࢊሻ࢏൯ ൌ࢓൫ሺࢇ൅ࢉሻ ൅ ሺ࢈൅ࢊሻ࢏൯; they are the same.

For multiplication property:

ࡸሺ࢑ࢠሻ ൌ࢑ࡸሺࢠሻ ൌ࢑࢓ࢠൌ࢑࢓ሺࢇ൅࢈࢏ሻ

ࡸሺ࢑ࢠሻ ൌࡸ൫࢑ሺࢇ൅࢈࢏ሻ൯ ൌ࢓࢑ሺࢇ൅࢈࢏ሻ; they are the same.

8. For complex numbers, the linear transformation requires ࡸሺ࢞൅࢟ሻ ൌࡸሺ࢞ሻ ൅ࡸሺ࢟ሻǡࡸሺࢇή࢞ሻ ൌࢇήࡸሺ࢞ሻ. Prove that ࢞ ࢇ࢈ ࢞ ࢞ in general ࡸቀ ቁൌቀ ቁቀ ቁ is a linear transformation, where ቀ ቁ represents ࢠൌ࢞൅࢟࢏. ࢟ ࢉࢊ࢟ ࢟ For the addition property:

࢞૚ ࢞૛ ࢞૚ ൅࢞૛ ࢇࢉ ࢞૚ ൅࢞૛ ࢇሺ࢞૚ ൅࢞૛ሻ ൅ࢉሺ࢟૚ ൅࢟૛ሻ ࡸሺࢠ૚ ൅ࢠ૛ሻ ൌࡸቆቀ ቁ൅ቀ ቁቇ ൌࡸቀ ቁൌቀ ቁቀ ቁൌ൬ ൰ ࢟૚ ࢟૛ ࢟૚ ൅࢟૛ ࢈ࢊ ࢟૚ ൅࢟૛ ࢈ሺ࢞૚ ൅࢞૛ሻ ൅ࢊሺ࢟૚ ൅࢟૛ሻ

ࢇࢉ ࢞૚ ࢇࢉ ࢞૛ ࢇ࢞૚ ൅ࢉ࢟૚ ࢇ࢞૛ ൅ࢉ࢟૛ ࡸሺࢠ૚ ൅ࢠ૛ሻ ൌࡸሺࢠ૚ሻ ൅ࡸሺࢠ૛ሻ ൌቀ ቁቀ ቁ൅ቀ ቁቀ ቁൌ൬ ൰൅൬ ൰ൌ ࢈ࢊ࢟૚ ࢈ࢊ ࢟૛ ࢈࢞૚ ൅ࢊ࢟૚ ࢈࢞૛ ൅ࢊ࢟૛ ࢇ࢞ ൅ࢉ࢟ ൅ࢇ࢞ ൅ࢉ࢟ ࢇሺ࢞ ൅࢞ ሻ ൅ࢉሺ࢟ ൅࢟ ሻ ൬ ૚ ૚ ૛ ૛ ൰ൌ൬ ૚ ૛ ૚ ૛ ൰; the result is the same. ࢈࢞૚ ൅ࢊ࢟૚ ൅࢈࢞૛ ൅ࢊ࢟૛ ࢈ሺ࢞૚ ൅࢞૛ሻ ൅ࢊሺ࢟૚ ൅࢟૛ሻ

Now we need to prove the multiplication property.

࢞ ࢑࢞ ࢇࢉ ࢑࢞ ࢇ࢑࢞ ൅ ࢉ࢑࢟ ࡸሺ࢑ࢠሻ ൌࡸቆ࢑ቀ ቁቇ ൌࡸ൬ ൰ൌቀ ቁ൬ ൰ൌ൬ ൰ ࢟ ࢑࢟ ࢈ࢊ ࢑࢟ ࢈࢑࢞ ൅ ࢊ࢑࢟

ࢇࢉ ࢞ ࢇ࢞ ൅ ࢉ࢟ ࢇ࢑࢞ ൅ ࢉ࢑࢟ ࡸሺ࢑ࢠሻ ൌ࢑ࡸሺࢠሻ ൌ࢑ቀ ቁቀ ቁൌ࢑൬ ൰ൌ൬ ൰; the result is the same. ࢈ࢊ࢟ ࢈࢞ ൅ ࢊ࢟ ࢈࢑࢞ ൅ ࢊ࢑࢟

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Lesson 5: Coordinates of Points in Space

Student Outcomes Students use parallelograms to interpret the sum of two points in Թଶ geometrically. Students make a similar interpretation for the sum of two points in Թଷ. Students recognize that scalar multiplication of points in Թଶ corresponds to a dilation. Students make a similar interpretation for scaling points in Թଷ.

Lesson Notes In the Opening Exercise, students perform addition and scalar multiplication, first on complex numbers and then on ordered pairs. This leads into the Discussion portion of the lesson in which students explore geometric interpretations for these operations, first for points in Թଶ and then for points in Թଷ. This lesson focuses on students constructing arguments and making use of structure while studying points in space. This is the first of several lessons that have students graph in three-dimensional space. This may be difficult for some students to visualize, but it can be made easier by using a corner of the room. Allow the wall seam to be the ݖ-axis and one seam between the wall and the floor to be the ݔ-axis while the other is the ݕ-axis. Consider setting up a tape with a number line in each direction. Have students experiment with plotting different points to get the feel of what graphing in 3-D looks like. It is important for students to understand that since they live in a 3-D world, they must have a way to explain points, objects, etc., in this 3-D space. There are many computer graphics programs that are available free of charge that can support student learning on 3-D graphing such as GeoGebra. Additionally, blank 3-D coordinate axes are supplied at the end of this lesson for use throughout this module. The study of vectors forms a vital part of this course; notation for vectors varies across different contexts and curricula. ,or in column format ۄͶǡ ͷۃ lowercase, bold, non-italicized) or) ܞ These materials refer to a vector as Ͷ Ͷ ቀ ቁ or ቂ ቃ. ͷ ͷ .ۄͶǡ ͷۃ to establish a name for the vector ”ۄͶǡͷۃൌܞ Use “let without the word “let” preceding the equation when naming a vector unless it is ۄͶǡͷۃൌܞ This curriculum avoids stating absolutely clear from the context that a vector is being named. However, the “ൌ” continues to be used to describe vector equations, like ܞ൅ܟൌܟ൅ܞ, as has been done with equations throughout all other grades.

The vector from ܣ to ܤ is referred to as “vector ሬሬሬሬሬԦܣܤ”—notice, this is a ray with a full arrow. This notation is consistent with the way vectors are introduced in Grade 8 and is also widely used in postsecondary textbooks to describe both a ray and a vector depending on the context. To avoid confusion in this curriculum, the context is provided or strongly implied, so it is clear whether the full arrow indicates a vector or a ray. For example, when referring to a ray from ܣ passing through ܤ, “ray ሬሬሬሬሬԦܣܤ” is used, and when referring to a vector from ܣ to ܤ, “vector ሬሬሬሬሬԦܣܤ” is used. Students should be encouraged to think about the context of the problem and not just rely on a hasty inference based on the symbol. The magnitude of a vector is signified as ԡܞԡ (lowercase, bold, non-italicized).

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Classwork In the Opening Exercise, students practice adding and multiplying complex numbers and then ordered pairs displayed as matrices. This allows students to review previously taught skills and then explore the geometric interpretation of these operations in the lesson.

Opening Exercise (2 minutes)

Opening Exercise

Compute:

a. ሺെ૚૙ ൅ ૢ࢏ሻ ൅ ሺૠെ૞࢏ሻ Scaffolding: െ૜ ൅ ૝࢏ To help students visualize the results, have them b. ૞ήሺ૛൅૜࢏ሻ color-code each vector, ૚૙ ൅ ૚૞࢏ always having the resulting vector a standard color. ૞ ૛ For example, the first c. ቀ ቁ൅ቀ ቁ െ૟ ૠ vector could always be ૠ ቀ ቁ blue, the second vector ૚ green, and so on, but the resulting vector always ૜ d. െ૛ ቀ ቁ red. This helps students െ૜ visualize. െ૟ ቀ ቁ ૟ When plotting in 3-D, continue color-coding. Give students only positive Discussion (8 minutes): Addition Viewed Geometrically values, and use the Here are two questions to ponder: corners of the classroom to represent 3-D space, When we add two points ݔ and ݕ, we produce a third point, ݖൌݔ൅ݕ. How having students create the do you think ݖ is related geometrically to ݔ and ݕ? ͵ number lines, ݔ, ݕ, and .When we multiply a point ݔ by a scalar (say, ͵), we produce another point, ݖ ,ݕൌ͵ݔ. How do you think ݕ is related to ݔ geometrically? For advanced learners Take a moment to write down any conjectures you may have about these once they have shown questions, and then share your thinking with a partner. We’ll explore these understanding of 2-D, questions in detail in the upcoming discussion. allow them to move to 3-D ͸ ͳ graphing, and try it .Let ݔൌቀ ቁ, and let ݕൌቀ ቁ. Compute ݖൌݔ൅ݕ ͵ ͷ without scaffolding. ͸ ͳ ͹ Challenge them to create a à ݖൌቀ ቁ൅ቀ ቁൌቀ ቁ ͵ ͷ ͺ physical model of 3-D space, using a shoe box or something in the classroom—this can be used later for all students.

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.To discover the geometric relationships between ݔ, ݕ, and ݖ, it’s helpful to picture the box shown below ,͵ Note that the width of this rectangle is ͸, which is the first coordinate of ݔ, and the height of the rectangle is which is the second coordinate of ݔ. We can associate ݔ with the diagonal of this box. In fact, we might even ,go so far as to say that the diagonal is ݔ. In some contexts, it’s useful to think of an ordered pair as a vector Ͳ ͸ which in this case means an arrow that originates at ቀ ቁ and terminates at ቀ ቁ. Ͳ ͵

When we add ܡ to ܠ, we are effectively performing a translation. Let’s draw a picture of this:

So, the question still remains: How exactly are these three arrows related? Take a moment to discuss your thoughts with a partner, and then we’ll continue.

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Let’s see what happens when we perform the addition the other way around, that is, when we compute ܡ൅ܠ :

To really see what’s going on, let’s look at ܠ൅ܡ and ܡ൅ܠ together in the same picture:

Now a clear picture is starting to emerge. Make a conjecture, and share it with a partner. What strikes you about this figure? à It looks as though the points in the figure form a parallelogram. Can you make an argument showing that this is indeed a parallelogram?

à The upper and lower sides both have a rise-to-run ratio of ͵ǣ ͸, which means they must be parallel. à The left and right sides both have a rise-to-run ratio of ͷǣ ͳ, which means they must also be parallel. à Since both pairs of opposite sides of the figure are parallel, the figure is indeed a parallelogram.

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Exercises 1–3 (3 minutes)

Exercises ૞ ૛ 1. Let ܠൌቀ ቁ, ܡൌቀ ቁ. Compute ܢൌܠ൅ܡ, and draw the associated parallelogram. ૚ ૜

ૠ ቀ ቁ ૝

െ૝ ૚ 2. Let ܠൌቀ ቁ, ܡൌቀ ቁ.Compute ܢൌܠ൅ܡ, and draw the associated parallelogram. ૛ ૜

െ૜ ቀ ቁ ૞

૜ െ૚ 3. Let ܠൌቀ ቁ, ܡൌቀ ቁ. Compute ܢൌܠ൅ܡ, and draw the associated parallelogram. ૛ െ૜

૛ ቀ ቁ െ૚

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Example 1 (3 minutes): Degenerate Parallelograms

We’ve seen that adding two points gives a third point, which lies at the vertex of a parallelogram. Can you think of a case where no parallelogram is produced? Think for a moment about this. à Answers will vary. Take two points that lie on a line through the origin. These cases do not produce proper parallelograms. We call these degenerate parallelograms. Let’s try a few examples. For each problem below, compute the sum of the given points, and draw the associated picture. Ͷ ʹ ܢൌቀ ቁ൅ቀ ቁ ʹ ͳ Ͷ ʹ ͸ à ܢൌቀ ቁ൅ቀ ቁൌቀ ቁ ʹ ͳ ͵

Ͷ െʹ ܢൌቀ ቁ൅ቀ ቁ ʹ െͳ Ͷ െʹ ʹ à ܢൌቀ ቁ൅ቀ ቁൌቀ ቁ ʹ െͳ ͳ

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Ͷ െͶ ܢൌቀ ቁ൅ቀ ቁ ʹ െʹ Ͷ െͶ Ͳ à ܢൌቀ ቁ൅ቀ ቁൌቀ ቁ ʹ െʹ Ͳ

Discussion (6 minutes): Scalar Multiplication ʹ Now let’s turn our attention to scalar multiplication. What is the geometric effect of taking ܠൌቀ ቁ and ͳ ʹ .ൌ͵ቀ ቁ, and then plot ݔ and ݖ in the planeܢ scaling it by a factor of ͵? Compute ͳ ʹ ͸ à We have ܢൌ͵ቀ ቁൌቀ ቁ. ͳ ͵

Describe the relationship between ܠ and ͵ܠ. à It appears that ܠ and ͵ܠ lie on a line through the origin and that the length of the arrow to ͵ܠ is ͵ times as long as the arrow to ܠ.

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Does the picture below add to your understanding of the nature of scalar multiplication? Discuss what you see with a partner. Try to make connections between the picture and the underlying arithmetic involved in ʹ computing ͵ቀ ቁ. ͳ

.à The ݔ-coordinate is scaled from ʹ to ʹή͵ൌ͸, so the widths of the triangles are in a ͵ǣ ͳ ratio .à The ݕ-coordinate is scaled from ͳ to ͳή͵ൌ͵, so the heights of the triangles are also in a ͵ǣ ͳ ratio à Since both of these triangles have a right angle, they are similar to each other by the SAS principle. This means that the lengths of the arrows to ͵ܠ and to ܠ are indeed in a ͵ǣ ͳ ratio and that Ͳ, ܠ, and ͵ܠ do lie on a common line. ʹ What do you suppose would happen if we scaled ܠൌቀ ቁ by െ͵ instead? Plot ܠ and െ͵ܠ in the plane, and ͳ then describe the geometric relationship between these two vectors. ʹ െ͸ à Now we have ܢൌെ͵ቀ ቁൌቀ ቁ. ͳ െ͵

à The vector ܢൌെ͵ܠ is ͵ times as long as ܠ, but it lies on the opposite side of the origin. We call the new vector produced the resultant. Can you state the previous result using the word resultant? à The resultant is ͵ times as long as ܠ, but it lies on the opposite side of the origin.

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Exercises 4–6 (2 minutes)

૜ 4. Let ܠൌቀ ቁ. Compute ܢൌ૛ܠ, and plot ܠ and ܢ in the plane. ૛

૟ ቀ ቁ ૝

െ૟ ૚ 5. Let ܠൌቀ ቁ. Compute ܢൌ ܠ, and plot ܠ and ܢ in the plane. ૜ ૜

૛ ቀ ቁ െ૚

૚ 6. Let ܠൌቀ ቁ. Compute ܢ ൌ െ૜ܠ, and plot ܠ and ܢ in the plane. െ૚

െ૜ ቀ ቁ ૜

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Example 2 (5 minutes): 3-D Addition Viewed Geometrically

Now that we’ve discussed the geometric interpretation of adding and scaling points in Թଶ, let’s turn our attention to Թଷ. What do you think adding points in Թଷ will mean from a geometric point of view? What will scaling points mean? Make a conjecture, and then quickly share your thinking with a partner. We’ll pursue these questions in detail in the upcoming Discussion. If graphing software is available, demonstrate plotting points in 3-D space for the class, and then allow students to try. Being able to visualize points in 3-D space is difficult for students, and they will need lots of practice. Blank 3-D axes are included for teacher and student use at the end of this lesson. ͷ െ͵ Let ܠൌ൭Ͷ൱ and ܡൌ൭ʹ ൱. Compute ܢൌܠ൅ܡ, and then plot each of these three points. ʹ ͵

z z

y y

x x

Can you tell how ܠ, ܡ, and ܢ are related? This picture should help:

à In the resultant, the ݔ–value of ͷ is translated in the ݔ direction െ͵ units, the ݔ–value of Ͷ is translated ʹ ൌ൭͸൱ܢ .in the ݕ direction ʹunits, and the ݔ–value of ʹ is translated in the ݖ direction ͵units ͷ

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Exercises 7–8 (2 minutes)

૜ ૚ 7. Let ܠൌ൭૚൱ and ܡൌ൭૜൱. Compute ܢൌܠ൅ܡ, and then plot each of these three points. ૚ ૚

૝ ൭૝൱ ૛

૜ ૙ 8. Let ܠൌ൭૙൱ and ܡൌ൭૜൱. Compute ܢൌܠ൅ܡ, and then plot each of these three points. ૙ ૙

૜ ൭૜൱ ૙

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Example 3 (5 minutes): Scalar Multiplication in 3-D

What do you suppose it means to perform scalar multiplication in a three-dimensional setting? Make a conjecture, and briefly share it with a partner. ͳ Let ܠൌ൭͵൱. Compute ܢൌ͵ܠ, and then plot each of these points. Describe what you see. ʹ

z z

x x

y y

à The resultant was dilated by a factor of ͵. The resultant lies on the same line through the origin as the vector. ͳ What do you suppose would happen if we scaled ܠൌ൭͵൱ by a factor of െͳ? What if the scale factor ʹ were െʹ? ͳ െͳ à When we scale by െͳ, we get െͳ ή ܠ ൌ െͳ ή ൭͵൱ൌ൭െ͵൱. The resultant is the same length as ܠ, but it ʹ െʹ points in the opposite direction. ͳ െʹ à When we scale by െʹ, we get െʹ ή ܠ ൌ െʹ ή ൭͵൱ൌ൭െ͸൱. The resultant is twice as long as ܠ, but ʹ െͶ again, it points in the opposite direction.

z z

x y

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Exercises 9–10 (2 minutes)

૚ 9. Let ܠൌ൭૚൱. Compute ܢൌ૝ܠ, and then plot each of the three points. ૚

૝ ൭૝൱ ૝

૛ ૚ ܠൌ൭ ൱ ܢൌെ ܠ 10. Let ૝ . Compute ૛ , and then plot each of the three points. Describe what you see. ૝

െ૚ ൭െ૛൱ െ૛ In the resultant, the direction has reversed, and the length of the ray is half the original.

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Closing (2 minutes) Write a brief response to the following questions in your notebook, and then share your responses with a partner: What did you learn about the geometry of vector addition in Թଶ and Թଷ? à We can use the diagonal of a parallelogram to visualize the resultant vector of the sum of two vectors. What did you learn about the geometry of scalar multiplication of vectors in Թଶ and Թଷ? à Multiplying a vector by a scalar produces a resultant that is a dilation of the original vector.

Exit Ticket (5 minutes)

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Name Date

Lesson 5: Coordinates of Points in Space

Exit Ticket

1. Find the sum of the following, and plot the points and the resultant. Describe the geometric interpretation. ͵ ͳ a. ቀ ቁ൅ቀ ቁ ͳ ͵

ʹ ͵ b. ቀ ቁ൅ቀ ቁ Ͳ ʹ

െʹ ͵ c. ቀ ቁ൅ቀ ቁ Ͷ െʹ

͵ d. െቀ ቁ ͳ

2. Find the sum of the following. ͵ ͳ a. ൭ͳ൱൅൭͵൱ ͵ ͳ

ʹ Ͳ b. ൭Ͳ൱൅൭ʹ൱ ͳ ͳ

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Exit Ticket Sample Solutions

1. Find the sum of the following, and plot the points and the resultant. Describe the geometric interpretation. ૜ ૚ a. ቀ ቁ൅ቀ ቁ ૚ ૜

૝ ቀ ቁ The two points, the resultant, and the origin form a parallelogram. ૝

૛ ૚ b. ቀ ቁ൅ቀ ቁ ૙ ૛

૜ ቀ ቁ The two points, the resultant, and the origin form a parallelogram. ૛

െ૛ ૜ c. ቀ ቁ൅ቀ ቁ ૝ െ૛

૚ ቀ ቁ The two points, the resultant, and the origin form a parallelogram. ૛

૜ d. െቀ ቁ ૚

െ૜ ቀ ቁ The resultant is a vector of the same magnitude in the opposite direction. െ૚

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2. Find the sum of the following.

૜ ૚ a. ൭૚൱൅൭૜൱ ૜ ૚ ૝ ൭૝൱ ૝

૛ ૙ b. ൭૙൱൅൭૛൱ ૚ ૚ ૛ ൭૛൱ ૛

Problem Set Sample Solutions

1. Find the sum of the following complex numbers, and graph them on the complex plane. Trace the parallelogram that is formed by those two complex numbers, the resultant, and the origin. Describe the geometric interpretation. ૛ ૜ a. ܠൌቀ ቁ, ܡൌቀ ቁ ૜ ૛

૞ ቀ ቁ The two points, the resultant, and the origin form a parallelogram. ૞

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૛ െ૝ b. ܠൌቀ ቁ, ܡൌቀ ቁ ૝ ૛

െ૛ ቀ ቁ The two points, the resultant, and the origin form a parallelogram. ૟

૛ െ૝ c. ܠൌቀ ቁ, ܡൌቀ ቁ ૚ െ૛

െ૛ ቀ ቁ The resultant is double the magnitude of the original vector in the opposite direction. െ૚

૚ ૛ d. ܠൌቀ ቁ, ܡൌቀ ቁ ૛ ૝

૜ ቀ ቁ The resultant is triple the length of the original vector in the same direction. ૟

2. Simplify and graph the complex number and the resultant. Describe the geometric effect on the complex number. ૚ a. ܠൌቀ ቁ, ࢑ൌ૛, ࢑ܠ ൌ ? ૛

૛ ቀ ቁ The point is dilated by a factor of ૛. ૝

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െ૟ ૚ b. ܠൌቀ ቁ, ࢑ൌെ , ࢑ܠ ൌ? ૜ ૜

૛ ૚ ቀ ቁ The point is dilated by a factor of and mapped to the other side of the origin on the same line. െ૚ ૜

૜ c. ܠൌቀ ቁ, ࢑ൌ૙, ࢑ܠ ൌ? െ૛

૙ ቀ ቁ The point is mapped to the origin, ሺ૙ǡ ૙ሻ. ૙

3. Find the sum of the following points, graph the points and the resultant on a three-dimensional coordinate plane, and describe the geometric interpretation. ૚ ૛ a. ܠൌ൭૚൱, ܡൌ൭૚൱ ૛ ૛

૜ ൭૛൱ The two points, the resultant, and the origin are on the same plane and form a parallelogram. ૝

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૚ ૛ b. ܠൌ൭૚൱, ܡൌ൭૛൱ ૚ ૛

૜ ൭૜൱ Three points all collapse onto the same line in space. ૜

૛ ૙ c. ܠൌ൭૙൱, ܡൌ൭૛൱ ૙ ૙

૛ ൭૛൱ The two points, the resultant, and the origin are on the same plane and formed a parallelogram. ૙

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4. Simplify the following, graph the point and the resultant on a three-dimensional coordinate plane, and describe the

geometric effect. ૛ a. ܠൌ൭૚൱, ࢑ൌ૛, ࢑ܠ ൌ? ૚

૝ ൭૛൱ The point is dilated by a factor of ૛. ૛

૛ b. ܠൌ൭૛൱, ࢑ൌെ૚, ࢑ܠ ൌ? ૛

െ૛ ൭െ૛൱ The point is mapped to the other side of the origin on the same line. െ૛

5. Find: ૙ a. Any two different points whose sum is ቀ ቁ ૙ ૛ െ૛ Answers vary. ቀ ቁ,ቀ ቁ ૛ െ૛

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૙ b. Any two different points in three dimensions whose sum is ൭૙൱ ૙ ૛ െ૛ Answers vary. ൭૛൱, ൭െ૛൱ ૛ െ૛

c. Any two different complex numbers and their sum will create the degenerate parallelogram.

૚ ૛ ࢟૚ ࢟૛ Answers vary. ቀ ቁ, ቀ ቁ, as long as the answers are in the relation of ൌ ૛ ૝ ࢞૚ ࢞૛

d. Any two different points in three dimensions and their sum lie on the same line ૚ ૛ Answers vary. ൭૚൱, ൭૛൱ ૚ ૛

૚ e. A point that is mapped to ቀ ቁ after multiplying Ȃ૛ െ૜ ૚ െ ൮ ૛൲ ૜ ૛

૚ ૛ ቌ ૛ ቍ െ f. A point that is mapped to െ૛ after multiplying ૜ ૝ ૜ െ ൮ ૝൲ ૜ െ૟

૛ െ૝ 6. Given ܠൌቀ ቁ and ܡൌቀ ቁ: ૚ െ૛ a. Find ܠ൅ܡ, and graph the parallelogram that is formed by ܠ, ܡ, ܠ൅ܡ, and the origin.

െ૛ ቀ ቁ െ૚

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૛െ૝ b. Transform the unit square by multiplying it by the matrix ቀ ቁ, and graph the result. ૚െ૛ ૛െ૝ ૙ ૙ ૛െ૝ ૚ ૛ ૛െ૝ ૚ െ૛ ૛െ૝ ૙ െ૝ ቀ ቁቀ ቁൌቀ ቁ, ቀ ቁቀ ቁൌቀ ቁ, ቀ ቁቀ ቁൌቀ ቁ, ቀ ቁቀ ቁൌቀ ቁ ૚െ૛ ૙ ૙ ૚െ૛ ૙ ૚ ૚െ૛ ૚ െ૚ ૚െ૛ ૚ െ૛

c. What did you find from parts (a) and (b)?

They both have degenerate parallelograms.

d. What is the area of the parallelogram that is formed by part (a)?

The area is ૙.

૛െ૝ e. What is the determinant of the matrix ቀ ቁ? ૚െ૛ ૛െ૝ ቚ ቚൌ૛ሺെ૛ሻ െ૚ሺെ૝ሻ ൌെ૝൅૝ൌ૙ The determinant is ૙. ૚െ૛

f. Based on observation, what can you say about the degenerate parallelograms in part (a) and part (b)?

࢈૚ ࢈૛ For two complex numbers, if ൌ and the determinant of the matrix is ૙, then they will produce ࢇ૚ ࢇ૛ degenerate parallelograms.

૜ 7. We learned that when multiplying െ૚ by a complex number ࢠ, for example, ࢠൌቀ ቁ, the resulting complex number ૛ െ૜ ࢠ ൌቀ ቁ will be on the same line but on the opposite side of the origin. What matrix will produce the same ૚ െ૛ effect? Verify your answer. െ૚ ૙ The matrix that will rotate ࣊ radians is ቂ ቃ. ૙െ૚ െ૚ ૙ ૜ െ૜ ቀ ቁቀ ቁൌቀ ቁ ૙െ૚૛ െ૛

ξ૛ െ૛ ૙ 8. A point ࢠൌ൬ ൰ is transformed to ቀ ቁ. The final step of the transformation is adding the complex number ቀ ቁ. ξ૛ ૙ െ૛ Describe a possible transformation that can get this result.

െ૛ ૙ െ૛ ξ૛ െ૛ ࣊ Going backwards: ቀ ቁ൅ሺെ૚ሻቀ ቁൌቀ ቁ. To move from ൬ ൰ to ቀ ቁ requires a radians counterclockwise ૙ െ૛ ૛ ξ૛ ૛ ૛ ૙െ૚ ξ૛ ૙ rotation: ቂ ቃ and a dilation with a factor of ξ૛: ൤ ൨. ૚૙ ૙ ξ૛

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A STORY OF FUNCTIONS Lesson 6 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 6: Linear Transformations as Matrices

Student Outcomes Students verify that ͵ൈ͵ matrices represent linear transformations from Թଷ ՜Թଷ. Students investigate the matrices associated with transformations such as dilations and reflections in a coordinate plane. Students explore properties of vector arithmetic, including the commutative, associative, and distributive properties.

Classwork Opening Exercise (3 minutes) In the Opening Exercise, students are reminded of their work with ʹൈʹ matrices and prove that the matrix given represents a linear transformation.

Opening Exercise

ૠെ૛ ࢞૚ ࢟૚ Let ࡭ൌቀ ቁ, ࢞ൌቀ ቁ, and ࢟ൌቀ ቁ. Does this represent a linear transformation? Explain how you know. ૞െ૜ ࢞૛ ࢟૛ A linear transformation satisfies the following conditions: ࡸሺ࢞൅࢟ሻ ൌࡸሺ࢞ሻ ൅ࡸሺ࢟ሻandࡸሺ࢑࢞ሻ ൌ࢑ࡸሺ࢞ሻ

ૠെ૛ ࢞ ൅࢟ ૠሺ࢞ ൅࢟ ሻ െ૛ሺ࢞ ൅࢟ ሻ ૠ࢞ ൅ૠ࢟ െ૛࢞ െ૛࢟ ሺ࢞൅࢟ሻ ൌቀ ቁቀ ૚ ૚ቁൌ൬ ૚ ૚ ૛ ૛ ൰ൌ൬ ૚ ૚ ૛ ૛൰ڄ࡭ ૞െ૜ ࢞૛ ൅࢟૛ ૞ሺ࢞૚ ൅࢟૚ሻ െ૜ሺ࢞૛ ൅࢟૛ሻ ૞࢞૚ ൅૞࢟૚ െ૜࢞૛ െ૜࢟૛

ૠെ૛ ࢞૚ ૠሺ࢞ ሻ െ૛ሺ࢞ ሻ ࡭ሺ࢞ሻ ൌቀ ቁቀ ቁൌ൬ ૚ ૛ ൰ ૞െ૜ ࢞૛ ૞ሺ࢞૚ሻ െ૜ሺ࢞૛ሻ

ૠെ૛ ࢟૚ ૠሺ࢟ ሻ െ૛ሺ࢟ ሻ ࡭ሺ࢟ሻ ൌቀ ቁቀ ቁൌ൬ ૚ ૛ ൰ ૞െ૜ ࢟૛ ૞ሺ࢟૚ሻ െ૜ሺ࢟૛ሻ ૠሺ࢞ ሻ െ૛ሺ࢞ ሻ ૠሺ࢟ ሻ െ૛ሺ࢟ ሻ ૠሺ࢞ ሻ െ૛ሺ࢞ ሻ ൅ૠሺ࢟ ሻ െ૛ሺ࢟ ሻ ࡭ሺ࢞ሻ ൅࡭ሺ࢟ሻ ൌ൬ ૚ ૛ ൰൅൬ ૚ ૛ ൰ൌ൬ ૚ ૛ ૚ ૛ ൰ ૞ሺ࢞૚ሻ െ૜ሺ࢞૛ሻ ૞ሺ࢟૚ሻ െ૜ሺ࢟૛ሻ ૞ሺ࢞૚ሻ െ૜ሺ࢞૛ሻ ൅૞ሺ࢟૚ሻ െ૜ሺ࢟૛ሻ

࡭ሺ࢞൅࢟ሻ ൌ࡭ሺ࢞ሻ ൅࡭ሺ࢟ሻ by the distributive property

Discussion (7 minutes): Linear Transformations Թ૜ ՜Թ૜ In previous lessons, we have seen that ʹൈʹ matrices represent linear transformations from Թଶ ՜Թଶ. Do you think that a ͵ൈ͵ matrix would represent a linear transformation from Թଷ ՜Թଷ? Can you prove it? ͹െʹെ͹ ݔଵ ݕଵ ଷ .ݕܣݔ൅ܣሺݔ൅ݕሻ ൌܣ ൌ൭ ͷെ͵െ͸൱, and let ݔൌ൭ݔଶ൱ and ݕൌ൭ݕଶ൱ be points in Թ . Show thatܣ Let െͳͲ െ͸ ͸ ݔଷ ݕଷ

͹െʹെ͹ݔଵ ൅ݕଵ ͹ሺݔଵ ൅ݕଵሻ െʹሺݔଶ ൅ݕଶሻ െ͹ሺݔଷ ൅ݕଷሻ ήሺݔ൅ݕሻ ൌ൭ ͷെ͵െ͸൱൭ݔଶ ൅ݕଶ൱ൌቌ ͷሺݔଵ ൅ݕଵሻ െ͵ሺݔଶ ൅ݕଶሻ െ͸ሺݔଷ ൅ݕଷሻ ቍܣ à െͳͲ െ͸ ͸ ݔଷ ൅ݕଷ െͳͲሺݔଵ ൅ݕଵሻ െ͸ሺݔଶ ൅ݕଶሻ ൅͸ሺݔଷ ൅ݕଷሻ

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ݔ ͹ݔ െʹݔ െ͹ݔ ͹െʹെ͹ ଵ ଵ ଶ ଷ ݔ ൌ ൭ ͷെ͵െ͸൱൭ݔଶ൱ൌ൭ ͷݔଵ െ͵ݔଶ െ͸ݔଷ ൱ܣ à െͳͲ െ͸ ͸ ݔଷ െͳͲݔଵ െ͸ݔଶ ൅͸ݔଷ

͹െʹെ͹ݕଵ ͹ݕଵ െʹݕଶ െ͹ݕଷ ݕ ൌ ൭ ͷെ͵െ͸൱൭ݕଶ൱ൌ൭ ͷݕଵ െ͵ݕଶ െ͸ݕଷ ൱ܣ à െͳͲ െ͸ ͸ ݕଷ െͳͲݕଵ െ͸ݕଶ ൅͸ݕଷ

͹ݔଵ െʹݔଶ െ͹ݔଷ ൅͹ݕଵ െʹݕଶ െ͹ݕଷ ݕ ൌ ൭ ͷݔଵ െ͵ݔଶ െ͸ݔଷ ൅ͷݕଵ െ͵ݕଶ െ͸ݕଷ ൱ܣ ݔ ൅ܣ à െͳͲݔଵ െ͸ݔଶ ൅͸ݔଷ ൅െͳͲݕଵ െ͸ݕଶ ൅͸ݕଷ .ݕ by the distributive propertyܣݔ൅ܣήሺݔ൅ݕሻ ൌܣ à We can see that .ݔ, where ݇ represents a real numberܣሺ݇ήݔሻ ൌ݇ήܣ Now show that

͹െʹെ͹݇ήݔଵ ͹݇ ή ݔଵെʹ݇ ή ݔଶ െ͹݇ήݔଷ ሺ݇ήݔሻ ൌ൭ ͷെ͵െ͸൱൭݇ήݔଶ൱ൌ൭ ͷ݇ ή ݔଵെ͵݇ ή ݔଶ െ͸݇ήݔଷ ൱ܣ à െͳͲ െ͸ ͸ ݇ήݔଷ െͳͲ݇ ή ݔଵെ͸݇ ή ݔଶ ൅͸݇ήݔଷ

͹െʹെ͹ݔଵ ͹ݔଵ െʹݔଶ െ͹ݔଷ ͹݇ ή ݔଵെʹ݇ ή ݔଶ െ͹݇ήݔଷ ݔൌ݇ή൭ ͷെ͵െ͸൱൭ݔଶ൱ൌ݇ή൭ ͷݔଵ െ͵ݔଶ െ͸ݔଷ ൱ൌ൭ ͷ݇ ή ݔଵെ͵݇ ή ݔଶ െ͸݇ήݔଷ ൱ܣà ݇ή െͳͲ െ͸ ͸ ݔଷ െͳͲݔଵ െ͸ݔଶ ൅͸ݔଷ െͳͲ݇ ή ݔଵെ͸݇ ή ݔଶ ൅͸݇ήݔଷ .ݔܣሺ݇ήݔሻ ൌ݇ήܣ à We see that

Let’s briefly take a closer look at the reasoning used here. What properties of Scaffolding: ?arithmetic are involved in comparing, for example, ͹ήሺ݇ήݔ ሻ with ݇ήሺ͹ήݔ ሻ ଵ ଵ For students still struggling In other words, how exactly do we know these are the same number? with matrix multiplication, à We can use the associative property and the commutative property to give them a matrix with know these are the same number. blanks to complete so they ሺݔሻ does indeed represent a linear can see how many termsܣ Now we’ve proved that the transformation transformation, at least for this particular ͵ൈ͵ matrix. We could use exactly they should have in each the same reasoning to show that this is also true for any ͵ൈ͵ matrix. In fact, row such as mathematicians showed in the 1800s that every linear transformation from ̴̴̴ ൅ ̴̴̴ ൅ ̴̴̴ Թଷ ՜Թଷ can be represented by a ͵ൈ͵ matrix! This is one reason why the ቆ̴̴̴ ൅ ̴̴̴ ൅ ̴̴̴ቇ. theory of matrices is so powerful. Students would fill in the entry for each blank. For Exploratory Challenge 1 (10 minutes): The Geometry of 3-D Matrix example, ͳ͵ Transformations ʹ ൭െͷ ʹ൱ቀ ቁൌ െͳ In this activity, students work in groups of four on each of the challenges. Monitor the Ͷ͹ progress of the class as they work independently. For each challenge, select a particular ͳήʹ൅͵ήെͳ group to present their findings to the class at the conclusion of the activity. Allow ቌെͷ ή ʹ ൅ʹήെͳቍ. approximately 7 minutes for students to work in their groups and approximately 3 minutes Ͷήʹ൅͹ήെͳ for students to present their findings. Advanced learners can instead work with a general ͵ൈ͵ matrix as linearity conditions are explored. The reasoning is exactly the same.

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Exploratory Challenge 1: The Geometry of 3-D Matrix Transformations

a. What matrix in Թ૛ serves the role of ૚ in the real number system? What is that role? ૚૙ ቀ ቁ serves the role of the multiplicative identity. ૙૚

.ݔ ൌ ݔ for each point ݔ in Թଷ. Describe the geometric effect of this transformationܣ such that ܣ Find a matrix How might we call such a matrix? ͳͲͲ à We can choose ܣൌ൭ͲͳͲ൱. ͲͲͳ ͳͲͲ ݔଵ ͳήݔଵ ൅Ͳήݔଶ ൅Ͳήݔଷ ݔଵ ݔ ൌ ൭ͲͳͲ൱൭ݔଶ൱ൌ൭Ͳήݔଵ ൅ͳήݔଶ ൅Ͳήݔଷ൱ൌ൭ݔଶ൱ܣ à ͲͲͳ ݔଷ Ͳήݔଵ ൅Ͳήݔଶ ൅ͳήݔଷ ݔଷ à The name “identity matrix” makes sense since the output is identical to the input.

b. What matrix in Թ૛ serves the role of ૙ in the real number system? What is that role? ૙૙ ቀ ቁ serves the role of the additive identity. ૙૙

.ݔ ൌ Ͳ for each point ݔ in Թଷ. Describe the geometric effect of this transformationܣ such that ܣ Find a matrix ͲͲͲ ݔଵ Ͳ .à ൭ͲͲͲ൱൭ݔଶ൱ൌ൭Ͳ൱ This transformation takes every point in space and maps it to the origin ͲͲͲ ݔଷ Ͳ

c. What is the result of scalar multiplication in Թ૛?

Multiplying by a scalar, ࢑, dilates each point in Թ૛ by a factor of ࢑.

.͵ ݔ dilates each point in Թଷ by a factor ofܣsuch that the mapping ݔհ ܣ Find a matrix

ͲͲ ݔଵ ͵ݔଵ ݔଵ͵ ଷ .͵ à ൭Ͳ͵Ͳ൱൭ݔଶ൱ൌ൭͵ݔଶ൱ൌ͵൭ݔଶ൱ Thus, each point in Թ is dilated by a factor of ͲͲ͵ ݔଷ ͵ݔଷ ݔଷ

d. Given a complex number ࢇ൅࢈࢏, what represents the transformation of that point across the real axis?

The conjugate, ࢇെ࢈࢏

ͳͲ Ͳ ݔଵ Investigate the geometric effects of the transformation ൭Ͳͳ Ͳ൱൭ݔଶ൱. Illustrate your findings with one ͲͲെͳ ݔଷ or more specific examples. ͳͲ Ͳ ͷ ͷ à ൭Ͳͳ Ͳ൱൭͵൱ൌ൭͵ ൱ ͲͲെͳ Ͷ െͶ

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.à This transformation induces a reflection across the ݔݕ-plane z

x y

Exploratory Challenge 2 (18 minutes): Properties of Vector Arithmetic In this activity, students work in groups of four on one of the challenges below. Assign different challenges to different groups. When a group finishes one of the challenges, they may choose one of the other challenges to work on as time allows. Monitor the progress of the class as they work independently. For each challenge, select a particular group to present their findings to the class at the conclusion of the activity. For each challenge, students should explore part (a) from an algebraic point of view and part (b) from a geometric point of view. Allow approximately 10 minutes for students to work in their groups and organize their presentations. Allow approximately 9 minutes for students to present their findings.

Exploratory Challenge 2: Properties of Vector Arithmetic

a. Is vector addition commutative? That is, does ࢞൅࢟ൌ࢟൅࢞ for each pair of points in Թ૛? What about points in Թ૜?

We want to show that ࢞൅࢟ൌ࢟൅࢞. First, let’s look at this problem algebraically. ࢞ ࢟ ࢞ ൅࢟ When we compute ࢞൅࢟, we get ቀ ૚ቁ൅ቀ ૚ቁൌቀ ૚ ૚ቁ. ࢞૛ ࢟૛ ࢞૛ ൅࢟૛ ࢟ ࢞ ࢟ ൅࢞ Now let’s compute ࢟൅࢞. This time, we get ቀ ૚ቁ൅ቀ ૚ቁൌቀ ૚ ૚ቁ. ࢟૛ ࢞૛ ࢟૛ ൅࢞૛ The commutative property guarantees that the two components of these vectors are equal. For example,

࢞૚ ൅࢟૚ ൌ࢟૚ ൅࢞૚. Since both components are equal, the vectors themselves must be equal. We can make a similar argument to show that vector addition in Թ૜ is commutative. ૞ ૜ Next, let’s see what all of this means geometrically. Let’s examine the points ࢞ൌቀ ቁ and ࢟ൌቀ ቁ. ૚ ૝

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૞ ૜ Thinking in terms of translations, the sum ࢞൅࢟ൌቀ ቁ൅ቀ ቁ amounts to moving ૞ right and ૚ up, followed ૚ ૝ by a movement that takes us ૜ right and ૝ up. ૜ ૞ On the other hand, the sum ࢟൅࢞ൌቀ ቁ൅ቀ ቁ amounts to moving ૜ right and ૝ up, followed by a ૝ ૚ movement that takes us ૞ right and ૚ up.

It should be clear that, in both cases, we’ve moved a total of ૡ units to the right and ૞ units up. So it makes sense that, when viewed as translations, these two sums are the same.

Lastly, let’s consider vector addition in Թ૜ from a geometric point of view.

૞ െ૜ Let’s consider the vectors ൭૝൱ and ൭ ૛ ൱. To show that addition is commutative, let’s imagine that Jack and ૛ ૜ Jill are moving around a building. We’ll send them on two different journeys and see if they reach the same destination. ૞ െ૜ Jack’s movements will model the sum ൭૝൱൅൭ ૛ ൱. He walks ૞ units east, ૝ units north, and then goes up ૛ ૛ ૜ flights of stairs. Next, he goes ૜ units west, ૛ units north, and then goes up ૜ flights of stairs. െ૜ ૞ Jill starts at the same place as Jack. Her movements will model the sum ൭ ૛ ൱൅൭૝൱. She walks ૜ units ૜ ૛ west, ૛ units north, and then goes up ૜ flights of stairs. Then, she goes ૞ units east, ૝ units north, and then climbs ૛ flights of stairs. It should be clear from this description that Jack and Jill both end up in a location that is ૛ units east, ૟ units north, and ૞ stories above their starting point. In particular, they end up in the same spot!

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b. Is vector addition associative? That is, does ሺ࢞൅࢟ሻ ൅࢘ൌ࢞൅ሺ࢟൅࢘ሻ for any three points in Թ૛? What

about points in Թ૜?

Let’s check to see if ሺ࢞൅࢟ሻ ൅࢘ൌ࢞൅ሺ࢟൅࢘ሻ for points in Թ૛.

࢞ ࢟ ࢘ ࢞ ൅࢟ ࢘ ሺ࢞൅࢟ሻ ൅࢘ൌቆቀ ૚ቁ൅ቀ ૚ቁቇ ൅ቀ ૚ቁൌቀ ૚ ૚ቁ൅ቀ ૚ቁ ࢞૛ ࢟૛ ࢘૛ ࢞૛ ൅࢟૛ ࢘૛

࢞ ࢟ ࢘ ࢞ ࢟ ൅࢘ ࢞൅ሺ࢟൅࢘ሻ ൌቀ ૚ቁ൅ቆቀ ૚ቁ൅ቀ ૚ቁቇ ൌቀ ૚ቁ൅ቀ ૚ ૚ቁ ࢞૛ ࢟૛ ࢘૛ ࢞૛ ࢟૛ ൅࢘૛

The associative property guarantees that each of the components is equal. Let’s look at the first coordinate.

For example, we have ሺ࢞૚ ൅࢟૚ሻ ൅࢘૚, which is indeed the same as ࢞૚ ൅ሺ࢟૚ ൅࢘૚ሻ. We can make a similar argument for vectors in Թ૜.

Next, let’s examine the problem from a geometric point of view.

These pictures make it clear that these two sums should be the same. In both cases, the overall journey is equivalent to following each of the three paths separately. Now let’s look at the three-dimensional case.

As in the two-dimensional case, we are simply reaching the same location in two different ways, both of which are equivalent to following the three individual paths separately.

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c. Does the distributive property apply to vector arithmetic? That is, does ࢑ήሺ࢞൅࢟ሻ ൌ࢑࢞൅࢑࢟ for each pair

of points in Թ૛? What about points in Թ૜?

We want to show that ࢑ήሺ࢞൅࢟ሻ ൌ࢑࢞൅࢑࢟.

࢞ ࢟ ࢞ ൅࢟ ࢑ήሺ࢞ ൅࢟ ሻ We have ࢑ήሺ࢞൅࢟ሻ ൌ࢑ήቆቀ ૚ቁ൅ቀ ૚ቁቇ ൌ࢑ήቀ ૚ ૚ቁൌ൬ ૚ ૚ ൰. ࢞૛ ࢟૛ ࢞૛ ൅࢟૛ ࢑ήሺ࢞૛ ൅࢟૛ሻ ࢞ ࢟ ࢑࢞ ࢑࢟ ࢑࢞ ൅࢑࢟ Next, we have ࢑࢞ ൅ ࢑࢟ ൌ ࢑ ቀ ૚ቁ൅࢑ቀ ૚ቁൌ൬ ૚൰൅൬ ૚൰ൌ൬ ૚ ૚൰. ࢞૛ ࢟૛ ࢑࢞૛ ࢑࢟૛ ࢑࢞૛ ൅࢑࢟૛ The distributive property guarantees that each of the components is equal. For example,

࢑ήሺ࢞૚ ൅࢟૚ሻ ൌ࢑࢞૚ ൅࢑࢟૚. This means that the vectors themselves are equal. A similar argument can be used to show that this property holds for vectors in Թ૜.

Now let’s examine this property geometrically.

Suppose that Jack walks to the point marked ࢞൅࢟ and then walks that distance again, ending at the spot marked ૛ሺ࢞ ൅ ࢟ሻ. Now suppose Jill walks to the spot marked ૛࢞ and then follows the path labeled ૛࢟. The picture makes it clear that Jack and Jill end up at the same spot.

Next, let’s examine the three-dimensional case. Again, the picture makes it clear that the distributive property holds.

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d. Is there an identity element for vector addition? That is, can you find a point ࢇ in Թ૛ such that ࢞൅ࢇൌ࢞ for

every point ࢞ in Թ૛? What about forԹ૜? ࢞ ࢇ ࢞ ൅ࢇ We have ࢞൅ࢇൌቀ ૚ቁ൅ቀ ૚ቁൌቀ ૚ ૚ቁ. ࢞૛ ࢇ૛ ࢞૛ ൅ࢇ૛

࢞૚ ൅ࢇ૚ ࢞૚ If this sum is equal to ࢞, then we have ቀ ቁൌቀ ቁ. This means that ࢞૚ ൅ࢇ૚ ൌ࢞૚, which implies that ࢞૛ ൅ࢇ૛ ࢞૛ ࢇ૚ ൌ૙. In the same way, we can show that ࢇ૛ ൌ૙.

૙ ૙ Thus, the identity element for Թ૛ is ቀ ቁ. Similarly, the identity element for Թ૜ is ቆ ቇ. ૙ ૙ ૙ ૛ ૙ Let’s think about this geometrically. ቀ ቁ൅ቀ ቁ means to go to ሺ૛ǡ ૞ሻ and translate by ሺ૙ǡ ૙ሻ. That is, we ૞ ૙ don’t translate at all.

e. Does each element in Թ૛ have an additive inverse? That is, if you take a point ࢇ in Թ૛, can you find a second point ࢈ such that ࢇ൅࢈ൌ૙?

ࢇ૚ ࢈ ࢇ ൅࢈ ࢇ ൅࢈ ૙ We have ࢇ൅࢈ൌቀ ቁ൅൬ ૚൰ൌ൬ ૚ ૚൰. If this sum is equal to ૙, then ൬ ૚ ૚൰ൌቀ ቁ, which means ࢇ૛ ࢈૛ ࢇ૛ ൅࢈૛ ࢇ૛ ൅࢈૛ ૙ that ࢇ૚ ൅࢈૚ ൌ૙. We can conclude that ࢈૚ ൌെࢇ૚. Similarly, ࢈૛ ൌെࢇ૛.

ࢇ૚ െࢇ૚ ૜ െ૜ Thus, the additive inverse of ቀ ቁ is ቀ ቁ. For instance, the additive inverse of ቀ ቁ is ቀ ቁ. Geometrically, ࢈૚ െ࢈૚ ૞ െ૞ we see that these vectors have the same length but point in opposite directions. Thus, their sum is ሺ૙ǡ ૙ሻ. This makes sense because if Jack walks toward a spot that is ૜ miles east and ૞ miles north of the origin and then walks toward the spot that is ૜ miles west and ૞ miles south of the origin, then he’ll end up right back at the origin!

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Closing (2 minutes) Students should write a brief response to the following questions in their notebooks. How can we represent linear transformations from Թଷ ՜Թଷ? à These transformations can be represented using ͵ൈ͵ matrices. Which properties of real numbers are true for vectors in Թଶ and Թଷ? à Vector addition is commutative and associative. There is a Ͳ vector, and every vector has an additive inverse. Scalar multiplication distributes over vector addition.

Exit Ticket (5 minutes)

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Name Date

Lesson 6: Linear Transformations as Matrices

Exit Ticket

ͳ Ͷ ͵ ʹGiven ݔൌቀ ቁ, ݕൌቀ ቁ, ݖൌቀ ቁ, and ݇ൌെ .1 ʹ ʹ ͳ .a. Verify the associative property holds: ݔ൅ሺݕ൅ݖሻ ൌ ሺݔ൅ݕሻ ൅ݖ

.b. Verify the distributive property holds: ݇ሺݔ൅ݕሻ ൌ݇ݔ൅݇ݕ

2. Describe the geometric effect of the transformation on the ͵ൈ͵ identity matrix given by the following matrices. ʹͲͲ െͳ Ͳ Ͳ a. ൥ͲʹͲ൩ b. ൥ ͲͳͲ൩ ͲͲʹ ͲͲͳ

ͳͲͲ ͳͲ Ͳ c. ൥ͲെͳͲ൩ d. ൥Ͳͳ Ͳ൩ ͲͲͳ ͲͲെͳ

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Exit Ticket Sample Solutions

૚ ૝ ૜ 1. Given ࢞ൌቀ ቁ, ࢟ൌቀ ቁ, ࢠൌቀ ቁ, and ࢑ൌെ૛ ૛ ૛ ૚ a. Verify the associative property holds: ࢞൅ሺ࢟൅ࢠሻ ൌ ሺ࢞൅࢟ሻ ൅ࢠ.

૚ ૝ ૜ ૚ ૠ ૡ ࢞൅ሺ࢟൅ࢠሻ ൌቀ ቁ൅ቆቀ ቁ൅ቀ ቁቇ ൌቀ ቁ൅ቀ ቁൌቀ ቁ ૛ ૛ ૚ ૛ ૜ ૞

૚ ૝ ૜ ૞ ૜ ૡ ሺ࢞൅࢟ሻ ൅ࢠൌቆቀ ቁ൅ቀ ቁቇ ൅ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁ ૛ ૛ ૚ ૝ ૚ ૞

b. Verify the distributive property holds: ࢑ሺ࢞൅࢟ሻ ൌ࢑࢞൅࢑࢟.

૚ ૝ ૞ െ૚૙ ࢑ሺ࢞൅࢟ሻ ൌെ૛ቆቀ ቁ൅ቀ ቁቇ ൌെ૛ቀ ቁൌቀ ቁ ૛ ૛ ૝ െૡ ૚ ૝ െ૛ െૡ െ૚૙ ࢑࢞ ൅ ࢑࢟ ൌ െ૛ ቀ ቁ൅ሺെ૛ሻቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁ ૛ ૛ െ૝ െ૝ െૡ

2. Describe the geometric effect of the transformation on the ૜ൈ૜ identity matrix given by the following matrices. ૛૙૙ a. ൥૙૛૙൩ ૙૙૛ It dilates the point by a factor of ૛.

െ૚ ૙ ૙ b. ൥ ૙૚૙൩ ૙૙૚ It reflects the point about the ࢟ࢠ-plane.

૚૙૙ c. ൥૙െ૚૙൩ ૙૙૚ It reflects the point about the ࢞ࢠ-plane.

૚૙ ૙ d. ൥૙૚ ૙൩ ૙૙െ૚ It reflects the point about the ࢞࢟-plane.

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Problem Set Sample Solutions

1. Show that the associative property, ࢞൅ሺ࢟൅ࢠሻ ൌ ሺ࢞൅࢟ሻ ൅ࢠ, holds for the following. ૜ െ૝ െ૚ a. ࢞ൌቀ ቁ, ࢟ൌቀ ቁ, ࢠൌቀ ቁ െ૛ ૛ ૞ ૜ െ૝ െ૚ ૜ െ૞ െ૛ ቀ ቁ൅ቆቀ ቁ൅ቀ ቁቇ ൌቀ ቁ൅ቀ ቁൌቀ ቁ െ૛ ૛ ૞ െ૛ ૠ ૞

૜ െ૝ െ૚ െ૚ െ૚ െ૛ ቆቀ ቁ൅ቀ ቁቇ ൅ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁ െ૛ ૛ ૞ ૙ ૞ ૞

૛ ૙ ૜ b. ࢞ൌ൭െ૛൱, ࢟ൌ൭૞ ൱, ࢠൌ൭૙ ൱ ૚ െ૛ െ૜ ૛ ૙ ૜ ૛ ૜ ૞ ൭െ૛൱൅ቌ൭ ૞ ൱൅൭ ૙ ൱ቍ ൌ൭െ૛൱൅൭ ૞ ൱ൌ൭૜ ൱ ૚ െ૛ െ૜ ૚ െ૞ െ૝

૛ ૙ ૜ ૛ ૜ ૞ ቌ൭െ૛൱൅൭ ૞ ൱ቍ ൅൭ ૙ ൱ൌ൭૜ ൱൅൭ ૙ ൱ൌ൭૜ ൱ ૚ െ૛ െ૜ െ૚ െ૜ െ૝

2. Show that the distributive property, ࢑ሺ࢞൅࢟ሻ ൌ࢑࢞൅࢑࢟, holds for the following. ૞ െ૛ a. ࢞ൌቀ ቁ, ࢟ൌቀ ቁ, ࢑ൌെ૛ െ૜ ૝ ૞ െ૛ ૜ െ૟ െ૛ ቆቀ ቁ൅ቀ ቁቇ ൌെ૛ቀ ቁൌቀ ቁ െ૜ ૝ ૚ െ૛ ૞ െ૛ െ૚૙ ૝ െ૟ െ૛ ቀ ቁ൅ሺെ૛ሻ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁ െ૜ ૝ ૟ െૡ െ૛

૜ െ૝ b. ࢞ൌ൭െ૛൱, ࢟ൌ൭૟ ൱, ࢑ൌെ૜ ૞ െૠ ૜ െ૝ െ૚ ૜ െ૜ ቌ൭െ૛൱൅൭ ૟ ൱ቍ ൌെ૜൭૝ ൱ൌ൭െ૚૛൱ ૞ െૠ െ૛ ૟ ૜ െ૝ െૢ ૚૛ ૜ െ૜ ൭െ૛൱൅ሺെ૜ሻ ൭ ૟ ൱ൌ൭ ૟ ൱൅൭െ૚ૡ൱ൌ൭െ૚૛൱ ૞ െૠ െ૚૞ ૛૚ ૟

3. Compute the following. ૚૙૛ ૛ a. ൭૙૚૛൱൭૚൱ ૛૛૜ ૜ ૡ ൭ ૠ ൱ ૚૞

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െ૚ ૛ ૜ ૚ b. ൭ ૜૚െ૛൱൭െ૛൱ ૚െ૛૜ ૚ െ૛ ൭െ૚൱ ૡ

૚૙૛ ૛ c. ൭૙૚૙൱൭૚൱ ૛૙૜ ૛ ૟ ൭ ૚ ൱ ૚૙

૜ ࢇࢊࢍ 4. Let ࢞ൌ൭૚൱. Compute ࡸሺ࢞ሻ ൌ ൥࢈ࢋࢎ൩ ή࢞, plot the points, and describe the geometric effect to ࢞. ૛ ࢉࢌ࢏ െ૚ ૙ ૙ a. ൥ ૙૚૙൩ ૙૙૚ െ૜ ൭ ૚ ൱ It is reflected about the ࢟ࢠ-plane. ૛

૛૙૙ b. ൥૙૛૙൩ ૙૙૛ ૟ ൭૛൱ It is dilated by a factor of ૛. ૝

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૙૚૙ c. ൥૚૙૙൩ ૙૙૚ ૚ ൭૜൱ It is reflected about the vertical plane through ૛ the line ࢟ൌ࢞ on the ࢞࢟-plane.

૚૙૙ d. ൥૙૙૚൩ ૙૚૙ ૜ ൭૛൱ It is reflected about the vertical plane through the line ૚ ࢟ൌࢠ on the ࢠ࢟-plane.

૜ ࢇࢊࢍ 5. Let ࢞ൌ൭૚൱. Compute ࡸሺ࢞ሻ ൌ ൥࢈ࢋࢎ൩ ή࢞. Describe the geometric effect to ࢞. ૛ ࢉࢌ࢏ ૙૙૙ a. ൥૙૙૙൩ ૙૙૙ ૙ ൭૙൱ It is mapped to the origin. ૙

૜૙૙ b. ൥૙૜૙൩ ૙૙૜ ૢ ൭૜൱ It is dilated by a factor of ૜. ૟

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െ૚ ૙ ૙ c. ൥ ૙െ૚૙൩ ૙૙െ૚ െ૜ ൭െ૚൱ It is mapped to the opposite side of the origin on the same line that is equal distance from the origin. െ૛

െ૚ ૙ ૙ d. ൥ ૙૚૙൩ ૙૙૚ െ૜ ൭ ૚ ൱ It is reflected about the ࢟ࢠ-plane. ૛

૚૙૙ e. ൥૙െ૚૙൩ ૙૙૚ ૜ ൭െ૚൱ It is reflected about the ࢞ࢠ-plane. ૛

૚૙ ૙ f. ൥૙૚ ૙൩ ૙૙െ૚ ૜ ൭ ૚ ൱ It is reflected about the ࢞࢟-plane. െ૛

૙૚૙ g. ൥૚૙૙൩ ૙૙૚ ૚ ൭૜൱ It is reflected about the vertical plane through the line ࢟ൌ࢞ on the ࢞࢟-plane. ૛

૚૙૙ h. ൥૙૙૚൩ ૙૚૙ ૜ ൭૛൱ It is reflected about the vertical plane through the line ࢟ൌࢠ on the ࢟ࢠ-plane. ૚

૙૙૚ i. ൥૙૚૙൩ ૚૙૙ ૛ ൭૚൱ It is reflected about the vertical plane through the line ࢞ൌࢠ on the ࢞ࢠ-plane. ૜

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૚ 6. Find the matrix that will transform the point ࢞ൌ൭૜൱ to the following point: ૛ െ૝ a. ൭െ૚૛൱ െૡ െ૝ ૙ ૙ ൥ ૙െ૝૙൩ ૙૙െ૝

૜ b. ൭૚൱ ૛ ૙૚૙ ൥૚૙૙൩ ૙૙૚

૛ 7. Find the matrix/matrices that will transform the point ܠൌ൭૜൱ to the following point: ૚ ૟ െ૚ a. ܠᇱ ൌ൭૝൱ b. ܠᇱ ൌ൭૜ ൱ ૛ ૛

૙૛૙ ૙૙െ૚ ൥૛૙૙൩ ൥૙૚ ૙൩ ૙૙૛ ૚૙ ૙

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Lesson 7: Linear Transformations Applied to Cubes

Student Outcomes ݔ ݔ ቈݕ቉ has a desired geometric ܣ ቆቈݕ቉ቇ ൌܮ so that the linear transformation ܣ Students construct ͵ൈ͵ matrices ݖ ݖ effect. ݔ ݔ ቈݕ቉ based on the structure of ܣ ቆቈݕ቉ቇ ൌܮ Students identify the geometric effect of the linear transformation ݖ ݖ the ͵ൈ͵ matrix ܣ.

Lesson Notes In this lesson, students examine the geometric effects of linear transformations in Թଷ induced by various ͵ൈ͵ matrices on the unit cube. This lesson extends work done in Module 1 in which students studied analogous transformations in Թଶ using ʹൈʹ matrices. This lesson is written for classes that have access to the GeoGebra demo TransformCubes (http://eureka-math.org/G12M2L7/geogebra-TransformCubes). This GeoGebra demo allows students to input values in a ͵ൈ͵ matrix between െͷ and ͷ in increments of ͲǤͳand visually see the effect of the transformation induced by the matrix on the unit cube. If there are not enough computers for students to access the demo in pairs or small groups, then the lesson needs to be modified accordingly. If the demo is not available, students can come to the same conclusion by plotting points in three-dimensional space. In this case, consider having students plot points, and then show visuals from the teacher pages of actual screenshots from the demo to help students visualize the transformations. In addition to the tasks included in the lesson, students are encouraged to explore and play with the demo file to discover the connections between the structure of a ͵ൈ͵ matrix and the geometric effect of the transformation induced by the matrix.

Classwork Opening Exercise (10 minutes) The Opening Exercise allows students to review the geometric effects of linear transformations in the plane induced by various ʹൈʹmatrices. Students then extend this idea to transformations in space induced by ͵ൈ͵ matrices.

Opening Exercise ૚૛ ૛૙ ૛െ૛ Consider the following matrices: ࡭ൌቂ ቃ, ࡮ൌቂ ቃ, and ࡯ൌቂ ቃ ૛૝ ૙૙ ૛૛ a. Compute the following determinants. i. ܌܍ܜሺ࡭ሻ ܌܍ܜሺ࡭ሻ ൌ૝െ૝ൌ૙

ii. ܌܍ܜሺ࡮ሻ ܌܍ܜሺ࡮ሻ ൌ૙െ૙ൌ૙

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iii. ܌܍ܜሺ࡯ሻ ܌܍ܜሺ࡯ሻ ൌ૝െሺെ૝ሻൌૡ

b. Sketch the image of the unit square after being transformed by each transformation. ࢞ ૚૛࢞ i. ࡸ ቀቂ ቃቁ ൌ ቂ ቃቂ ቃ ࡭ ࢟ ૛૝࢟

࢞ ૛૙࢞ ii. ࡸ ቀቂ ቃቁ ൌ ቂ ቃቂ ቃ ࡮ ࢟ ૙૙࢟

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࢞ ૛െ૛࢞ iii. ࡸ ቀቂ ቃቁ ൌ ቂ ቃቂ ቃ ࡯ ࢟ ૛૛࢟

Scaffolding: Pair students so that students who can easily c. Find the area of each image of the unit square in Part b. see the transformations

The first two have no area because each image is a line segment. The third one is a are paired with students ૛ square with sides of length ૛ξ૛, so its area is ൫૛ξ૛൯ ൌૡ. who struggle. Give students with spatial

difficulties pictures of the d. Explain the connection between the responses to Parts a and b. transformed images, and The determinant of the matrix ࡭ is the area of the image of the unit square under the have them label the transformation induced by ࡭. transformed points. Have advanced learners Discussion (2 minutes) create transformations and the accompanying Use this Discussion to have students share the connections they made in part (d) of the matrix on their own and Opening Exercise. present different ݔ ݔ transformations to the ቂ ቃ when ڄ ܣ ቀቂ ቃቁ ൌܮ What happens to the unit square under a transformation .ݕ ݕ class the matrix ܣhas a determinant of Ͳ? à The square is transformed into a line segment. Does such a transformation have an inverse? à No, there is no way to undo this transformation because we don’t know where it came from. In the same way, we want to explore the geometric effect of a linear transformation induced by matrix multiplication on the unit cube.

Example (5 minutes)

If the program is not available, have students plot the original image and the transformation in three dimensions and color-code them. Also, show the images in the teacher materials below to help students visualize the transformations at first, and then have them draw different transformations.

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ݔ ͳͲ Ͳ ݔ ቈݕ቉ will be on theڄቆቈݕ቉ቇ ൌ ൥ͲെͳͲ൩ܮ What do we expect the geometric effect of the transformation ݖ ͲͲെͳ ݖ unit cube? One way to find out would be to transform the vertices of the cube. If we do that, we find the following transformed points.

ሺͲǡͲǡͲሻ ՜ ሺͲǡͲǡͲሻ ሺͳǡͳǡͲሻ ՜ሺͳǡെͳǡͲሻ ሺͳǡͲǡͲሻ ՜ ሺͳǡͲǡͲሻ ሺͳǡͲǡͳሻ ՜ሺെͳǡͲǡെͳሻ ሺͲǡͳǡͲሻ ՜ሺͲǡെͳǡͲሻ ሺͲǡͳǡͳሻ ՜ሺͲǡെͳǡെͳሻ ሺͲǡͲǡͳሻ ՜ሺͲǡͲǡെͳሻ ሺͳǡͳǡͳሻ ՜ ሺͳǡ െͳǡ െͳሻ

It can be hard to truly see the image when plotting these points in three dimensions. Instead, let’s use GeoGebra. Project the GeoGebra demo TransformCubes.ggb so that all students can see it. Use the sliders to set ܽൌͳ, ݁ൌെͳ, ݅ൌെͳ, and the remaining entries in the matrix to Ͳ as shown. The image above is an exact copy of the file screen that students and teachers can use to make sure they are using the correct settings. Using the software, we can see the blue cube before the transformation using matrix ܣ, and the green figure is the image of the original cube after the transformation using matrix ܣ. What appears to be the geometric effect of multiplication by the matrix ܣ? .à The green cube is the result of rotating ͳͺͲι about the ݔ-axis

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It may not be obvious that the green cube is the result of rotating and not from translating, but remind students that a Ͳ Ͳ linear transformation never changes the origin. In this case, ܮ൭൥Ͳ൩൱ ൌ ൥Ͳ൩. Knowing that the origin does not change Ͳ Ͳ ݔ ͳͲ Ͳ ݔ ቈݕ቉ has the geometric effect of rotationڄቆቈݕ቉ቇ ൌ ൥ͲെͳͲ൩ܮ makes it more obvious that the transformation ݖ ͲͲെͳ ݖ .about the ݔ-axis

Exploratory Challenge 1 (10 minutes) For this Exploratory Challenge, students use the GeoGebra demo TransformCubes.ggb to explore the geometric effects of a linear transformation defined by matrix multiplication for different matrices ܣ. Screenshots of the GeoGebra file are shown in the steps below so students can see transformations and matrices. If possible, allow students to access the software themselves. If there is no way for students to access the software themselves, have them plot the transformed vertices of the unit cube, color coding the original image and its transformation, and draw conclusions from there. Students plotted points in three dimensions in the previous lessons. Students may need a quick reminder that if a transformation has an inverse, then each point in the image came from exactly one point on the cube so that the function can be undone in a well-defined way.

Exploratory Challenge 1

For each matrix ࡭ given below:

i. Plot the image of the unit cube under the transformation. ii. Find the volume of the image of the unit cube from part (i). iii. Does the transformation have an inverse? If so, what is the matrix that induces the inverse transformation? ૛૙૙ a. ࡭ൌ൥૙૛૙൩ ૙૙૛

ii. The image of the unit cube is a cube with sides of length ૛. Thus, the volume of the image of the unit .૛ൌૡڄ૛ڄcube is ૛ ૚ iii. This transformation has an inverse. To undo dilation by ૛, we need to dilate by . The matrix that ૛ ૚ ૙૙ ې ૛ۍ ۑ ૚ ێ represents the inverse transformation is ࡮ൌ ૙ ૙ . ۑ ૛ ێ ۑ૚ ێ ૙૙ ے૛ ۏ

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૝૙ ૙ b. ࡭ൌ൥૙૚ ૙൩ ૙૙െ૛ i.

ii. The image of the unit cube is a box with sides of length ૝, ૚, and ૛. Thus, the volume of the image of .૛ൌૡڄ૚ڄthe unit cube is ૝ ૚ iii. This transformation is also invertible. To undo this transformation, we need to scale by in the ૝ ૚ ࢞ ૚ ࢟ െ ࢠ -direction, by in the -direction, and by ૛ in the -direction. The matrix that will induce this ૚ ૙૙ ې ૝ۍ .ۑ૙૚ ૙ێinverse transformation is ࡮ൌ ۑ૚ ێ ૙૙െ ے૛ ۏ

૝૙ ૙ c. ࡭ൌ൥૙૙ ૙൩ ૙૙െ૛ i.

ii. The image of the unit cube is a rectangle with sides of length ૛ and ૝. Thus, the volume of the image of .૛ൌ૙ڄ૙ڄthe unit cube is ૝ iii. This transformation is not invertible since the image was collapsed from a cube onto a rectangle. Many points get sent to the same point on the rectangle, so we cannot undo the transformation in a clear way.

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࢞ ࢇ૙૙ ࢞ .ቈ࢟቉ for numbers ࢇ, ࢈, and ࢉ ڄ d. Describe the geometric effect of a transformation ࡸ ቆቈ࢟቉ቇ ൌ ൥૙࢈૙൩ ࢠ ૙૙ࢉ ࢠ Describe when such a transformation is invertible.

The transformation ࡸ scales the side of the cube along the ࢞-axis by ࢇ, the side of the cube along the ࢟-axis by ࢈, and the side of the cube along the ࢠ-axis by ࢉ. If any of the numbers ࢇ, ࢈, and ࢉ are zero, then the image is no longer a three-dimensional figure, so the transformation will not be invertible. Thus, the transformation is invertible only when ࢇ്૙, ࢈്૙, and ࢉ്૙.

Exploratory Challenge 2 (10 minutes) For this challenge, students continue to use the GeoGebra demo TransformCubes.ggb to explore the geometric effects of a linear transformation defined by matrix multiplication. A screenshot of the file is shown below. In this case, students are asked to discover how matrix multiplication can produce rotation about an axis. Encourage students to play with the software and explore how different geometric effects arise from different matrix structures.

Exploratory Challenge 2

a. Make a prediction: What would be the geometric effect of the transformation ࢞ ૚૙ ૙ ࢞ ቈ࢟቉ on the unit cube? Use the GeoGebra demo to test your ڄ ሺ૝૞ιሻ൩ܖܑܛሺ૝૞ιሻെܛܗ܋ࡸ ቆቈ࢟቉ቇ ൌ ൥૙ ࢠ ૙ ܛܑܖሺ૝૞ιሻ ܋ܗܛሺ૝૞ιሻ ࢠ conjecture.

Students should recognize that this transformation should have something to do with rotation. However, they most likely need to see the resulting image before identifying it as rotation by ૝૞ι about the ࢞-axis.

࢞ ࢞ ቈ࢟቉ isڄb. For each geometric transformation below, find a matrix ࡭ so that the geometric effect of ࡸ ቆቈ࢟቉ቇ ൌ࡭ ࢠ ࢠ the specified transformation. i. Rotation by െ૝૞ι about the ࢞-axis ૚૙ ૙ ې ۍ ૚૙ ૙ ξ૛ ξ૛ ۑ ૙ێ ۑ ૛ ૛ ێ ሺെ૝૞ιሻ൩ ൌܖܑܛሺെ૝૞ιሻ െܛܗ܋࡭ൌ൥૙ ۑξ૛ ξ૛ ێ ሺെ૝૞ιሻܛܗ܋ ሺെ૝૞ιሻܖܑܛ૙ ૙െ ے ૛ ૛ ۏ

ii. Rotation by ૝૞ι about the ࢟-axis ξ૛ ξ૛ ې ૙െ ۍ ۑ ૛ ૛ ێ ሺ૝૞ιሻܖܑܛሺ૝૞ιሻ ૙ െܛܗ܋ ۑ૙૚ ૙ ێ ࡭ൌ൥ ૙૚ ૙൩ ൌ ۑ ξ૛ ξ૛ێ ሺ૝૞ιሻܛܗ܋ሺ૝૞ιሻ ૙ܖܑܛ ૙ ے ૛ ૛ ۏ

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iii. Rotation by ૝૞ι about the ࢠ-axis ξ૛ ξ૛ ېെ ૙ ۍ ۑ ૛ ૛ ێ ሺ૝૞ιሻ ૙ܖܑܛሺ૝૞ιሻ െܛܗ܋ ۑ ξ૛ ξ૛ێ ሺ૝૞ιሻ ૙൩ ൌܛܗ܋ ሺ૝૞ιሻܖܑܛൌ൥ۯ ૙ ۑ ૛ ૛ ێ૙૙૚ ے૙૙૚ ۏ

iv. Rotation by ૢ૙ι about the ࢞-axis ૚૙ ૙ ૚૙ ૙ ࡭ൌ൥૙܋ܗܛሺૢ૙ιሻ െܛܑܖሺૢ૙ιሻ൩ ൌ ൥૙૙െ૚൩ ૙ܛܑܖሺૢ૙ιሻ ܋ܗܛሺૢ૙ιሻ ૙૚ ૙

v. Rotation by ૢ૙ι about the ࢟-axis ܋ܗܛሺૢ૙ιሻ ૙ െܛܑܖሺૢ૙ιሻ ૙૙െ૚ ࡭ൌ൥ ૙૚ ૙൩ ൌ ൥૙૚ ૙൩ ܛܑܖሺૢ૙ιሻ ૙܋ܗܛሺૢ૙ιሻ ૚૙ ૙

vi. Rotation by ૢ૙ι about the ࢠ-axis ܋ܗܛሺૢ૙ιሻ െܛܑܖሺૢ૙ιሻ ૙ ૙െ૚૙ ࡭ൌ൥ܛܑܖሺૢ૙ιሻ ܋ܗܛሺૢ૙ιሻ ૙൩ ൌ ൥૚૙૙൩ ૙૙૚૙૙૚

vii. Rotation by ࣂ about the ࢞-axis ૚૙ ૙ ࡭ൌ൥૙܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ൩ ૙ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ

viii. Rotation by ࣂ about the ࢟-axis ܋ܗܛሺࣂሻ ૙ െܛܑܖሺࣂሻ ࡭ൌ൥ ૙૚ ૙൩ ܛܑܖሺࣂሻ ૙܋ܗܛሺࣂሻ

ix. Rotation by ࣂ about the ࢠ-axis ܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ ૙ ࡭ൌ൥ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ૙൩ ૙૙૚

Exploratory Challenge 3 (Optional) This challenge is for students who have completed the other challenges quickly. Allow students to play with the ݔ ݔ ቈݕ቉ for the following matrices ڄ ܣ ቆቈݕ቉ቇ ൌܮ software and try to describe the geometric transformations that arise from ݖ ݖ ܣ. Exact pictures of file screens are shown below to help students and teachers.

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Exploratory Challenge 3 ࢞ ࢞ ቈ࢟቉ for the given matrices ࡭. Be as specific as youڄDescribe the geometric effect of each transformation ࡸ ቆቈ࢟቉ቇ ൌ࡭ ࢠ ࢠ can. ૚૙૙ a. ࡭ൌ൥૙૙૚൩ ૙૚૙

The images of the two cubes coincide; this transformation reflects across the plane through the points ሺ૚ǡ ૙ǡ ૙ሻ, ሺ૚ǡ ૚ǡ ૚ሻ, ሺ૙ǡ ૚ǡ ૚ሻ, and ሺ૙ǡ ૙ǡ ૙ሻ.

૛૚૙ b. ࡭ൌ൥૛૚૙൩ ૛૚૙

This transformation collapses the cube onto a line segment from ሺ૙ǡ ૙ǡ ૙ሻ to ሺ૜ǡ ૜ǡ ૜ሻ.

૛૛૙ c. ࡭ൌ൥െ૛ ૛ ૙൩ ૙૙૚

This transformation stretches the cube in the ࢞ and ࢟ directions and then rotates by െ૝૞ι.

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Closing (3 minutes) Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements are listed below.

Lesson Summary ࢞ ࢞ ቈ࢟቉ is a function from points in space to points inڄFor a matrix ࡭, the transformation ࡸ ቆቈ࢟቉ቇ ൌ࡭ ࢠ ࢠ space. Different matrices induce transformations such as rotation, dilation, and reflection. ࢇ૙૙ The transformation induced by a diagonal matrix ࡭ൌ൥૙࢈૙൩ scales by ࢇ in the direction ૙૙ࢉ

parallel to the ࢞-axis, by ࢈ in the direction parallel to the ࢟-axis, and by ࢉ in the direction parallel to the ࢠ-axis. ૚૙ ૙ ܋ܗܛሺࣂሻ ૙െܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ ૙ The matrices ൥૙܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ൩, ൥ ૙૚ ૙൩, and ൥ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ૙൩ ૙ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ܛܑܖሺࣂሻ ૙܋ܗܛሺࣂሻ ૙૙૚

induce rotation by ࣂ about the ࢞, ࢟, and ࢠ axes, respectively.

Exit Ticket (5 minutes)

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Name Date

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Exit Ticket

ݔ ͵ͲͲ ݔ .ቈݕ቉ on the axes providedڄቆቈݕ቉ቇ ൌ ൥ͲെʹͲ൩ܮ Sketch the image of the unit cube under the transformation .1 ݖ ͲͲͲ ݖ

2. Does the transformation from Problem 1 have an inverse? Explain how you know.

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Exit Ticket Sample Solutions

࢞ ૜૙૙ ࢞ .ቈ࢟቉ on the axes provided ڄ Sketch the image of the unit cube under the transformation ࡸ ቆቈ࢟቉ቇ ൌ ൥૙െ૛૙൩ .1 ࢠ ૙૙૙ ࢠ

2. Does the transformation from Problem 1 have an inverse? Explain how you know.

No, this transformation does not have an inverse. The entry in the matrix in the lower right corner is ૙, leading to a collapse of the image of the cube onto a rectangle. Since multiple points on the cube were transformed to the same point on the rectangle, there is no way to undo this transformation.

Problem Set Sample Solutions Problems 1 and 2 continue to develop the theory of linear transformations represented by matrix multiplication. Problem 3 can be done with or without access to the GeoGebra demo used in the lesson.

࢞ ࢇࢊࢍ ࢞ ࢇࢊࢍ .ቈ࢟቉, for some matrix ࡭ൌ൥࢈ࢋࢎ൩ ڄ Suppose that we have a linear transformation ቆቈ࢟቉ቇ ൌ ൥࢈ࢋࢎ൩ .1 ࢠ ࢉࢌ࢏ ࢠ ࢉࢌ࢏ ૚ a. Evaluate ࡸ൭൥૙൩൱. How does the result relate to the matrix ࡭? ૙ ૚ ࢇ ࡸ൭൥૙൩൱ൌቈ࢈቉ It is the first column of matrix ࡭. ૙ ࢉ

૙ b. Evaluate ࡸ൭൥૚൩൱. How does the result relate to the matrix ࡭? ૙ ૙ ࢊ ࡸ൭൥૚൩൱ൌ൥ࢋ൩ It is the second column of matrix ࡭. ૙ ࢌ

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૙ c. Evaluate ࡸ൭൥૙൩൱. How does the result relate to the matrix ࡭? ૚ ૙ ࢍ ࡸ൭൥૙൩൱ൌቈࢎ቉ It is the third column of matrix ࡭. ૚ ࢏

d. James correctly said that if you know what a linear transformation does to the three points ሺ૚ǡ ૙ǡ ૙ሻ, ሺ૙ǡ ૚ǡ ૙ሻ, and ሺ૙ǡ ૙ǡ ૚ሻ, you can find the matrix of the transformation. Explain how you can find the matrix of the transformation given the image of these three points. ૚ ૙ ૙ Create a matrix with ࡸ൭൥૙൩൱ in the first column, ࡸ൭൥૚൩൱ in the second column, and ࡸ൭൥૚൩൱ in the third ૙ ૙ ૙ column.

2. Use the result from Problem 1(d) to answer the following questions. ૚ ૜ ૙ ૙ ૙ ૙ a. Suppose a transformation ࡸǣ Թ૜ ՜Թ૜ satisfies ࡸ൭൥૙൩൱ൌ൥૙൩, ࡸ൭൥૚൩൱ൌ൥૜൩, and ࡸ൭൥૙൩൱ൌ൥૙൩. ૙ ૙ ૙ ૙ ૚ ૜ i. What is the matrix ࡭ that represents the transformation ࡸ? ૜૙૙ ࡭ൌ൥૙૜૙൩ ૙૙૜

ii. What is the geometric effect of the transformation ࡸ?

The transformation ࡸ has the geometric effect of dilation by a factor of ૜.

iii. Sketch the image of the unit cube after the transformation by ࡸ.

૚ ૚ ૙ ૙ ૙ ૙ b. Suppose a transformation ࡸǣ Թ૜ ՜Թ૜ satisfies ࡸ൭൥૙൩൱ൌ൥૙൩,ࡸ ൭൥૚൩൱ൌ൥૚൩, and ࡸ൭൥૙൩൱ൌ൥ ૙ ൩. ૙ ૙ ૙ ૙ ૚ െ૝ i. What is the matrix ࡭ that represents the transformation ࡸ? ૚૙ ૙ ࡭ൌ൥૙૚ ૙൩ ૙૙െ૝

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ii. What is the geometric effect of the transformation ࡸ?

The transformation ࡸ has the geometric effect of stretching by a factor of െ૝ in the ࢠ-direction.

iii. Sketch the image of the unit cube after the transformation by ࡸ.

૚ െ૛ ૙ ૙ ૙ ૙ c. Suppose a transformation ࡸǣ Թ૜ ՜Թ૜ satisfies ࡸ൭൥૙൩൱ൌ൥ ૙ ൩,ࡸ ൭൥૚൩൱ൌ൥૚൩, and ࡸ൭൥૙൩൱ൌ൥ ૙ ൩. ૙ ૙ ૙ ૙ ૚ െ૛ i. What is the matrix ࡭ that represents the transformation ࡸ? െ૛ ૙ ૙ ࡭ൌ൥ ૙૚૙൩ ૙૙െ૛

ii. What is the geometric effect of the transformation ࡸ?

The transformation ࡸ has the geometric effect of rotating about the ࢟-axis by ૚ૡ૙ι and stretching by a factor of ૛ in the ࢞ and ࢠ directions.

iii. Sketch the image of the unit cube after transformation by ࡸ.

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3. Find the matrix of the transformation that will produce the following images of the unit cube. Describe the geometric effect of the transformation.

The geometric effect of this transformation is reflection across the ࢞࢟-plane. The matrix that represents ૚૙ ૙ this transformation is ࡭ൌ൥૙૚ ૙൩. ૙૙െ૚

The geometric effect of this transformation is a stretch in the ࢟-direction by a factor of ૜. The matrix that ૚૙૙ represents this transformation is ࡭ൌ൥૙૜૙൩. ૙૙૚

The geometric effect of this transformation is a stretch in the ࢞- and ࢠ-directions by a factor of ૛ and ૛૙૙ projection onto the ࢞ࢠ-plane. The matrix that represents this transformation is ࡭ൌ൥૙૙૙൩. ૙૙૛

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The geometric effect of this transformation is a shear parallel to the ࢞ࢠ-plane. The matrix that represents ૚૙૙ this transformation is ࡭ൌ൥૙૚૙൩. ૙૚૚

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Lesson 8: Composition of Linear Transformations

Student Outcomes Students compose two linear transformations in the plane by multiplying the associated matrices. Students visualize composition of linear transformations in the plane.

Lesson Notes In this lesson, students examine the geometric effects of performing a sequence of two linear transformations in Թଶ produced by various ʹൈʹ matrices on the unit square. The GeoGebra demo TransformCube (http://eurmath.link/ g12m2/transformcube) may be used in this lesson but is not necessary. This GeoGebra demo allows students to input values in ʹൈʹ matrices ܣ between െͷ and ͷ in increments of ͲǤͳand visually see the effect of the transformation produced by the matrix ܣ on the unit square. If there are not enough computers for students to access the demos in pairs or small groups, modify the lesson. Consider having students work in groups on transformations. Then, as parts of the Exploratory Challenge are finished, show specific transformations to the entire class using the teacher computer. Another option would be to have different groups come up and show the transformation that they created by hand and then use the teacher computer to show that transformation using software.

Classwork Opening (3 minutes) In this lesson, the transformation that arises from doing two linear transformations in sequence is considered. Display each transformation matrix below on the board, and ask students what transformation the matrix represents. ʹͲ ቂ ቃ A dilation with a scale factor of ʹ Ͳʹ Ͳͳ ቂ ቃ A reflection across the line ݕൌݔ ͳͲ ʹെͳ ʹ ቂ ቃ A rotation about the origin by ߠൌିଵ ቀ ቁ and a dilation with a scale factor of ξʹଶ ൅ͳଶ ൌ ξͷ ͳʹ ͳ How could we represent a composition of two of these transformations?

Opening Exercise (5 minutes) The focus of this lesson is on students recognizing that a matrix produced by a composition of two linear transformations in the plane is the product of the matrices of each transformation. Thus, students need to review the process of multiplying ʹൈʹ matrices. The products used in this Opening Exercise appear later in the lesson.

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Opening Exercise

Compute the product ࡭࡮ for the following pairs of matrices. ૙െ૚ ૙െ૚ a. ࡭ൌቂ ቃ, ࡮ൌቂ ቃ ૚૙ ૚૙ െ૚ ૙ ࡭࡮ ൌ ቂ ቃ ૙െ૚

૚૙ െ૚ ૙ b. ࡭ൌቂ ቃ, ࡮ൌቂ ቃ ૙െ૚ ૙૚ െ૚ ૙ ࡭࡮ ൌ ቂ ቃ ૙െ૚

ඥ૛ ඥ૛ െ ૝૙ c. ࡭ൌ൦૛ ૛ ൪, ࡮ൌቂ ቃ ඥ૛ ඥ૛ ૙૝ ૛ ૛ ૛ ૛ െ૛ ૛ ࡭࡮ ൌ ൤ ξ ξ ൨ ૛ξ૛ ૛ξ૛

૚૚ ૛૙ d. ࡭ൌቂ ቃ, ࡮ൌቂ ቃ ૙૚ ૙૛ ૛૛ ࡭࡮ ൌ ቂ ቃ ૙૛

૚ ඥ૜ ඥ૜ ૚ െ e. ࡭ൌ൦ ૛ ૛ ൪, ࡮ൌ൦૛ ૛൪ ඥ૜ ૚ ૚ ඥ૜ െ ૛ ૛ ૛ ૛ ඥ૜ ૚ Scaffolding: ࡭࡮ ൌ ൦ ૛ ૛ ൪ െ ૚ ඥ૜ Provide struggling ૛ ૛ students with colored pencils or markers to use to identify the three Exploratory Challenge (25 minutes) figures. For example, use For this challenge, students use the GeoGebra demo TransformSquare.ggb to transform a blue for the original unit square in the plane in order to explore the geometric effects of performing a square, green for the sequence of two linear transformations defined by matrix multiplication for different square transformed by ܮ஻, matrices ܣ and ܤ. While the software illustrates a single transformation via matrix and red for the square

multiplication, students have to reason through the effects of doing two transformations transformed by ܮ஻ and in sequence. If possible, allow students to access the software themselves. If there is no then ܮ஺. way for students to access the software themselves, have them plot the transformed Ask early finishers to figure vertices of the unit square and draw conclusions from there. out whether or not the geometric effect of performing first ܮ஺ and then ܮ஻ is the same as performing first ܮ஻ and then ܮ஺.

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Exploratory Challenge

a. For each pair of matrices ࡭ and ࡮ given below: ࢞ ࢞ .ቂ࢟ቃ ڄ Describe the geometric effect of the transformation ࡸ࡮ ቀቂ࢟ቃቁ ൌ ࡮ .1 ࢞ ࢞ .ቂ࢟ቃ ڄ Describe the geometric effect of the transformation ࡸ࡭ ቀቂ࢟ቃቁ ൌ ࡭ .2 ࢞ ࢞ .ቂ࢟ቃ ڄ Draw the image of the unit square under the transformation ࡸ࡮ ቀቂ࢟ቃቁ ൌ ࡮ .3 ࢞ ࢞ .ቂ࢟ቃ ڄ Draw the image of the transformed square under the transformation ࡸ࡭ ቀቂ࢟ቃቁ ൌ ࡭ .4

5. Describe the geometric effect on the unit square of performing first ࡸ࡮ and then ࡸ࡭. 6. Compute the matrix product ࡭࡮. ࢞ ࢞ .ቂ࢟ቃ ڄ Describe the geometric effect of the transformation ࡸ࡭࡮ ቀቂ࢟ቃቁ ൌ ࡭࡮ .7

૙െ૚ ૙െ૚ i. ࡭ൌቂ ቃ, ࡮ൌቂ ቃ ૚૙ ૚૙ 1. The transformation produced by matrix ࡮has the effect of rotation by ૢ૙ι counterclockwise.

2. The transformation produced by matrix ࡭ has the effect of rotation by ૢ૙ι counterclockwise.

3. The green square in the second quadrant is the image of the original square under the transformation produced by ࡮.

4. The red square in the third quadrant is the image of the square after transforming with matrix ࡭and then matrix ࡮.

5. If we rotate twice by ૢ૙ι, then the net effect is a rotation by ૚ૡ૙ι. െ૚ ૙ 6. The matrix product is ࡭࡮ ൌ ቂ ቃ. ૙െ૚ 7. The transformation produced by matrix ࡭࡮ has the effect of rotation by ૚ૡ૙ι.

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૚૙ െ૚ ૙ ii. ࡭ൌቂ ቃ, ࡮ൌቂ ቃ ૙െ૚ ૙૚ 1. The transformation produced by matrix ࡮has the effect of reflection across the ࢟-axis.

2. The transformation produced by matrix ࡭ has the effect of reflection across the ࢞-axis.

3. The green square in the second quadrant is the image of the original square under the transformation produced by ࡮.

4. The red square in the third quadrant is the image of the square after transforming with matrix ࡭and then matrix ࡮.

5. If we reflect across the ࢟-axis and then reflect across the ࢞-axis, the net effect is a rotation by ૚ૡ૙ι. െ૚ ૙ 6. The matrix product is ࡭࡮ ൌ ቂ ቃ. ૙െ૚ 7. The transformation produced by matrix ࡭࡮ has the effect of rotation by ૚ૡ૙ι.

ඥ૛ ඥ૛ െ ૝૙ iii. ࡭ൌ൦૛ ૛ ൪, ࡮ൌቂ ቃ ඥ૛ ඥ૛ ૙૝ ૛ ૛ 1. The transformation produced by matrix ࡮has the effect of dilation from the origin with scale factor ૝.

2. The transformation produced by matrix ࡭ has the effect of rotation by ૝૞ι.

3. The large green square is the image of the original unit square under the transformation produced by ࡮.

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4. The tilted red square is the image of the green square under the transformation produced by ࡭.

5. If we dilate with factor ૝ and then rotate by ૝૞ι, the net effect is a rotation by ૝૞ι and dilation with scale factor ૝.

૛ξ૛ െ૛ξ૛ 6. The matrix product is ࡭࡮ ൌ ൤ ൨. ૛ξ૛ ૛ξ૛ 7. The transformation produced by matrix ࡭࡮ has the effect of rotation by ૝૞ι while scaling by ૝.

૚૚ ૛૙ iv. ࡭ൌቂ ቃ, ࡮ൌቂ ቃ ૙૚ ૙૛ 1. The transformation produced by matrix ࡮has the effect of dilation with scale factor ૛.

2. The transformation produced by matrix ࡭ has the effect of shearing parallel to the ࢟-axis.

3. The large green square is the image of the original unit square under the transformation produced by ࡮.

4. The red parallelogram is the image of the green square under the transformation produced by ࡭.

5. If we dilate and then shear, the net effect is a dilated shear.

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૛૛ 6. The matrix product is ࡭࡮ ൌ ቂ ቃ. ૙૛ 7. The transformation produced by matrix ࡭࡮ has the effect of a dilated shear.

૚ ඥ૜ ඥ૜ െ ૚ v. ࡭ൌ൦ ૛ ૛ ൪, ࡮ൌ൦૛ ૛൪ ඥ૜ ૚ ૚ ඥ૜ െ ૛ ૛ ૛ ૛ 1. The transformation produced by matrix ࡮has the effect of rotation by ૜૙ι.

2. The transformation produced by matrix ࡭ has the effect of rotation by െ૟૙ι.

3. The tilted green square is the image of the original unit square under the transformation produced by ࡮.

4. The tilted red square is the image of the green square under the transformation produced by ࡭.

5. If we rotate by ૜૙ι counterclockwise and then rotate by ૟૙ι clockwise, the net result is a rotation by ૜૙ι counterclockwise.

ඥ૜ ૚ 6. The matrix product is ࡭࡮ ൌ ൦ ૛ ૛ ൪. ૚ ඥ૜ െ ૛ ૛ 7. The transformation produced by matrix ࡭࡮ has the effect of rotation by െ૜૙ι.

b. Make a conjecture about the geometric effect of the linear transformation produced by the matrix ࡭࡮. Justify your answer. The linear transformation produced by matrix ࡭࡮ has the same geometric effect of the sequence of the linear transformation produced by matrix ࡮ followed by the linear transformation produced by matrix ࡭. We have seen in previous work that if we multiply by ࡭࡮, we get the same transformation as when we multiplied by ࡭ first and then multiplied the result by ࡮.

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Discussion (5 minutes) This Discussion justifies why the conjecture students made at the end of Exploratory Challenge is valid. What was the conjecture you made at the end of the Exploratory Challenge? à The transformation that comes from the matrix ܣܤ has the same geometric effect as the sequence of transformations that come from the matrix ܤ first and then ܣ. Why does matrix ܣܤ represent ܤ first and then ܣ? Doesn’t that seem backward? If we look at this as multiplication, we are multiplying ݔ ݔ ቂݕቃڄ ሻܤܣ஺஻ ቀቂݕቃቁ ൌ ሺܮ ݔ ቂݕቃቁڄܤቀܣൌ ݔ , ஻ ቀቂݕቃቁቇܮ஺ ቆܮൌ

஺. It looks backward, butܮ ஻, and then the result is transformed byܮ so the point ሺݔǡ ݕሻ is transformed first by the transformation on the right happens to our point first. We saw this phenomenon in Geometry when we

represented a rotation by ߠ about the origin by ܴ଴ǡఏ and a reflection across line κ by ݎκ. Then, doing the ைǡఏ, which represents the combined effect of firstܴל κݎ rotation and then the reflection is denoted by performing the rotation ܴைǡఏ and then the reflection ݎκ.

Closing (3 minutes) Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements are listed below.

Lesson Summary

The linear transformation produced by matrix ࡭࡮ has the same geometric effect as the sequence of the linear transformation produced by matrix ࡮ followed by the linear transformation produced by matrix ࡭.

That is, if matrices ࡭ and ࡮produce linear transformations ࡸ࡭ and ࡸ࡮ in the plane, respectively, then the linear transformation ࡸ࡭࡮ produced by the matrix ࡭࡮ satisfies ࢞ ࢞ ࡸ࡭࡮ ቀቂ࢟ቃቁ ൌ ࡸ࡭ ቆࡸ࡮ ቀቂ࢟ቃቁቇ.

Exit Ticket (4 minutes)

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Name Date

Lesson 8: Composition of Linear Transformations

Exit Ticket

Let ܣ be the matrix representing a rotation about the origin ͳ͵ͷι and ܤ be the matrix representing a dilation of ͸. Let െͳ .ݔൌቈͳ ቉ ʹ a. Write down ܣ and ܤ.

.b. Find the matrix representing a dilation of ݔ by ͸, followed by a rotation about the origin of ͳ͵ͷι

.c. Graph and label ݔ, ݔ after a dilation of ͸, and ݔ after both transformations have been applied

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Exit Ticket Sample Solutions

Let ࡭ be the matrix representing a rotation about the origin ૚૜૞ι and ࡮ be the matrix representing a dilation of ૟. Let െ૚ ࢞ൌቈ ૚ ቉. ૛ a. Write down ࡭ and ࡮.

ξ૛ ξ૛ ې െ െۍ ૟૙ ۑ ૛ ૛ ێ ࡮ ൌ ቂ ቃ, ۑ ێ࡭ൌ ξ૛ ξ૛ ૙૟ ۑ െ ێ ے ૛ ૛ ۏ

b. Find the matrix representing a dilation ࢞ by ૟, followed by a rotation about the origin of ૚૜૞ι.

െ૜ ૛ െ૜ ૛ ࡭࡮ ൌ ൤ ξ ξ ൨ ૜ξ૛ െ૜ξ૛

c. Graph and label ࢞, ࢞ after a dilation of ૟, and ࢞ after both transformations have been applied.

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Problem Set Sample Solutions

૚ 1. Let ࡭ be the matrix representing a dilation of , and let ࡮ be the matrix representing a reflection across the ࢟-axis. ૛ a. Write ࡭ and ࡮. ૚ ૙ െ૚ ૙ ࡭ൌ൦૛ ൪ ,࡮ ൌ ቂ ቃ ૚ ૙૚ ૙ ૛

b. Evaluate ࡭࡮. What does this matrix represent? ૚ െ ૙ ࡭࡮ ൌ ൦ ૛ ൪ ૚ ૙ ૛ ૚ ࡭࡮ is a reflection across the ࢟-axis followed by a dilation of . ૛

૞ െ૚ ૡ c. Let ࢞ൌቂ ቃ,࢟ ൌ ቂ ቃ, and ࢠൌቂ ቃ. Find ሺ࡭࡮ሻ࢞,ሺ࡭࡮ሻ࢟, and ሺ࡭࡮ሻࢠ. ૟ ૜ െ૛ ૚ ૞ െ૝ ሺ࡭࡮ሻ࢞ൌ൥െ ൩ ,ሺ࡭࡮ሻ࢟ൌ൦૛൪ ,ሺ࡭࡮ሻࢠൌቂ ቃ ૛ ૜ െ૚ ૜ ૛

2. Let ࡭ be the matrix representing a rotation of ૜૙ι, and let ࡮ be the matrix representing a dilation of ૞. a. Write ࡭ and ࡮.

ξ૜ ૚ ې െ ۍ ૞૙ ۑ૛ ૛ ێ ࡮ ൌ ቂ ቃ, ۑ ێ࡭ൌ ૙૞ ۑ૚ ξ૜ ێ ے ૛ ૛ ۏ

b. Evaluate ࡭࡮. What does this matrix represent?

૞ξ૜ ૞ ې െ ۍ ۑ ૛ ૛ ێ ۑ ێ ࡭࡮ ൌ ۑ૞ ૞ξ૜ ێ ے ૛ ૛ ۏ ࡭࡮ is a dilation of ૞ followed by a rotation of ૜૙ι.

૚ c. Let ࢞ൌቂ ቃ. Find ሺ࡭࡮ሻ࢞. ૙

ې૞ξ૜ۍ ۑ ૛ ێሺ࡭࡮ሻ࢞ൌ ۑ ૞ ێ ے ૛ ۏ

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3. Let ࡭ be the matrix representing a dilation of ૜, and let ࡮ be the matrix representing a reflection across the line ࢟ൌ࢞. a. Write ࡭ and ࡮. ૜૙ ૙૚ ࡭ൌቂ ቃ ,࡮ ൌ ቂ ቃ ૙૜ ૚૙

b. Evaluate ࡭࡮. What does this matrix represent? ૙૜ ࡭࡮ ൌ ቂ ቃ ૜૙ ࡭࡮ is a reflection across the line ࢟ൌ࢞ followed by a dilation of ૜.

െ૛ c. Let ࢞ൌቂ ቃ. Find ሺ࡭࡮ሻ࢞. ૠ ૛૚ ሺ࡭࡮ሻ࢞ൌቂ ቃ െ૟

૜૙ ૙െ૚ 4. Let ࡭ൌቂ ቃ܉ܖ܌࡮ൌቂ ቃ. ૜૜ ૚૙ a. Evaluate ࡭࡮. ૙െ૜ ቂ ቃ ૜െ૜

െ૛ b. Let ࢞ൌቂ ቃ. Find ሺ࡭࡮ሻ࢞. ૛ െ૟ ቂ ቃ െ૚૛

c. Graph ࢞ and ሺ࡭࡮ሻ࢞.

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૚ ૙ ૜૚ ࡭ൌ቎૜ ቏ ࡮ൌቂ ቃ 5. Let ૚ and . ૛ ૚െ૜ ૜ a. Evaluate ࡭࡮. ૚ ૚ ൦ ૜൪ ૚ૢ ૚ ૜

૙ b. Let ࢞ൌቂ ቃ. Find ሺ࡭࡮ሻ࢞. ૜ ૚ ቂ ቃ ૜

c. Graph ࢞ and ሺ࡭࡮ሻ࢞.

૛૛ ૙૛ 6. Let ࡭ൌቂ ቃ and ࡮ൌቂ ቃ. ૛૙ ૛૛ a. Evaluate ࡭࡮. ૝ૡ ቂ ቃ ૙૝

૜ b. Let ࢞ൌቂ ቃ. Find ሺ࡭࡮ሻ࢞. െ૛ െ૝ ቂ ቃ െૡ

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c. Graph ࢞ and ሺ࡭࡮ሻ࢞.

7. Let ࡭, ࡮, ࡯ be ૛ൈ૛ matrices representing linear transformations. a. What does ࡭ሺ࡮࡯ሻ represent?

࡭ሺ࡮࡯ሻ represents the linear transformation of applying the transformation that ࡯ represents, followed by the transformation that ࡮ represents, followed by the transformation that ࡭ represents.

b. Will the pattern established in part (a) be true no matter how many matrices are multiplied on the left?

Yes, in general. When you multiply by a matrix on the left, you are applying a linear transformation after all linear transformations to the right have been applied.

c. Does ሺ࡭࡮ሻ࡯ represent something different from ࡭ሺ࡮࡯ሻ? Explain.

No, it does not. This is the linear transformation obtained by applying ࡯ and then ࡭࡮, which is ࡮ followed by ࡭.

૙ 8. Let ࡭࡮ represent any composition of linear transformations in Թ૛. What is the value of ሺ࡭࡮ሻ࢞ where ࢞ൌቂ ቃ? ૙ Since a composition of linear transformations in Թ૛ is also a linear transformation, we know that applying it to the origin will result in no change.

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Lesson 9: Composition of Linear Transformations

Student Outcomes Students use technology to perform compositions of linear transformations in Թଷ.

Lesson Notes

In Lesson 8, students discovered that if they compose two linear transformations in the plane, ܮ஺ and ܮ஻, represented by matrices ܣ and ܤ, respectively, then the resulting transformation can be produced using a single matrix ܣܤ. In this lesson, this result from transformations in the plane is extended to transformations in three-dimensional space, Թଷ. This lesson was designed to be implemented using the GeoGebra applet TransformCubes (http://eurmath.link/g12m2. transformcube), which allows students to visualize the transformation of the cube.

Classwork Opening Exercise (10 minutes) The Opening Exercise reviews the discovery from the Problem Set in Lesson 7 that the matrix of a transformation in Թଷ is determined by the images of the three points ሺͳǡͲǡͲሻ, ሺͲǡͳǡͲሻ, and ሺͲǡͲǡͳሻ. This fact is needed to construct matrices of transformations in this lesson before composing them. Students may question what is meant by counterclockwise rotation in space; if this happens, let them know that a rotation about the ݖ-axis is considered to be a counterclockwise rotation if it rotates the ݔݕ-plane counterclockwise when placed in its standard orientation as shown to the right. Students should work these exercises with pencil and paper.

Opening Exercise

Recall from Problem 1, part (d), of the Problem Set of Lesson 7 that if you know what a linear Scaffolding: transformation does to the three points ሺ૚ǡ ૙ǡ ૙ሻ, ሺ૙ǡ ૚ǡ ૙ሻ, and ሺ૙ǡ ૙ǡ ૚ሻ, you can find the matrix of the transformation. How do the images of these three points lead to the matrix of the Encourage struggling students transformation? to draw the image of a cube

a. Suppose that a linear transformation ࡸ૚ rotates the unit cube by ૢ૙ι before and after the counterclockwise about the ࢠ-axis. Find the matrix ࡭૚ of the transformation ࡸ૚. transformation to find the Since this transformation rotates by ૢ૙ι counterclockwise in the ࢞࢟-plane, a vector images of the points ሺͳǡͲǡͲሻ, along the positive ࢞-axis will be transformed to lie along the positive ࢟-axis, a vector ሺͲǡͳǡͲሻ, and ሺͲǡͲǡͳሻ in space. along the positive ࢟-axis will be transformed to lie along the negative ࢞-axis, and a vector along the ࢠ-axis will be left alone. Thus, ૚ ૙ ૙ െ૚ ૙ ૙ ࡸ૚ ൭൥૙൩൱ൌ൥૚൩,ࡸ૚ ൭൥૚൩൱ൌ൥ ૙ ൩, and ࡸ૚ ൭൥૙൩൱ൌ൥૙൩, ૙ ૙ ૙ ૙ ૚ ૚ ૙െ૚૙ so the matrix of the transformation is ࡭૚ ൌ ൥૚૙૙൩. ૙૙૚

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b. Suppose that a linear transformation ࡸ૛ rotates the unit cube by ૢ૙ι counterclockwise about the ࢟-axis. Find the matrix ࡭૛ of the transformation ࡸ૛. Since this transformation rotates by ૢ૙ι counterclockwise in the ࢞ࢠ-plane, a vector along the positive ࢞-axis will be transformed to lie along the positive ࢠ-axis, a vector along the ࢟-axis will be left alone, and a vector along the positive ࢠ-axis will be transformed to lie along the negative ࢞-axis. Thus, ૚ ૙ ૙ ૙ ૙ െ૚ ࡸ૛ ൭൥૙൩൱ൌ൥૙൩,ࡸ૛ ൭൥૚൩൱ൌ൥૚൩, and ࡸ૛ ൭൥૙൩൱ൌ൥ ૙ ൩, ૙ ૚ ૙ ૙ ૚ ૙ ૙૙െ૚ so the matrix of the transformation is ࡭૛ ൌ ൥૙૚ ૙൩. ૚૙ ૙

c. Suppose that a linear transformation ࡸ૜ scales by ૛ in the ࢞-direction, scales by ૜ in the ࢟-direction, and scales by ૝ in the ࢠ-direction. Find the matrix ࡭૜ of the transformation ࡸ૜. Since this transformation scales by ૛ in the ࢞-direction, by ૜ in the ࢟-direction, and by ૝ in theࢠ-direction, a vector along the ࢞-axis will be multiplied by ૛, a vector along the ࢟-axis will be multiplied by ૜, and a vector along the ࢠ-axis will be multiplied by ૝. Thus, ૚ ૛ ૙ ૙ ૙ ૙ ࡸ૜ ൭൥૙൩൱ൌ൥૙൩,ࡸ૜ ൭൥૚൩൱ൌ൥૜൩, and ࡸ૜ ൭൥૙൩൱ൌ൥૙൩, ૙ ૙ ૙ ૙ ૚ ૝ ૛૙૙ so the matrix of the transformation is ࡭૜ ൌ ൥૙૜૙൩. ૙૙૝

d. Suppose that a linear transformation ࡸ૝ projects onto the ࢞࢟-plane. Find the matrix ࡭૝ of the transformation ࡸ૝. Since this transformation projects onto the ࢞࢟-plane, a vector along the ࢞-axis will be left alone, a vector along the ࢟-axis will be left alone, and a vector along the ࢠ-axis will be transformed into the zero vector. Thus, ૚ ૚ ૙ ૙ ૙ ૙ ࡸ૝ ൭൥૙൩൱ൌ൥૙൩,ࡸ૝ ൭൥૚൩൱ൌ൥૚൩, and ࡸ૝ ൭൥૙൩൱ൌ൥૙൩, ૙ ૙ ૙ ૙ ૚ ૙ ૚૙૙ so the matrix of the transformation is ࡭૝ ൌ ൥૙૚૙൩. ૙૙૙

e. Suppose that a linear transformation ࡸ૞ projects onto the ࢞ࢠ-plane. Find the matrix ࡭૞ of the transformation ࡸ૞. Since this transformation projects onto the ࢞ࢠ-plane, a vector along the ࢞-axis will be left alone, a vector along the ࢟-axis will be transformed into the zero vector, and a vector along the ࢠ-axis will be left alone. Thus, ૚ ૚ ૙ ૙ ૙ ૙ ࡸ૞ ൭൥૙൩൱ൌ൥૙൩, ࡸ૞ ൭൥૚൩൱ൌ൥૙൩, and ࡸ૞ ൭൥૙൩൱ൌ൥૙൩, ૙ ૙ ૙ ૙ ૚ ૚ ૚૙૙ so the matrix of the transformation is ࡭૞ ൌ ൥૙૙૙൩. ૙૙૚

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f. Suppose that a linear transformation ࡸ૟ reflects across the plane with equation ࢟ൌ࢞. Find the matrix ࡭૟ of the transformation ࡸ૟. Since this transformation reflects across the plane ࢟ൌ࢞, a vector along the positive ࢞-axis will be transformed into a vector along the positive ࢟-axis with the same length, a vector along the positive ࢟-axis will be transformed into a vector along the positive ࢞-axis, and a vector along the ࢠ-axis will be left alone. Thus, ૚ ૙ ૙ ૚ ૙ ૙ ࡸ૟ ൭൥૙൩൱ൌ൥૚൩, ࡸ૟ ൭൥૚൩൱ൌ൥૙൩, and ࡸ૟ ൭൥૙൩൱ൌ൥૙൩, ૙ ૙ ૙ ૙ ૚ ૚ ૙૚૙ so the matrix of the transformation is ࡭૟ ൌ ൥૚૙૙൩. ૙૙૚

૚ ૛ ૙ ૙ ૙ ૙ g. Suppose that a linear transformation ࡸ satisfies ࡸ ൭൥૙൩൱ൌ൥૙൩, ࡸ ൭൥૚൩൱ൌ൥૚൩, and ࡸ ൭൥૙൩൱ൌ቎૙቏. Find ૠ ૠ ૠ ૠ ૚ ૙ ૙ ૙ ૙ ૚ ૛ the matrix ࡭ૠ of the transformation ࡸૠ. What is the geometric effect of this transformation? ૛૙૙ The matrix of this transformation is ࡭ ൌ቎૙૚૙቏. The geometric effect of ࡸ is to stretch by a factor of ૛ ૠ ૚ ૠ ૙૙ ૛ ૚ in the ࢞-direction and scale by a factor of in the ࢠ-direction. ૛

૚ ૚ ૙ െ૚ ૙ ૙ h. Suppose that a linear transformation ࡸૡ satisfies ࡸૡ ൭൥૙൩൱ൌ൥૚൩ ǡࡸૡ ൭൥૚൩൱ൌ൥ ૚ ൩, and ࡸૡ ൭൥૙൩൱ൌ൥૙൩. ૙ ૙ ૙ ૙ ૚ ૚ Find the matrix of the transformation ࡸૡ. What is the geometric effect of this transformation? ૚െ૚૙ The matrix of this transformation is ࡭ૡ ൌ ൥૚૚૙൩. The geometric effect of ࡸૡ is to rotate by ૝૞ι in the ૙૙૚ ࢞࢟-plane, to scale by ξ૛ in both the ࢞ and ࢟ directions, and to not change in the ࢠ-direction.

Once students have completed the Opening Exercise with pencil and paper, allow them to use the GeoGebra applet

TransformCubes.ggb to check their work before continuing. The remaining exercises rely on the matrices ܣଵ–ܣ଼, so ensure that students have found the correct matrices before proceeding with the lesson.

Discussion (2 minutes)

In Lesson 8, we saw that for linear transformations in the plane, if ܮ஺ is a linear transformation represented by a ʹൈʹ matrix ܣ, and ܮ஻ is a linear transformation represented by a ʹൈʹ matrix ܤ, then the ʹൈʹ matrix ܣܤ is the matrix of the composition of ܮ஻ followed by ܮ஺. Today we explore composition of linear transformations in Թଷ to see whether the same result extends to the case when ܣ, ܤ, and ܣܤare ͵ൈ͵ matrices.

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Exploratory Challenge 1 (25 minutes) In the Exploratory Challenge, students predict the geometric effect of composing pairs of transformations from the Opening Exercise and then check their predictions with the GeoGebra applet TransformCubes.ggb.

Exploratory Challenge 1

Transformations ࡸ૚–ࡸૡ refer to the linear transformations from the Opening Exercise. For each pair: i. Make a conjecture to predict the geometric effect of performing the two transformations in the order specified. ii. Find the product of the corresponding matrices in the order that corresponds to the indicated order of

composition. Remember that if we perform a transformation ࡸ࡮ with matrix ࡮ and then ࡸ࡭ with ࡸ࡮ is ࡭࡮. That is, ࡸ࡮ is applied first, butל matrix ࡭, the matrix that corresponds to the composition ࡸ࡭ matrix ࡮ is written second. iii. Use the GeoGebra applet TransformCubes.ggb to draw the image of the unit cube under the transformation induced by the matrix product in part (ii). Was your conjecture in part (i) correct?

a. Perform ࡸ૟ and then ࡸ૟.

i. Since ࡸ૟ reflects across the plane through ࢟ൌ࢞ that is perpendicular to the ࢞࢟-plane, performing ࡸ૟ twice in succession will result in the identity transformation. ૙૚૙ ૙૚૙ ૚૙૙ ࡭૟ is the identity matrix, we know thatڄ ൥૚૙૙൩ ൌ ൥૙૚૙൩. Since ࡭૟ ڄ ࡭૟ ൌ ൥૚૙૙൩ڄ ii. ࡭૟ ૙૙૚ ૙૙૚ ૙૙૚ .ࡸ૟ is the identity transformationל ࡸ૟ iii. The conjecture was correct.

b. Perform ࡸ૚ and then ࡸ૛.

i. Sample student response: Since ࡸ૚ rotates ૢ૙ι about the ࢠ-axis and ࡸ૛ rotates ૢ૙ι about the ࢟-axis, .ࡸ૚ should rotate ૚ૡ૙ι about the line ࢟ൌെ࢞ in the ࢞࢟-planeל the composition ࡸ૛ ૙૙െ૚ ૙െ૚૙ ૙૙െ૚ ൥૚૙૙൩ ൌ ൥૚૙ ૙൩ ڄ ࡭૚ ൌ ൥૙૚ ૙൩ڄ ii. ࡭૛ ૚૙ ૙ ૙૙૚ ૙െ૚૙ iii. The conjecture in part (i) is not correct. While it appears that

ࡸ૚ is a rotation by ૚ૡ૙ι about the lineל the composition ࡸ૛ ࢟ൌെ࢞, it is not because, for example, point ሺ૙ǡ ૚ǡ ૙ሻ is transformed to point ሺ૙ǡ ૙ǡ െ૚ሻ and does not remain on the ࢟-axis after the transformation. Thus, this cannot be a rotation around the ࢟-axis.

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c. Perform ࡸ૝ and then ࡸ૞.

ࡸ૝ willל i. Since ࡸ૝ projects onto the ࢞࢟-plane and ࡸ૞ projects onto the ࢞ࢠ-plane, the composition ࡸ૞ project onto the ࢞-axis. ૚૙૙ ૚૙૙ ૚૙૙ ൥૙૚૙൩ ൌ ൥૙૙૙൩ ڄ ࡭૝ ൌ ൥૙૙૙൩ڄ ii. ࡭૞ ૙૙૚ ૙૙૙ ૙૙૙ iii. The conjecture in part (i) is correct. The cube is first transformed to a square in the ࢞࢟-plane and then transformed onto a segment on the ࢞-axis.

d. Perform ࡸ૝ and then ࡸ૜.

ࡸ૝ל i. Since ࡸ૝ projects onto the ࢞࢟-plane and ࡸ૜ scales in the ࢞, ࢟, and ࢠ directions, the composition ࡸ૜ will project onto the ࢞࢟-plane and scale in the ࢞ and ࢟ directions. ૛૙૙ ૚૙૙ ૛૙૙ ൥૙૚૙൩ ൌ ൥૙૜૙൩ ڄ ࡭૝ ൌ ൥૙૜૙൩ڄ ii. ࡭૜ ૙૙૝ ૙૙૙ ૙૙૙ iii. The conjecture in part (i) is correct.

e. Perform ࡸ૜ and then ࡸૠ.

i. Since ࡸ૜ scales in the ࢞, ࢟, and ࢠ directions and so does ࡸૠ, the composition will be a transformation that also scales in all three directions but with different scale factors. ૛૙૙ ૛૙૙ ૝૙૙ ൥૙૜૙൩ ൌ ൥૙૜૙൩ڄ࡭ ൌ቎૙૚૙቏ڄ ii. ࡭ ૠ ૜ ૚ ૙૙ ૛ ૙૙૝ ૙૙૛ iii. The conjecture from (i) is correct.

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f. Perform ࡸૡ and then ࡸ૝.

i. Transformation ࡸૡ rotates the unit cube by ૝૞ι about the ࢠ-axis and stretches by a factor of ξ૛ in both the ࢞ and ࢟ directions, while ࡸ૝ projects the image onto the ࢞࢟-plane. The composition will transform the unit cube into a larger square that has been rotated ૝૞ι in the ࢞࢟-plane. ૚૙૙ ૚െ૚૙ ૚െ૚૙ ൥૚૚૙൩ ൌ ൥૚૚૙൩ ڄ ࡭ૡ ൌ ൥૙૚૙൩ڄ ii. ࡭૝ ૙૙૙ ૙૙૚ ૙૙૙ iii. The conjecture from part (i) is correct.

g. Perform ࡸ૝ and then ࡸ૟.

i. Transformation ࡸ૝ projects onto the ࢞࢟-plane, and transformation ࡸ૟ reflects across the plane through ࡸ૝ willל the line ࢟ൌ࢞ in the ࢞࢟-plane and is perpendicular to the ࢞࢟-plane, so the composition ࡸ૟ appear to be the reflection in the ࢞࢟-plane across the line ࢟ൌ࢞. ૙૚૙ ૚૙૙ ૙૚૙ ൥૙૚૙൩ ൌ ൥૚૙૙൩ ڄ ࡭૝ ൌ ൥૚૙૙൩ڄ ii. ࡭૟ ૙૙૚ ૙૙૙ ૙૙૙ iii. The conjecture in part (i) is correct.

h. Perform ࡸ૛ and then ࡸૠ.

ࡸ૛ל i. Since ࡸ૛ rotates ૢ૙ι around the ࢟-axis and ࡸૠ scales in the ࢞ and ࢠ directions, the composition ࡸૠ will rotate and scale simultaneously. ૛૙૙ ૙૙െ૚ ૙૙െ૛ ൥૙૚ ૙൩ ൌ቎૙૚ ૙቏ڄ࡭ ൌ቎૙૚૙቏ڄ ii. ࡭ ૠ ૛ ૚ ૚ ૙૙ ૙૙ ૛ ૚૙ ૙ ૛ iii. The conjecture in part (i) is correct.

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i. Perform ࡸૡ and then ࡸૡ.

i. Since ࡸૡ rotates by ૝૞ι about the ࢠ-axis and scales by ξ૛ in the ࢞ and ࢟ directions, performing this transformation twice will rotate by ૢ૙ι about the ࢠ-axis and scale by ૛ in the ࢞ and ࢟ directions. ૚െ૚૙ ૚െ૚૙ ૙െ૛૙ ൥૚૚૙൩ ൌ ൥૛૙૙൩ ڄ ࡭ૡ ൌ ൥૚૚૙൩ڄ ii. ࡭ૡ ૙૙૚ ૙૙૚ ૙૙૚ iii. The conjecture in part (i) is correct.

Do a 30-second Quick Write on what we have discovered in Exploratory Challenge 1, and share with your neighbor.

Exploratory Challenge 2 (Optional) This optional challenge is for students who finished Exploratory Challenge 1 early. The challenge is designed to prompt the question of whether or not order matters when composing two linear transformations, a question that is definitively answered in the next lesson and demonstrates that matrix multiplication is in general not commutative. Students are

஻ܮל ஺ܮ respectively, and to compare ,ܤ and ܣ ஻, with matricesܮ ஺ andܮ asked to compose two linear transformations ஺. The directions for this challenge are left intentionally vague so that students may use either an algebraic orܮל ஻ܮ with a geometric approach to answer the question.

Exploratory Challenge 2

Transformations ࡸ૚–ࡸૡ refer to the transformations from the Opening Exercise. For each of the following pairs of .ࡸ࡭ל ࡸ࡮ and ࡸ࡮ל matrices ࡭ and ࡮ below, compare the transformations ࡸ࡭

a. ࡸ૝ and ࡸ૞ ૚૙૙ ૚૙૙ ૚૙૙ ൥૙૙૙൩ ൌ ൥૙૙૙൩, and ڄ ࡭૞ ൌ ൥૙૚૙൩ڄ ࡸ૞ has matrix representation ࡭૝ל Transformation ࡸ૝ ૙૙૙ ૙૙૚ ૙૙૙ ૚૙૙ ૚૙૙ ૚૙૙ .൥૙૚૙൩ ൌ ൥૙૙૙൩ ڄ ࡭૝ ൌ ൥૙૙૙൩ל ࡸ૝ has matrix representation ࡭૞ל transformation ࡸ૞ ૙૙૚ ૙૙૙ ૙૙૙ Since the two transformations have the same matrix representation, they are the same transformation:

ࡸ૞ל ࡸ૝ ൌࡸ૝ל ࡸ૞

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b. ࡸ૛ and ࡸ૞ ૙૙െ૚ ૚૙૙ ૙૙െ૚ ൥૙૙૙൩ ൌ ൥૙૙ ૙൩, and ڄ ࡭૞ ൌ ൥૙૚ ૙൩ڄ ࡸ૞ has matrix representation ࡭૛ל Transformation ࡸ૛ ૚૙ ૙ ૙૙૚ ૚૙ ૙ ૚૙૙ ૙૙െ૚ ૙૙െ૚ .൥૙૚ ૙൩ ൌ ൥૙૙ ૙൩ ڄ ࡭૛ ൌ ൥૙૙૙൩ڄ ࡸ૛ has matrix representation ࡭૞ל transformation ࡸ૞ ૙૙૚ ૚૙ ૙ ૚૙ ૙ Since the two transformations have the same matrix representation, they are the same transformation:

ࡸ૛ל ࡸ૞ ൌࡸ૞ל ࡸ૛

c. ࡸ૜ and ࡸૠ ૛૙૙ ૛૙૙ ૝૙૙ ቎૙૚૙቏ൌ൥૙૜૙൩, andڄ ࡭ ൌ ൥૙૜૙൩ڄ ࡸ has matrix representation ࡭ל Transformation ࡸ ૜ ૠ ૜ ૠ ૚ ૙૙ ૙૙૝ ૛ ૙૙૛ ૛૙૙ ૛૙૙ ૝૙૙ .൥૙૜૙൩ ൌ ൥૙૜૙൩ڄ࡭ ൌ቎૙૚૙቏ڄ ࡸ has matrix representation ࡭ל transformation ࡸ ૠ ૜ ૠ ૜ ૚ ૙૙ ૛ ૙૙૝ ૙૙૛ Since the two transformations have the same matrix representation, they are the same transformation:

ࡸ૜ל ࡸૠ ൌࡸૠל ࡸ૜

d. ࡸ૜ and ࡸ૟ ૛૙૙ ૙૚૙ ૙૛૙ ൥૚૙૙൩ ൌ ൥૜૙૙൩, and ڄ ࡭૟ ൌ ൥૙૜૙൩ڄ ࡸ૟ has matrix representation ࡭૜ל Transformation ࡸ૜ ૙૙૝ ૙૙૚ ૙૙૝ ૙૚૙ ૛૙૙ ૙૜૙ .൥૙૜૙൩ ൌ ൥૛૙૙൩ ڄ ࡭૜ ൌ ൥૚૙૙൩ڄ ࡸ૜ has matrix representation ࡭૟ל transformation ࡸ૟ ૙૙૚ ૙૙૝ ૙૙૝ Since the two transformations have different matrix representations, they are not the same transformation:

ࡸ૜ל ࡸ૟ ്ࡸ૟ל ࡸ૜

e. ࡸૠ and ࡸ૚

૛૙૙ ૙െ૚૙ ૙െ૛૙ ൥૚૙૙൩ ൌ቎૚૙૙቏, andڄ࡭ ൌ቎૙૚૙቏ڄ ࡸ has matrix representation ࡭ל Transformation ࡸ ૠ ૚ ૠ ૚ ૚ ૚ ૙૙ ૙૙ ૛ ૙૙૚ ૛ ૙െ૚૙ ૛૙૙ ૙െ૚૙ .቎૙૚૙቏ൌ቎૛૙૙቏ڄ ࡭ ൌ ൥૚૙૙൩ڄ ࡸ has matrix representation ࡭ל transformation ࡸ ૚ ૠ ૚ ૠ ૚ ૚ ૙૙ ૙૙ ૙૙૚ ૛ ૛ Since the two transformations have different matrix representations, they are not the same transformation:

ࡸ૚ל ࡸૠ ്ࡸૠל ࡸ૚

f. What can you conclude about the order in which you compose two linear transformations?

In some cases, the order of composition of two linear transformations matters. For two matrices ࡭ and ࡮, the

.ࡸ࡭ל ࡸ࡮ is not always the same transformation as ࡸ࡮ל transformation ࡸ࡭

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Closing (4 minutes) Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements are listed below.

Lesson Summary

The linear transformation induced by a ૜ൈ૜ matrix ࡭࡮ has the same geometric effect as the sequence of the linear transformation induced by the ૜ൈ૜ matrix ࡮ followed by the linear transformation induced by the ૜ൈ૜ matrix ࡭. ૜ That is, if matrices ࡭ and ࡮induce linear transformations ࡸ࡭ and ࡸ࡮ in Թ , respectively, then the linear ࢞ ࢞ transformation ࡸ࡭࡮ induced by the matrix ࡭࡮ satisfies ࡸ࡭࡮ ቆቈ࢟቉ቇ ൌࡸ࡭ ቌࡸ࡮ ቆቈ࢟቉ቇቍ. ࢠ ࢠ

Exit Ticket (4 minutes)

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Name Date

Lesson 9: Composition of Linear Transformations

Exit Ticket

.ʹ be the matrix representing a dilation of ܤ be the matrix representing a rotation about the ݖ-axis of Ͷͷι and ܣ Let a. Write down ܣ and ܤ.

͵ .b. Let ݔൌ൥െͳ൩. Find the matrix representing a dilation of ݔ by ʹ followed by a rotation about the ݖ-axis of Ͷͷι ʹ

c. Do your best to sketch a picture of ݔ, ݔ after the first transformation, and ݔ after both transformations. You may use technology to help you.

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Exit Ticket Sample Solutions

Let ࡭ be the matrix representing a rotation about the ࢠ-axis of ૝૞ι and ࡮ be the matrix representing a dilation of ૛.

a. Write down ࡭ and ࡮.

ξ૛ ξ૛ ېെ ૙ ۍ ૛૙૙ ۑ ૛ ૛ ێ ࡮ ൌ ൥૙૛૙൩, ۑ ξ૛ ξ૛ێ࡭ൌ ૙ ૙૙૛ ۑ ૛ ૛ ێ ے૙૙૚ ۏ

૜ b. Let ࢞ൌ൥െ૚൩. Find the matrix representing a dilation of ࢞ by ૛ followed by a rotation about the ࢠ-axis of ૛ ૝૞ι.

ξ૛ െξ૛ ૙ ࡭࡮ ൌ ቎ξ૛ ξ૛ ૙቏ ૙૙૛

c. Do your best to sketch a picture of ࢞, ࢞ after the first transformation, and ࢞ after both transformations. You may use technology to help you.

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Problem Set Sample Solutions

૚ 1. Let ࡭ be the matrix representing a dilation of , and let ࡮ be the matrix representing a reflection across the ૛ ࢟ࢠ-plane. a. Write ࡭ and ࡮. ૚ ې૙૙ ۍ ۑ ૛ێ ૚ െ૚ ૙ ૙ ࡮ ൌ ൥ ૙૚૙൩, ۑ૙ ૙ێ࡭ൌ ૙૙૚ ۑ ૛ ێ ۑ૚ ێ ૙૙ ے૛ ۏ

b. Evaluate ࡭࡮. What does this matrix represent? ૚ ېെ ૙૙ۍ ۑ ૛ ێ ૚ ۑ૙ ૙ ێ ࡭࡮ ൌ ۑ ૛ ێ ۑ૚ ێ ૙૙ ے૛ ۏ ૚ ࡭࡮ is a reflection across the ࢟ࢠ-plane followed by a dilation of . ૛

૞ െ૚ ૡ c. Let ࢞ൌ൥૟൩, ࢟ൌ൥ ૜ ൩, and ࢠൌ൥െ૛൩. Find ሺ࡭࡮ሻ࢞, ሺ࡭࡮ሻ࢟, and ሺ࡭࡮ሻࢠ. ૝ ૛ െ૝ ૚ ૞ െ૝ ې ۍ ۑ૛ێ െ ሺ࡭࡮ሻ࢞ൌ൦ ૛൪ ,ሺ࡭࡮ሻ࢟ൌ ૜ ,ሺ࡭࡮ሻࢠൌ൥െ૚൩ ۑ ێ ૜ െ૛ ۑ ێ ૛ ૛ ے૚ۏ

2. Let ࡭ be the matrix representing a rotation of ૜૙ι about the ࢞-axis, and let ࡮ be the matrix representing a dilation of ૞. a. Write ࡭ and ࡮. ૚૙ ૙ ې ۍ ξ૜ ૚ ૞૙૙ ۑ ૙ െێ ࡮ ൌ ൥૙૞૙൩, ۑ૛ ૛ ێ࡭ൌ ૙૙૞ ۑ૚ ξ૜ ێ ૙ ے ૛ ૛ ۏ

b. Evaluate ࡭࡮. What does this matrix represent? ૞૙ ૙ ې ۍ ૞ξ૜ ૞ ۑ ૙ െێ ۑ ૛ ૛ ێ ࡭࡮ ൌ ۑ૞ ૞ξ૜ ێ ૙ ے ૛ ૛ ۏ ࡭࡮ is a dilation of ૞ followed by a rotation of ૜૙ι about the ࢞-axis.

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૚ ૙ ૙ c. Let ࢞ൌ൥૙൩, ࢟ൌ൥૚൩, ࢠൌ൥૙൩. Find ሺ࡭࡮ሻ࢞, ሺ࡭࡮ሻ࢟, and ሺ࡭࡮ሻࢠ. ૙ ૙ ૚ ૞ ሺ࡭࡮ሻ࢞ൌ൥૙൩ ૙ ૙ ې ۍ ૞ξ૜ ۑ ێ ۑ ૛ ێሺ࡭࡮ሻ࢟ൌ ۑ ૞ ێ ے ૛ ۏ ૙ ې ૞ ۍ ۑ െ ێ ۑ ૛ ێሺ࡭࡮ሻࢠൌ ۑ૞ξ૜ێ ے ૛ ۏ

3. Let ࡭ be the matrix representing a dilation of ૜, and let ࡮ be the matrix representing a reflection across the plane ࢟ൌ࢞. a. Write ࡭ and ࡮. ૜૙૙ ૙૚૙ ࡭ൌ൥૙૜૙൩ ,࡮ ൌ ൥૚૙૙൩ ૙૙૜ ૙૙૚

b. Evaluate ࡭࡮. What does this matrix represent? ૙૜૙ ࡭࡮ ൌ ൥૜૙૙൩ ૙૙૜ ࡭࡮ is a reflection across the ࢟ൌ࢞ plane followed by a dilation of ૜.

െ૛ c. Let ࢞ൌ൥ ૠ ൩. Find ሺ࡭࡮ሻ࢞. ૜ ૛૚ ሺ࡭࡮ሻ࢞ൌ൥െ૟൩ ૢ

૜૙૙ ૙െ૚૙ 4. Let ࡭ൌ൥૜૜૙൩, ࡮ൌ൥૚૙૙൩. ૙૙૚ ૙૙૚ a. Evaluate ࡭࡮. ૙െ૜૙ ൥૜െ૜૙൩ ૙૙૚

െ૛ b. Let ࢞ൌ൥ ૛ ൩. Find ሺ࡭࡮ሻ࢞. ૞ െ૟ ൥െ૚૛൩ ૞

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c. Graph ࢞ and ሺ࡭࡮ሻ࢞.

ሻݔܤܣሺ

૚ ૙૙ ૜૚૙ ې ૜ۍ .࡮ൌ൥૚െ૜૙൩ ,ۑ૙૚૙ێLet ࡭ൌ .5 ۑ૚ ێ ૛૙ ૙૙૚ ے૜ ۏ a. Evaluate ࡭࡮. ૚ ې૚ ૙ۍ ۑ ૜ ێ ۑ૚െ૜૙ێ ۑ૚ ێ ૟૛ ے૜ ۏ

૙ b. Let ࢞ൌ൥૜൩. Find ሺ࡭࡮ሻ࢞. ૛ ૚ െૢ ൦૛૙൪ ૜

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c. Graph ࢞ and ሺ࡭࡮ሻ࢞.

૛૙૙ ૚૙૙ 6. Let ࡭ൌ൥૙૜૙൩, ࡮ൌ൥૙૚૙൩. ૙૙ૡ ૙૙૙ a. Evaluate ࡭࡮. ૛૙૙ ൥૙૜૙൩ ૙૙૙

૚ b. Let ࢞ൌ൥െ૛൩. Find ሺ࡭࡮ሻ࢞. ૝ ૛ ൥െ૟൩ ૙

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c. Graph ࢞ and ሺ࡭࡮ሻ࢞.

d. What does ࡭࡮ represent geometrically?

࡭࡮ represents a projection onto the ࢞࢟-plane, and a dilation of ૛ in the ࢞-direction and ૜ in the ࢟-direction.

7. Let ࡭, ࡮, ࡯ be ૜ൈ૜ matrices representing linear transformations. a. What does ࡭ሺ࡮࡯ሻ represent?

The linear transformation of applying the linear transformation that ࡯ represents followed by the transformation that ࡮ represents followed by the transformation that ࡭ represents

b. Will the pattern established in part (a) be true no matter how many matrices are multiplied on the left?

Yes, in general. When you multiply by a matrix on the left, you are applying a linear transformation after all linear transformations to the right have been applied.

c. Does ሺ࡭࡮ሻ࡯ represent something different from ࡭ሺ࡮࡯ሻ? Explain.

No, it does not. This is the linear transformation obtained by applying C and then ࡭࡮, which is ࡮ followed by ࡭.

૙ 8. Let ࡭࡮ represent any composition of linear transformations in Թ૜. What is the value of ሺ࡭࡮ሻ࢞ where ࢞ൌ൥૙൩? ૙ Since a composition of linear transformations in Թ૜ is also a linear transformation, we know that applying it to the origin will result in no change.

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A STORY OF FUNCTIONS Lesson 10 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 10: Matrix Multiplication Is Not Commutative

Student Outcomes Students understand that, unlike multiplication of numbers, matrix multiplication is not commutative.

Lesson Notes In this lesson, students first demonstrate that linear transformations in the coordinate plane do not commute. Since each linear transformation corresponds to a ʹൈʹ matrix, students see that the corresponding matrix multiplication must also fail to commute. Students verify this fact algebraically by multiplying matrices in both orders. Work is then extended to coordinates in 3-D space to prove that multiplication of ͵ൈ͵ matrices is also not commutative.

Classwork Opening Exercise (7 minutes) Allow students time to work on the Opening Exercise independently before discussing as a Scaffolding: class. Solicit volunteers to demonstrate each transformation in order to illustrate that the Consider providing result is not the same. students who struggle with transformations a Opening Exercise coordinate grid, a ૙ transparency, and a fine- Consider the vector ܞൌቀ ቁ. ૚ point dry erase marker to If ܞ is rotated ૝૞ι counterclockwise and then reflected across the ࢟-axis, what is the resulting perform each vector? transformation. ξ૛ Students place the transparency on the grid ۊ ૛ ۇ ۋ ۈ ξ૛ and draw vector ሺͲǡ ͳሻ; ૛ then, they perform the ی ۉ rotation with the If ܞ is reflected across the ࢟-axis and then rotated ૝૞ι counterclockwise about the origin, what is transparency to help the resulting vector? determine the location of ξ૛ the transformed vector. െ ۊ ૛ ۇ ۋ ۈ ξ૛ ی ૛ ۉ

Did these linear transformations commute? Explain.

ࡾ૙ǡ૝૞ιל ࡾ࢟ ്ࡾ࢟ל No ࡾ૙ǡ૝૞ι

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Do you think linear transformations, in general, commute? Explain why or why not. à No, the resulting vector varies depending on the order in which the transformations are applied.

଴ǡସହιܴל ௬ ്ܴ௬ܴל Write this on the board: ܴ଴ǡସହι What matrix corresponds to a Ͷͷι counterclockwise rotation? ሺͶͷιሻ െሺͶͷιሻ à ൬ ൰ ሺͶͷιሻ ሺͶͷιሻ ?What matrix corresponds to a reflection across the ݕ-axis െͳ Ͳ à ቀ ቁ Ͳͳ What does the fact that the linear transformations failed to commute tell us about the corresponding matrix multiplication? à The multiplication of the two transformation matrices must also fail to commute.

Exercise 1 (5 minutes) Allow students time to verify algebraically that the matrix multiplication fails to commute. Share results as a class.

Exercises

1. Let ࡭ equal the matrix that corresponds to a ૝૞ι rotation counterclockwise and ࡮ equal the matrix that corresponds to a reflection across the ࢟-axis. Verify that matrix multiplication does not commute by finding the products ࡭࡮ and ࡮࡭.

ξ૛ ξ૛ ξ૛ ξ૛ െ െ െ ۊ ૛ ૛ ۇ െ૚ ૙ ۊ ૛ ૛ ۇ ۋ ۈቀ ቁൌ ۋ ۈ ࡭࡮ ൌ ξ૛ ξ૛ ૙૚ ξ૛ ξ૛ െ ی ૛ ૛ ۉ ی ૛ ૛ ۉ

ξ૛ ξ૛ ξ૛ ξ૛ െ െ ۊ ૛ ૛ ۇ ۊ ૛ ૛ ۇ െ૚ ૙ ۋ ۈ ൌ ۋ ۈ ࡮࡭ ൌ ቀ ቁ ૙૚ξ૛ ξ૛ ξ૛ ξ૛ ی ૛ ૛ ۉ ی ૛ ૛ ۉ ࡭࡮ ് ࡮࡭

What can be said about multiplication of two ʹൈʹ matrices? à Unlike multiplication of two numbers, it is not commutative. What do you think about multiplication of two ͵ൈ͵ matrices? Why? à I do not think the multiplication would be commutative either. Each ͵ൈ͵ matrix represents a transformation in 3-D space. These transformations will not be commutative; therefore, the corresponding multiplication will not be commutative.

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Exercise 2 (7 minutes) Allow students time to verify algebraically that the matrix multiplication fails to commute. Share results as a class.

૙െ૚૙ ૚૙ ૙ 2. Let ࡭ൌ൭૚૙૙൱ and ࡮ൌ൭૙૙െ૚൱. Verify that matrix multiplication does not commute by finding the ૙૙૚ ૙૚ ૙ products ࡭࡮ and ࡮࡭. ૙૙૚ ࡭࡮ ൌ ൭૚૙૙൱ ૙૚૙ ૙െ૚૙ ࡮࡭ ൌ ൭૙૙െ૚൱ ૚૙ ૙ ࡭࡮ ് ࡮࡭

Summarize to a partner what can be said about linear transformations and, therefore, matrix multiplication. à Neither of them commutes. The order in which transformations are applied affects the outcome; therefore, the order in which square matrices are multiplied affects the product.

Exercise 3 (7 minutes) In this exercise, students find that the products are equal. Students should realize that although there are special cases when matrix multiplication appears to be commutative, matrix multiplication is not commutative, because, as seen in previous examples, there are many cases where ܣܤ does not equal ܤܣ (for square matrices ܣ and ܤ). Allow students time to work with a partner. Share results as a class.

Scaffolding: ૙ Exercise 4 could be varied 3. Consider the vector ܞൌ൭૙൱. by providing each student ૚ a. If ܞ is rotated ૝૞ι counterclockwise about the ࢠ-axis and then reflected across the with a card that has a ࢞࢟-plane, what is the resulting vector? matrix in the form ܽെܾ ૙ ቀ ቁ. Students could ൭ ૙ ൱ ܾܽ െ૚ then pair up with another student and work the b. If ܞ is reflected across the ࢞࢟-plane and then rotated ૝૞ι counterclockwise about the exercise. ࢠ-axis, what is the resulting vector? Advanced students may ૙ try working with the ൭ ૙ ൱ ܽെܾ general matrices ቀ ቁ െ૚ ܾܽ ܿെ݀ and ቀ ቁ. ݀ܿ ʹെ͵ Show the matrix ቀ ቁ, ͵ʹ and ask students, “Explain how you know this is in ܽെܾ the form ቀ ቁ.” ܾܽ

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c. Verify algebraically that the product of the two corresponding matrices is the same regardless of the order in which they are multiplied.

ඥ૛ ඥ૛ െ ૙ ۊ ૛ ૛ ۇ Let ࡭ൌ ඥ૛ ඥ૛ (the matrix representing ૝૞ι rotation about the ࢠ-axis). ۋ૙ ۈ ૛ ૛ ی૙૙૚ ۉ ૚૙ ૙ Let ࡮ൌ൭૙૚ ૙൱ (the matrix representing a reflection across the ࢞࢟-plane). ૙૙െ૚ ξ૛ ξ૛ െ ૙ ۊ ૛ ૛ ۇ ۋ ξ૛ ξ૛ۈ ࡭࡮ ൌ ࡮࡭ ൌ ૙ ૛ ૛ ی૙૙െ૚ ۉ

What can be said about the order in which the transformations are applied in this particular example? à The resulting point is the same regardless of the order in which the transformations are applied. What can be said about the corresponding matrix multiplication? à In this case, ܣܤ ൌ ܤܣ, but this is a particular case. Summarize with a partner what can be said about matrix multiplication. à Matrix multiplication is not commutative. However, there are instances where the products are the same. This would correspond geometrically to cases in which the order that transformations are applied does not affect the location of the point.

Exercise 4 (9 minutes) Allow students time to work with a partner. Have several groups show their results from part (a) in order to establish that multiplication of any two matrices in this form has the same product. Stress that this is a special case and does not mean that matrix multiplication is commutative.

ࢇെ࢈ 4. Write two matrices in the form ቀ ቁ. ࢈ࢇ a. Verify algebraically that the products of these two matrices are equal. ૛െ૚ ૜െ૛ Let ࡭ൌቀ ቁ and ࡮ൌቀ ቁ. ૚૛ ૛૜ ૝െૠ ૝െૠ ࡭࡮ ൌ ቀ ቁ and ࡮࡭ ൌ ቀ ቁ ૠ૝ ૠ૝ ࡭࡮ ൌ ࡮࡭

b. Write each of your matrices as a complex number. Find the product of the two complex numbers.

ሺ૛൅࢏ሻሺ૜൅૛࢏ሻ ൌ૝൅ૠ࢏

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ࢇെ࢈ c. Why is it the case that any two matrices in the form ቀ ቁ have products that are equal regardless of the ࢈ࢇ order in which they are multiplied?

Matrices in this form represent the geometric effect of complex multiplication. Multiplying a complex number ࢠ by a complex number ࢻ and then by a complex number ࢼ gives the same answer as multiplying by ࢼ and then ࢻ; that is, ࢼሺࢻࢠሻ ൌࢻሺࢼࢠሻ; thus, the corresponding matrix multiplication yields the same product.

What did you discover about the matrices above? (Allow several groups to share their work.) à ܣܤ ൌ ܤܣ Does this mean matrix multiplication is commutative? Explain. ܽെܾ à No, this is a special case because the matrices are in the form ቀ ቁ. ܾܽ What is the relationship between these matrices and complex numbers? à Matrices in this form can be used to represent a corresponding complex number. Multiplying these matrices is the same as multiplying two complex numbers. Is the multiplication of two complex numbers commutative? ܽെܾ à Yes, two matrices in the form ቀ ቁ have the same product, but this does not mean that matrix ܾܽ multiplication is commutative.

Closing (5 minutes) Ask students to summarize in writing what can be said about multiplication of two square matrices and then share with a partner. Discuss key points as a class. Is matrix multiplication commutative? à No How does what we know about linear transformations support this notion? à The product of the two matrices represents two linear transformations. Since linear transformations do not commute, matrix multiplication must also fail to commute. Are there instances when, for two matrices ܣ and ܤ, ܣܤ ൌ ܤܣ? Explain. à Yes, particularly when the two matrices represent complex numbers. The product of the two matrices is also a complex number. Since complex multiplication is commutative, matrix multiplication appears to be commutative in this instance.

Exit Ticket (5 minutes)

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Name Date

Lesson 10: Matrix Multiplication Is Not Commutative

Exit Ticket

1. Let ܣ be the matrix representing a rotation about the origin by ͸Ͳι and ܤ be the matrix representing a reflection .across the ݔ-axis a. Give two reasons why ܣܤ ് ܤܣ.

ͳ .ݔሻܣሺܤ ݔሻ andܤሺܣ b. Let ݔൌቀ ቁ. Evaluate െͳ

͵ ݔሻ where ݔൌቀ ቁ. Explainܣሺܤ ݔሻ andܤሺܣ that represent linear transformations where ,ܤ ,ܣ ,Write two matrices .2 ʹ why the products are the same.

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Exit Ticket Sample Solutions

1. Let ࡭ be the matrix representing a rotation about the origin by ૟૙ι and ࡮ be the matrix representing a reflection across the ࢞-axis. a. Give two reasons why ࡭࡮ ് ࡮࡭.

The linear transformations these matrices represent do not commute, and the matrices themselves do not commute.

૚ ξ૜ ૚ ξ૜ ૚ ξ૜ െ െ ۊ ૛ ૛ ۇ ۊ ૛ ૛ ۇ ૚૙ ۊ ૛ ૛ ۇ ۋ ۈ ࡮࡭ ൌ; ۋ ۈ ࡮ ൌ ቀ ቁ ;࡭࡮ ൌ; ۋ ۈ࡭ൌ ξ૜ ૚ ૙െ૚ ξ૜ ૚ ξ૜ ૚ െ െ െ ی ૛ ૛ ۉ ی૛ ૛ ۉ ی ૛ ૛ ۉ

૚ b. Let ࢞ൌቀ ቁ. Evaluate ࡭ሺ࡮࢞ሻ and ࡮ሺ࡭࢞ሻ. െ૚ ૚െξ૜ ۊ ૛ ۇ ۋ ۈ࡭ሺ࡮࢞ሻ ൌ ሺ࡭࡮ሻ࢞ൌ ૚൅ξ૜ ی ૛ ۉ ૚൅ξ૜ ۊ ૛ ۇ ۋ ۈ࡮ሺ࡭࢞ሻ ൌ ሺ࡮࡭ሻ࢞ൌ ૚െξ૜ ی ૛ ۉ

૜ 2. Write two matrices, ࡭, ࡮, that represent linear transformations where ࡭ሺ࡮࢞ሻ and ࡮ሺ࡭࢞ሻ where ࢞ൌቀ ቁ. Explain ૛ why the products are the same.

Answers may vary. Examples include any two rotations about the origin, any two matrices representing complex numbers, and a dilation with any other matrix (since scalars commute with all matrices). For example: ૛૙ ૚െ૚ ࡭ൌቀ ቁ ,࡮ ൌ ቀ ቁ ૙૛ ૚૚ ૛െ૛ ࡭࡮ ൌ ࡮࡭ ൌ ቀ ቁ ૛૛ ૛ ࡭ሺ࡮࢞ሻ ൌ࡮ሺ࡭࢞ሻ ൌቀ ቁ ૚૙ ࢇെ࢈ Matrices in the form ቀ ቁ represent the geometric effect of complex multiplication. Multiplying a complex ࢈ࢇ number ࢠby a complex number ࢻ and then by a complex number ࢼ gives the same answer as multiplying by ࢼ and then ࢻ; that is, ࢼሺࢻࢠሻ ൌࢻሺࢼࢠሻ; thus, the corresponding matrix multiplication yields the same product.

Problem Set Sample Solutions

െ૛ 1. Let ࡭ be the matrix representing a dilation of ૛, ࡮ the matrix representing a rotation of ૜૙ι, and ࢞ൌቀ ቁ. ૜ a. Evaluate ࡭࡮.

૜ െ૚ ൬ξ ൰ ૚ ξ૜

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b. Evaluate ࡮࡭.

૜ െ૚ ൬ξ ൰ ૚ ξ૜

c. Find ࡭࡮࢞.

െ૛ ૜ െ૜ ൬ ξ ൰ െ૛ ൅ ૜ξ૜

d. Find ࡮࡭࢞.

െ૛ ૜ െ૜ ൬ ξ ൰ െ૛ ൅ ૜ξ૜

2. Let ࡭ be the matrix representing a reflection across the line ࢟ൌ࢞, ࡮ the matrix representing a rotation of ૢ૙ι, and ૚ ࢞ൌቀ ቁ. ૙ a. Evaluate ࡭࡮. ૚૙ ቀ ቁ ૙െ૚

b. Evaluate ࡮࡭. െ૚ ૙ ቀ ቁ ૙૚

c. Find ࡭࢞. ૙ ቀ ቁ ૚

d. Find ࡮࢞. ૙ ቀ ቁ ૚

e. Find ࡭࡮࢞. ૚ ቀ ቁ ૙

f. Find ࡮࡭࢞. െ૚ ቀ ቁ ૙

g. Describe the linear transformation represented by ࡭࡮.

࡭࡮ represents a reflection across the ࢞-axis.

h. Describe the linear transformation represented by ࡮࡭.

࡮࡭ represents a reflection across the ࢟-axis.

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3. Let matrices ࡭ǡ ࡮ represent scalars. Answer the following questions. a. Would you expect ࡭࡮ ൌ ࡮࡭? Explain why or why not.

Yes, scalars are real numbers, and real numbers commute.

ࢇ૙ ࢈૙ b. Let ࡭ൌቀ ቁ and ࡮ൌቀ ቁ. Show ࡭࡮ ൌ ࡮࡭ through matrix multiplication, and explain why. ૙ࢇ ૙࢈ ࢇ࢈ ૙ ࢈ࢇ ૙ ࡭࡮ ൌ ቀ ቁൌቀ ቁൌ࡮࡭ ૙ࢇ࢈ ૙࢈ࢇ ࢇെ࢈ These matrices represent complex numbers and are both of the form ቀ ቁ, so their products will be the ࢈ࢇ same regardless of the order.

4. Let matrices ࡭ǡ ࡮ represent complex numbers. Answer the following questions. ࢇെ࢈ ࢉെࢊ a. Let࡭ ൌ ቀ ቁ and ࡮ൌቀ ቁ. Show ࡭࡮ ൌ ࡮࡭ through matrix multiplication. ࢈ࢇ ࢊࢉ ࢇࢉ െ ࢈ࢊ െࢇࢊ െ ࢈ࢉ ࡭࡮ ൌ ቀ ቁ ࢈ࢉ ൅ ࢇࢊ െ࢈ࢊ ൅ ࢇࢉ ࢉࢇ െ ࢊ࢈ െࢉ࢈ െ ࢊࢇ ࡮࡭ ൌ ቀ ቁ ࢊࢇ ൅ ࢉ࢈ െࢊ࢈ ൅ ࢉࢇ They are equivalent.

b. Would you expect ࡭࡯ ൌ ࡯࡭ for any matrix ࡯? Explain.

No, matrix multiplication is not commutative.

࢞࢟ c. Let ࡯ be any ૛ൈ૛ matrix, ቀ ቁ. Show ࡭࡯ ് ࡯࡭ through matrix multiplication. ࢠ࢝ ࢇ࢞ െ ࢈ࢠ ࢇ࢟ െ ࢈࢝ ࡭࡯ ൌ ൬ ൰ ࢈࢞൅ࢇࢠ ࢈࢟൅ࢇ࢝ ࢇ࢞൅࢈ࢠ ࢇ࢟൅࢈࢝ ࡯࡭ ൌ ቀ ቁ ࢇࢠ ൅ ࢈࢞ െ࢈ࢠ ൅ ࢇ࢝

d. Summarize your results from Problems 3 and 4.

Matrices representing scalars commute with any other matrix, and matrices representing complex number multiplication only commute when multiplying by other complex numbers (or if ࢈ൌ૙).

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5. Quaternions are a number system that extends to complex numbers discovered by William Hamilton in 1843. Multiplication of quaternions is not commutative and is defined as the quotient of two vectors. They are useful in three-dimensional rotation calculations for computer graphics. Quaternions are formed the following way: ࢏૛ ൌ࢐૛ ൌ࢑૛ ൌ࢏࢐࢑ൌെ૚ Multiplication by െ૚ and ૚ works normally. We can represent all possible multiplications of quaternions through a table:

ൈ ࢏ ࢐ ࢑

࢏ െ૚ ࢑ െ࢐

࢐ െ࢑ െ૚ ࢏

࢑ ࢐ െ࢏ െ૚

a. Is multiplication of quaternions commutative? Explain why or why not.

No, it is not commutative. For example, ࢏࢐ ൌ ࢑, but ࢐࢏ ൌ െ࢑.

b. Is multiplication of quaternions associative? Explain why or why not.

Yes, it is associative. ࢏ሺ࢐࢑ሻ ൌ࢏ሺ࢏ሻ ൌെ૚, and ሺ࢏࢐ሻ࢑ൌሺ࢑ሻ࢑ൌെ૚ ૚ and െ૚ work normally.

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Lesson 11: Matrix Addition Is Commutative

Student Outcomes Students prove both geometrically and algebraically that matrix addition is commutative.

Lesson Notes In Topic A, matrices were interpreted as representing network diagrams. In that context, an arithmetic system for matrices was natural. Given two matrices ܣ and ܤ of equal dimensions, the matrix product ܣܤ and the matrix sum ܣ൅ܤ were defined. Both of these operations had a meaning within the context of networks. Topic B returned to the interpretation of matrices as representing the geometric effect of linear transformations from Module 1. The matrix product ܣܤ has meaning in this context; it is the composition of transformations. This lesson explores the question, “Can we give matrix addition meaning in the setting of geometric transformations?”

Classwork Opening Exercise (5 minutes) Allow students time to work on the Opening Exercise independently before discussing as a class.

Opening Exercise

Kiamba thinks ࡭൅࡮ൌ࡮൅࡭ for all ૛ൈ૛ matrices. Rachel thinks it is not always true. Who is correct? Explain. Scaffolding: ࢇ ࢇ ࢈ ࢈ Provide students with concrete Let ࡭ൌቀ ૚૚ ૚૛ቁ and ࡮ൌ൬ ૚૚ ૚૛൰. ࢇ૛૚ ࢇ૛૛ ࢈૛૚ ࢈૛૛ examples if necessary. What is the sum of ࡭൅࡮? െʹ ͳ ͶͲ ܣൌቀ ቁ and ܤൌቀ ቁ ࢇ ൅࢈ ࢇ ൅࢈ ͵ʹ െͳ ͸ ࡭൅࡮ൌ൬ ૚૚ ૚૚ ૚૛ ૚૛൰ ࢇ૛૚ ൅࢈૛૚ ࢇ૛૛ ൅࢈૛૛ ʹͳ ܣ൅ܤൌܤ൅ܣൌቀ ቁ ࢈૚૚ ൅ࢇ૚૚ ࢈૚૛൅ࢇ૚૛ ʹͺ ࡮൅࡭ൌ൬ ൰ ࢈૛૚൅ࢇ૛૚ ࢈૛૛ ൅ࢇ૛૛ The two matrices must be equal because each of the sums must be equal according to the commutative property of addition of real numbers. Kiamba is correct.

Can we say that matrix addition is commutative? à Yes, the order in which we add the matrices does not change the sum. Will this hold true if we change the dimensions of the matrices being added? à Yes, regardless of the size of the matrices, the two sums (ܣ൅ܤ and ܤ൅ܣ) would still be the same. (Note: Demonstrate with two ͵ൈ͵ matrices if students seem unsure.) So, we see that matrix addition is commutative, but we still have not determined the geometric meaning of matrix addition.

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Exercise 1 (12 minutes) Allow students time to work in groups on Exercise 1. Optionally, give students colored pencils or a transparency and fine-point dry erase marker to mark their points.

Exercises

1. In two-dimensional space, let ࡭ be the matrix representing a rotation about the origin through an angle of ૝૞ι, and let ࡮ be the matrix representing a reflection about the ࢞-axis. ૚ Let ࢞ be the point ቀ ቁ. ૚ a. Write down the matrices࡭,࡮,and ࡭൅࡮.

ξ૛ ξ૛ െ ૚૙ ۊ ૛ ૛ ۇ ࡮ ൌ ቀ ቁ ۋ ۈ࡭ൌ ξ૛ ξ૛ ૙െ૚ ی ૛ ૛ ۉ ξ૛ ξ૛ ൅૚ െ ۊ ૛ ૛ ۇ ۋ ۈ࡭൅࡮ൌ ξ૛ ξ૛ െ૚ ی ૛ ૛ ۉ

b. Write down the image points of ࡭࢞, ࡮࢞, and ሺ࡭൅࡮ሻ࢞, and plot them on graph paper. ૙ ૚ ૚ ࡭࢞ ൌ ൬ ൰ ࡮࢞ ൌ ቀ ቁ ሺ࡭൅࡮ሻ࢞ൌ൬ ൰ ξ૛ െ૚ ξ૛ െ૚

c. What do you notice about ሺ࡭൅࡮ሻ࢞ compared to ࡭࢞ and ࡮࢞?

The point ሺ࡭൅࡮ሻ࢞ is equal to the sum of the points ࡭࢞ and ࡮࢞ by the distributive property.

?ሻሺݔሻܤ൅ܣሻሺݔሻ ൌ ሺܣ൅ܤݔ ൌ ሺܣ ݔ ൅ܤ ሻሺݔሻ, does it follow thatܤ൅ܣݔ ൌ ሺܤ ݔ ൅ܣ Since ݔ must both map to the same point since we know that the addition of pointsܣ ݔ ൅ܤ ݔ andܤ ݔ ൅ܣ ,à Yes ݔܣ ݔ ൅ܤ ,ሻሺݔሻ by the distributive propertyܤ൅ܣݔ maps to the point ሺܤ ݔ ൅ܣ is commutative. Since .ݔܣ ݔ ൅ܤ ݔ ൌܤ ݔ ൅ܣ ሻሺݔሻ sinceܤ൅ܣmust also map to the point ሺ What does this say about matrix addition? à It confirms what we saw in the Opening Exercise. Matrix addition is commutative.

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Exercise 2 (13 minutes) Allow students time to work in groups on Exercise 2 before discussing as a class.

2. For three matrices of equal size, ࡭, ࡮, and ࡯, does it follow that ࡭൅ሺ࡮൅࡯ሻ ൌ ሺ࡭൅࡮ሻ ൅࡯? a. Determine if the statement is true geometrically. Let ࡭ be the matrix representing a reflection across the ࢟-axis. Let ࡮ be the matrix representing a counterclockwise rotation of ૜૙ι. Let ࡯ be the matrix representing a reflection about the ࢞-axis. ૚ Let ࢞ be the point ቀ ቁ. ૚

ξ૜ ૚ െ ۊ૛ ૛ ۇ െ૚ ૙ ۋ ۈ ࡭ൌቀ ቁ ࡮ ൌ ૙૚ ૚ ξ૜ ی ૛ ૛ ۉ ૚૙ ࡯ ൌ ቀ ቁ ૙െ૚

ξ૜ െ૛ ૚ െ ۊ ૛ ૛ ۇ ۋ ۈ࡭൅࡮ൌ ૚ ξ૜ ൅૛ ی ૛ ૛ ۉ ξ૜ ൅૛ ૚ െ ۊ ૛ ૛ ۇ ۋ ۈ࡮൅࡯ൌ ૚ ξ૜ െ૛ ی ૛ ૛ ۉ

From the graph, we see that ࡭࢞ ൅ ሺ࡮࢞ ൅ ࡯࢞ሻ ൌ ሺ࡭࢞ ൅ ࡮࢞ሻ ൅ ࡯࢞.

b. Confirm your results algebraically.

ξ૜ ൅૛ ૚ ξ૜ ૚ െ െ ۊ૛ ૛ ۇ ۊ ૛ ૛ ۇ െ૚ ૙ ۋ ۈ ൌ ۋ ۈ࡭൅ሺ࡮൅࡯ሻ ൌቀ ቁ൅ ૙૚ ૚ ξ૜ െ૛ ૚ ξ૜ ی ૛ ૛ ۉ ی ૛ ૛ ۉ

ξ૜ െ૛ ૚ ξ૜ ૚ െ െ ۊ૛ ૛ ۇ ૚૙ ۊ ૛ ૛ ۇ ۋ ۈ൅ቀ ቁൌ ۋ ۈሺ࡭൅࡮ሻ ൅࡯ൌ ૚ ξ૜ ൅૛ ૙െ૚ ૚ ξ૜ ی ૛ ૛ ۉ ی ૛ ૛ ۉ

c. What do your results say about matrix addition?

Matrix addition is associative.

What does this exercise tell us about matrix addition? à It is associative.

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How did the graph support the idea that matrix addition is associative? .൯ݔ mapped to the same pointܥሻ ൅ܤ൅ܣሻ൯ݔ and ൫ሺܥ൅ܤ൅ ሺ ܣà ൫ Would this still be true if we used matrices of different (but still equal) dimensions? à Yes, the addition would still be associative. For example, we could do the same type of operations using ͵ൈ͵ matrices, but geometrically it would represent points in 3-D space rather than on a coordinate plane. Could we say ܣ൅ሺܤ൅ܥሻ ൌܥ൅ሺܣ൅ܤሻ? à Yes, we just demonstrated that ܣ൅ሺܤ൅ܥሻ ൌ ሺܣ൅ܤሻ ൅ܥ. We can then say that ሺܣ൅ܤሻ ൅ܥൌܥ൅ሺܣ൅ܤሻbecause we know that matrix addition is commutative.

Exercises 3–6 (5 minutes) Allow students time to work in groups on Exercises 3–6 before discussing as a class.

࢞ ૙ 3. If ࢞ൌቆ࢟ቇ, what are the coordinates of a point ࢟ with the property ࢞൅࢟ if the origin ࡻൌ൭૙൱? ࢠ ૙ െ࢞ ࢟ൌቆെ࢟ቇ െࢠ

૚૚ െ૞ ૛ 4. Suppose ࡭ൌ൭െ૜૝ ૟ ૚ૢ൱ and matrix ࡮ has the property that ࡭࢞ ൅ ࡮࢞ is the origin. What is the matrix ࡮? ૡ െ૞૝૛ ૙ െ૚૚ ૞ െ૛ ࡮ൌ൭૜૝ െ૟ െ૚ૢ൱ െૡ ૞૝૛ ૙

5. For three matrices of equal size, ࡭, ࡮, and ࡯, where ࡭ represents a reflection across the line ࢟ൌ࢞, ࡮ represents a ૚ counterclockwise rotation of ૝૞ι, ࡯ represents a reflection across the ࢟-axis, and ࢞ൌቀ ቁ: ૛ a. Show that matrix addition is commutative: ࡭࢞ ൅ ࡮࢞ ൌ ࡮࢞ ൅ ࡭࢞. ૙െ૚ ૚ െ૛ ࡭࢞ ൌ ቀ ቁቀ ቁൌቀ ቁ ૚૙ ૛ ૚ ξ૛ ξ૛ ξ૛ െ െ ξ૛ ۊ ૛ ۇ ૚ ۊ ૛ ૛ ۇ ۋ ۈቀ ቁൌ ۋ ۈ ࡮࢞ ൌ ξ૛ ξ૛ ૛ ξ૛ ൅ ξ૛ ی ૛ ۉ ی ૛ ૛ ۉ ξ૛ െ૛ ൅ െ ξ૛ ۊ ૛ ۇ ൌ࡮࢞൅࡭࢞ ۋ ۈ ࡭࢞ ൅ ࡮࢞ ൌ ξ૛ ૚൅ ൅ ξ૛ ی ૛ ۉ

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b. Show that matrix addition is associative: ࡭࢞ ൅ ሺ࡮࢞ ൅ ࡯࢞ሻ ൌ ሺ࡭࢞ ൅ ࡮࢞ሻ ൅࡯࢞. െ૚ ૙ ૚ െ૚ ࡯࢞ ൌ ቀ ቁቀ ቁൌቀ ቁ ૙૚૛ ૛ ξ૛ െ ξ૛ െ૚ :Scaffolding ۊ ૛ ۇ െ૛ ۋ ۈ࡭࢞ ൅ ሺ࡮࢞ ൅ ࡯࢞ሻ ൌቀ ቁ൅ ૚ ξ૛ ൅ ξ૛ ൅૛ Provide early finishers with this ૛ :challenge question ی ۉ ξ૛ ݔ ൌ Ͳ for all ݔ, must everyܣ െ૛ ൅ െ ξ૛ If െ૚ ۊ ૛ ۇ ; ൅ቀ ቁ ۋ ۈሺ࡭࢞ ൅ ࡮࢞ሻ ൅࡯࢞ൌ ξ૛ ૛ entry of ܣ be Ͳ? ૚൅ ൅ ξ૛ ی ૛ ۉ ξ૛ െ૜ ൅ െ ξ૛ ۊ ૛ ۇ ൌ ሺ࡭࢞ ൅ ࡮࢞ሻ ൅࡯࢞ ۋ ۈ ࡭࢞ ൅ ሺ࡮࢞ ൅ ࡯࢞ሻ ൌ ξ૛ ૜൅ ൅ ξ૛ ی ૛ ۉ

6. Let ࡭, ࡮, ࡯, andࡰ be matrices of the same dimensions. Use the commutative property of addition of two matrices to prove ࡭൅࡮൅࡯ൌ࡯൅࡮൅࡭.

If we treat ࡭൅࡮ as one matrix, then: ሺ࡭൅࡮ሻ ൅࡯ൌ࡯൅ሺ࡭൅࡮ሻ ൌ࡯൅ሺ࡮൅࡭ሻ ൌ࡯൅࡮൅࡭

Closing (5 minutes) Discuss the key points of the lesson as a class. What can be said about matrix addition? à Matrix addition, like addition of numbers, is both commutative and associative. What is the geometric meaning of matrix addition? Illustrate with an example. .ݔܤ ݔ ൅ܣ ሻݔ maps to the same point asܤ ൅ ܣà ሺ ͳͲ ͳͲ ͳ ʹͲ ,ൌቀ ቁܤ൅ܣ .ൌቀ ቁ, and ݔൌቀ ቁܤ ,ൌቀ ቁܣ .à Answers may vary Ͳെͳ Ͳെͳ ͳ Ͳെʹ ʹ ͳ ͳ ʹ ,ݔ ൌ ቀ ቁ. Thereforeܤ ݔ ൅ܣ ݔ ൌ ቀ ቁ, andܤ ,ݔ ൌ ቀ ቁܣ ,ሻݔൌቀ ቁܤ൅ܣሺ െʹ െͳ െͳ െʹ .ݔܤݔ൅ܣሻݔൌܤ൅ܣሺ

Exit Ticket (5 minutes)

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Name Date

Lesson 11: Matrix Addition Is Commutative

Exit Ticket

͵ ʹʹ െͳ Ͳ .ൌ ቀ ቁ ܤൌቀ ቁ, andܣ ,Let ݔൌቀ ቁ .1 െͳ ʹͲ Ͳെͳ .ሻݔ on the axes belowܤ൅ܣݔ, and ሺܤ ,ݔܣ a. Find and plot the points

.ݔܣݔ൅ܤݔൌܤݔ൅ܣ :b. Show algebraically that matrix addition is commutative

ݔݕ ܾܽ 2. Let ܣൌቀ ቁandܤ ൌ ቀ ቁ. Prove ܣ൅ܤൌܤ൅ܣ. ݖݓ ݀ܿ

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Exit Ticket Sample Solutions

૜ ૛૛ െ૚ ૙ 1. Let ࢞ൌቀ ቁ, ࡭ൌቀ ቁ, and࡮ ൌ ቀ ቁ. െ૚ ૛૙ ૙െ૚ a. Find and plot the points ࡭࢞, ࡮࢞, and ሺ࡭൅࡮ሻ࢞ on the axes below.

b. Show algebraically that matrix addition is commutative: ࡭࢞ ൅ ࡮࢞ ൌ ࡮࢞ ൅ ࡭࢞. ૝ െ૜ ૚ ૚ ࡭࢞ ൌ ቀ ቁ, ࡮࢞ ൌ ቀ ቁ, ࡭࢞ ൅ ࡮࢞ ൌ ቀ ቁ, and࡮࢞ ൅ ࡭࢞ ൌ ቀ ቁ ࡭࢞ ൅ ࡮࢞ ൌ ࡮࢞ ൅ ࡭࢞ ૟ ૚ ૠ ૠ

ࢇ࢈ ࢞࢟ 2. Let ࡭ൌቀ ቁ and ࡮ൌቀ ቁ. Prove ࡭൅࡮ൌ࡮൅࡭. ࢉࢊ ࢠ࢝ ࢇ࢈ ࢞࢟ ࡭൅࡮ൌቀ ቁ൅ቀ ቁ ࢉࢊ ࢠ࢝ ࢇ൅࢞ ࢈൅࢟ ൌቀ ቁ ࢉ൅ࢊ ࢠ൅࢝ ࢞൅ࢇ ࢟൅࢈ ൌቀ ቁ ࢊ൅ࢉ ࢝൅ࢠ ࢞࢟ ࢇ࢈ ൌቀ ቁ൅ቀ ቁ ࢠ࢝ ࢉࢊ ൌ࡮൅࡭

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Problem Set Sample Solutions

1. Let ࡭ be a matrix transformation representing a rotation of ૝૞ι about the origin and ࡮ be a reflection across the ࢟-axis. Let ࢞ൌሺ૜ǡ૝ሻ. a. Represent ࡭and࡮ as matrices, and find ࡭൅࡮.

ξ૛ ξ૛ ξ૛ ξ૛ െ െ૚ ൅ െ ۊ ૛ ૛ ۇ െ૚ ૙ ۊ ૛ ૛ ۇ ۋ ۈ ࡮ ൌ ቀ ቁ ;࡭ ൅ ࡮ ൌ; ۋ ۈ࡭ൌ ξ૛ ξ૛ ૙૚ ξ૛ ξ૛ ૚൅ ی ૛ ૛ ۉ ی ૛ ૛ ۉ

b. Represent ࡭࢞and ࡮࢞ as matrices, and find ሺ࡭ ൅ ࡮ሻ࢞.

ξ૛ ξ૛ െ െ૜ െ ۊ ૛ ۇ െ૜ ۊ ૛ ۇ ۋ ۈ࡮࢞ ൌ ቀ ቁ ;ሺ࡭൅࡮ሻ࢞ൌ; ۋ ۈ ࡭࢞ ൌ ૠξ૛ ૝ ૠξ૛ ૝൅ ی ૛ ۉ ی ૛ ۉ

c. Graph your answer to part (b).

See graph in part (d).

d. Draw the parallelogram containing ࡭࢞, ࡮࢞, and ሺ࡭ ൅ ࡮ሻ࢞.

2. Let ࡭ be a matrix transformation representing a rotation of ૜૙૙ι about the origin and ࡮ be a reflection across the ࢞-axis. Let ࢞ൌሺ૛ǡെ૞ሻ. a. Represent ࡭and࡮ as matrices, and find ࡭൅࡮.

૚ ξ૜ ૜ ξ૜ ۊ ૛ ૛ ۇ ૚૙ ۊ ૛ ૛ ۇ ۋ ۈ ࡮ ൌ ቀ ቁ ;࡭ ൅ ࡮ ൌ; ۋ ۈ࡭ൌ ξ૜ ૚ ૙െ૚ ξ૜ ૚ െ െ െ ی૛ ૛ ۉ ی ૛ ૛ ۉ

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b. Represent ࡭࢞and ࡮࢞ as matrices, and find ሺ࡭ ൅ ࡮ሻ࢞.

૞ξ૜ ૞ξ૜ ૚െ ૜െ ۊ ૛ ۇ ૛ ۊ ૛ ۇ ۋ ۈ࡮࢞ ൌ ቀ ቁ ; ሺ࡭൅࡮ሻ࢞ൌ; ۋ ۈ ࡭࢞ ൌ െ૞െ૛ξ૜ ૞ ૞െ૛ξ૜ ی ૛ ۉ ی ૛ ۉ

c. Graph your answer to part (b).

See graph in part (d).

d. Draw the parallelogram containing ࡭࢞, ࡮࢞, and ሺ࡭ ൅ ࡮ሻ࢞.

3. Let ࡭, ࡮, ࡯, and ࡰ be matrices of the same dimensions. a. Use the associative property of addition for three matrices to prove ሺ࡭൅࡮ሻ ൅ ሺ࡯൅ࡰሻ ൌ࡭൅ሺ࡮൅࡯ሻ ൅ࡰ.

If we treat ࡯൅ࡰ as one matrix, then: ሺ࡭൅࡮ሻ ൅ ሺ࡯൅ࡰሻ ൌ ൌ࡭൅൫࡮൅ሺ࡯൅ࡰሻ൯ ൌ࡭൅൫ሺ࡮൅࡯ሻ ൅ࡰ൯ ൌ࡭൅ሺ࡮൅࡯ሻ ൅ࡰ

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b. Make an argument for the associative and commutative properties of addition of matrices to be true for finitely many matrices being added.

For finitely many matrices, we can always use the formula that has been proven already and break the rest of the problem down into that many pieces, just like we did in part (a). Since this is true for any finite number, we could also use the formula that has been proven for one less than whatever number we are trying to prove the property true for.

th th 4. Let ࡭ be a ࢓ൈ࢔ matrix with element in the ࢏ row, ࢐ column ࢇ࢏࢐, and ࡮ be a ࢓ൈ࢔ matrix with element in the th th ࢏ row, ࢐ column ࢈࢏࢐. Present an argument that ࡭൅࡮ൌ࡮൅࡭.

th th The entry in the ࢏ row, ࢐ column of ࡭൅࡮ will be ࢇ࢏࢐ ൅࢈࢏࢐, which is equal to ࢈࢏࢐ ൅ࢇ࢏࢐, which is the entry in the ࢏th row, ࢐th column of ࡮൅࡭. Since this is true for any element of ࡭൅࡮, we have that ࡭൅࡮ൌ࡮൅࡭ for any two matrices of equal dimensions.

.࢟ is defined normallyڄ࢟൅૚; read “࢞ plus ࢟” where ࢞ڄFor integers ࢞, ࢟, define ْ࢞࢟ൌ࢞ .5 a. Is this form of addition commutative? Explain why or why not.

Yes ْ࢞࢟ൌ࢞࢟൅૚ൌ࢟࢞൅૚ൌْ࢟࢞

b. Is this form of addition associative? Explain why or why not.

ْ࢞ሺْ࢟ࢠሻ ൌْ࢞ሺ࢟ࢠ ൅ ૚ሻ ൌ࢞ሺ࢟ࢠ ൅ ૚ሻ ൅૚ൌ࢞࢟ࢠ൅࢞൅૚ ሺْ࢞࢟ሻ ْࢠൌሺ࢞࢟ ൅ ૚ሻ ْࢠൌሺ࢞࢟ ൅ ૚ሻࢠ൅૚ൌ࢞࢟ࢠ൅ࢠ൅૚

.For integers ࢞, ࢟, define ْ࢞࢟ൌ࢞ .6 a. Is this form of addition commutative? Explain why or why not.

No ْ࢞࢟ൌ࢞, and ْ࢟࢞ൌ࢟

b. Is this form of addition associative? Explain why or why not.

Yes ْ࢞ሺْ࢟ࢠሻ ൌْ࢞࢟ൌ࢞, and ሺْ࢞࢟ሻ ْࢠൌْ࢞ࢠൌ࢞

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A STORY OF FUNCTIONS Lesson 12 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 12: Matrix Multiplication Is Distributive and Associative

Student Outcomes Students discover and verify that matrix multiplication is distributive. Students discover and verify that matrix multiplication is associative.

Lesson Notes In this lesson, students use specific matrix transformations on points to show that matrix multiplication is distributive and associative. They then revisit some of the properties of matrices to prove that these properties hold for all matrices under multiplication.

Classwork Opening Exercise (10 minutes) Students review transformations represented by matrices studied in previous lessons in preparation for their work in Lesson 13. Students write the matrix that represents the given transformation. Each transformation should be written on the board, and as the class agrees on the correct matrix that represents that transformation, a student should write the matrix under the transformation heading. Students can do this on paper at their desks, or this can be completed as a Rapid White Board Exchange. This could also be done in pairs or groups, and each group could present one transformation and explain it. The exercise can be modified to a matching activity if students are struggling.

Opening Exercise

Write the ૜ൈ૜ matrix that would represent the transformation listed.

a. No change when multiplying (the multiplicative identity matrix) ૚૙૙ ൥૙૚૙൩ ૙૙૚

b. No change when adding (the additive identity matrix)

૙૙૙ ൥૙૙૙൩ ૙૙૙

c. A rotation about the ࢞-axis of ࣂ degrees ૚૙ ૙ ൥૙܋ܗܛሺࣂሻെܛܑܖሺࣂሻ൩ ૙ ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ

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d. A rotation about the ࢟-axis of ࣂdegrees ܋ܗܛሺࣂሻ ૙ ܛܑܖሺࣂሻ ൥ ૙૚૙൩ െܛܑܖሺࣂሻ ૙ ܋ܗܛሺࣂሻ

e. A rotation about the ࢠ-axis of ࣂdegrees ܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ ૙ ൥ܛܑܖሺࣂሻ ܋ܗܛሺࣂ ૙൩ ૙૙૚

f. A reflection over the ࢞࢟-plane Scaffolding: ૚૙ ૙ ൥૙૚ ૙൩ Students could create a graphic ૙૙െ૚ organizer listing the matrix in one column and the g. A reflection over the ࢟ࢠ-plane transformation represented in െ૚ ૙ ૙ the next column. ൥ ૙૚૙൩ ૙૙૚

h. A reflection over the ࢞ࢠ-plane ૚૙૙ ൥૙െ૚૙൩ ૙૙૚

i. A reflection over ࢟ൌ࢞ in the ࢞࢟-plane ૙૚૙ ൥૚૙૙൩ ૙૙૚

Scaffolding: Example (15 minutes) Students with spatial issues may not be able to This example should be completed by students in pairs for parts (a) through (g). Part (h) see the transformations in should be completed as a teacher-led discussion after debriefing together the results of three dimensions. parts (a) through (g). Students look at transformations in two-dimensional space on a Consider using a graphing point. They predict the transformation, write a matrix that would represent that program to help them transformation, and apply the transformation to the point. Through a series of steps, they visualize the see that matrix multiplication is distributive and associative, revisit properties of matrices, transformations or and prove that these properties hold for all matrices under matrix multiplication. demonstrating each one What are the dimensions of a matrix that represents two-dimensional space? visually in some other way. à ʹൈʹ For advanced learners, have students work in Using your geometric intuition, can you tell me how the point ሺͳǡ ͳሻ would groups to complete this transform if it was rotated ͻͲι clockwise? What are the new coordinates? example with no leading à The ݔ-coordinate would stay the same, and the ݕ-coordinate would questions. They should change signs. ሺͳǡ െͳሻ write up a summary and present their findings during the class discussion.

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What would happen to the point ሺͳǡ ͳሻ if it was rotated ͳͺͲι counterclockwise? What are the new coordinates? à The ݔ- and ݕ-coordinates would change signs. ሺെͳǡ െͳሻ If matrix ܣrepresents a rotation of ͻͲι clockwise, write matrix ܣ. Show your steps. ሺെͻͲιሻ െ ሺെͻͲιሻ Ͳͳ à ܣൌ൤ ൨ൌቂ ቃ ሺെͻͲιሻ ሺെͻͲιሻ െͳ Ͳ .ܤ represents a reflection about the ݔ-axis, write matrix ܤ If matrix ͳͲ à ܤൌቂ ቃ Ͳെͳ .Show your steps .ܥ represents a rotation of ͳͺͲι about the ݕ-axis, write matrix ܥ If matrix ሺͳͺͲιሻ െሺͳͺͲιሻ െͳͲ à ܥൌ൤ ൨ൌቂ ቃ ሺͳͺͲιሻ ሺͳͺͲιሻ Ͳെͳ ͳ ܺൌቂ ቃ; confirm your answers above by performing the transformations ܤܺ and ܥܺ. Were your instincts ͳ correct? If not, analyze your mistakes. ͳ െͳ à ܤܺ ൌ ቂ ቃ; ܥܺ ൌ ቂ ቃ; answers will vary. െͳ െͳ What is ܤܺ ൅ ܥܺ? What did you notice about the results of part (e) and part (g)? Ͳ à ܤܺ ൅ ܥܺ ൌ ቂ ቃ. The results are the same matrix. െʹ Write out the mathematical statement. à ܣሺܤܺ ൅ ܥܺሻ ൌ ሺܣܤሻܺ൅ሺܣܥሻܺ Do you think matrix multiplication is distributive? If so, what would ܣሺܤܺ ൅ ܥܺሻ be equivalent to? à ܣሺܤܺ ൅ ܥܺሻ ൌܣሺܤܺሻ ൅ܣሺܥܺሻ What property would be necessary in order for us to prove matrix multiplication is distributive given what we have already proven? à We have proven for this set of matrices thatܣሺܤܺ ൅ ܥܺሻ ൌ ሺܣܤሻܺ൅ሺܣܥሻܺ, but we need ܣሺܤܺ ൅ ܥܺሻ ൌܣሺܤܺሻ൅ܣሺܥܺሻ. That means ሺܣܤሻܺ൅ሺܣܥሻܺൌܣሺܤܺሻ ൅ܣሺܥܺሻ, which means the associative property must hold. The sum of two matrices is the sum of corresponding elements. What would be the value of ሺܤ൅ܥሻܺ? à ܤܺ ൅ ܥܺ We also know that applying a matrix ܤ and then matrix ܣ is the same as applying the matrix product ܣܤ. So, ܣሺܤܺሻ ൌ? And ܣሺܥܺሻ ൌ? à ሺܣܤሻܺ à ሺܣܥሻܺ Let’s use what we have just stated to explain why ܣሺܤ൅ܥሻ and ܣܤ ൅ ܣܥ have the same geometric effect on a point or points for any matrices ܣ, ܤ, and ܥ. à Applying the product of ܣ and ሺܤ ൅ ܥሻ has the same effect as applying ܣܤ and then ܣܥ and then adding them together. How do we know that ൫ܣሺܤ൅ܥሻ൯ܺ ൌ ܣ൫ሺܤ൅ܥሻܺ൯? à Applying the product of ܣ and ሺܤ ൅ ܥሻ has the same effect as applying ሺܤ ൅ ܥሻ and then ܣ.

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Now, ܣ൫ሺܤ൅ܥሻܺ൯ ൌ ܣሺܤܺ ൅ ܥܺሻ. Why? à The sum of two matrices acts by summing the images. We also know that ܣሺܤܺ ൅ ܥܺሻ ൌܣሺܤܺሻ ൅ܣሺܥܺሻ. Why? à Each matrix represents a linear transformation. Continuing, ܣሺܤܺሻ ൅ܣሺܥܺሻ ൌ ሺܣܤሻܺ൅ሺܣܥሻܺ. Explain. à Applying ܤ and thenܣ is the same as applying ܣܤ. à Applying ܥ and thenܣ is the same as applying ܣܥ. And finally, why does ሺܣܤሻܺ൅ሺܣܥሻܺൌሺܣܤ ൅ ܣܥሻܺ? à The sum of two matrices acts by summing the images. Therefore, what must be true? à ܣሺܤ൅ܥሻ ൌܣܤ൅ܣܥ We have just shown that matrix multiplication is distributive and associative. Take a few minutes, and discuss what we have just proven with your neighbor. Write a summary.

Example

In three-dimensional space, let ࡭ represent a rotation of ૢ૙ι about the ࢞-axis, ࡮ represent a reflection about the ૚ ࢟ࢠ-plane, and ࡯ represent a rotation of ૚ૡ૙ι about the ࢠ-axis. Let ࢄൌ൥૚൩. ૚ a. As best you can, sketch a three-dimensional set of axes and the location of the point ࢄ.

b. Using only your geometric intuition, what are the coordinates of ࡮ࢄ? ࡯ࢄ? Explain your thinking. െ૚ െ૚ ࡮ࢄ ൌ ൥ ૚ ൩; ࡯ࢄ ൌ ൥െ૚൩; answers will vary but could include that when rotating about the ࢞-axis ૢ૙ι, only ૚ ૚ the ࢞-coordinate would change signs; however, when rotating about the ࢠ-axis ૚ૡ૙ι, the ࢞- and ࢟-coordinates would change signs.

c. Write down matrices ࡮ and ࡯, and verify or disprove your answers to part (b). െ૚ ૙ ૙ െ૚ ૙ ૙ െ૚ െ૚ ࡮ൌ൥ ૙૚૙൩;࡯ ൌ ൥ ૙െ૚૙൩;࡮ࢄ ൌ ൥ ૚ ൩;࡯ࢄ ൌ ൥െ૚൩ ૙૙૚ ૙૙૚ ૚ ૚

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d. What is the sum of ࡮ࢄ ൅ ࡯ࢄ? െ૛ ࡮ࢄ ൅ ࡯ࢄ ൌ ൥ ૙ ൩ ૛

e. Write down matrix ࡭, and compute ࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ. ૚૙ ૙ ࡭ൌ൥૙૙െ૚൩ ૙૚ ૙ ૚૙ ૙ െ૛ െ૛ ࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ ൌ ൥૙૙െ૚൩൥ ૙ ൩ ൌ ൥െ૛൩ ૙૚ ૙ ૛ ૙

f. Compute ࡭࡮ and ࡭࡯. ૚૙ ૙ െ૚ ૙ ૙ െ૚ ૙ ૙ ࡭࡮ ൌ ൥૙૙െ૚൩൥ ૙૚૙൩ ൌ ൥ ૙૙െ૚൩ ૙૚ ૙ ૙૙૚ ૙૚૙ ૚૙ ૙ െ૚ ૙ ૙ െ૚ ૙ ૙ ࡭࡯ ൌ ൥૙૙െ૚൩൥ ૙െ૚૙൩ ൌ ൥ ૙૙െ૚൩ ૙૚ ૙ ૙૙૚ ૙െ૚૙

g. Compute ሺ࡭࡮ሻࢄ, ሺ࡭࡯ሻࢄ, and their sum. Compare your result to your answer to part (e). What do you notice? െ૚ ૙ ૙ ૚ െ૚ ሺ࡭࡮ሻࢄൌ൥ ૙૙െ૚൩൥૚൩ ൌ ൥െ૚൩ ૙૚૙૚ ૚ െ૚ ૙ ૙ ૚ െ૚ ሺ࡭࡯ሻࢄൌ൥ ૙૙െ૚൩൥૚൩ ൌ ൥െ૚൩ ૙െ૚૙૚ െ૚ െ૚ െ૚ െ૛ ሺ࡭࡮ሻࢄ൅ሺ࡭࡯ሻࢄൌ൥െ૚൩൥െ૚൩ ൌ ൥െ૛൩ ૚ െ૚ ૙

࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ ൌ ሺ࡭࡮ሻࢄ൅ሺ࡭࡯ሻࢄ

h. In general, must ࡭ሺ࡮ ൅ ࡯ሻ and ࡭࡮ ൅ ࡭࡯ have the same geometric effect on a point, no matter what matrices ࡭, ࡮, and ࡯ are? Explain.

Yes. See full explanation in questions above.

Exercises 1–2 (10 minutes) Allow students to complete the exercises in pairs, each doing the work individually and then comparing answers. Some groups may need more help or one-on-one instructions. Exercise 1, part (b) and Exercise 2, part (a) are fully discussed as a class and are the focus of the Lesson Summary. Have early finishers write the results of Exercise 1, part (c) and Exercise 2, parts (a)–(b) on large paper and prepare an explanation to present to the class as part of the Lesson Summary.

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Exercises ࢞ࢠ ࢇࢉ ࢋࢍ 1. Let ࡭ൌቂ ቃ, ࡮ൌቂ ቃ, and ࡯ൌቂ ቃ. ࢟࢝ ࢈ࢊ ࢌࢎ a. Write down the products ࡭࡮, ࡭࡯, and ࡭ሺ࡮ ൅ ࡯ሻ. ࢇ࢞ ൅ ࢈ࢠ ࢉ࢞ ൅ ࢊࢠ ࢋ࢞ ൅ ࢌࢠ ࢍ࢞ ൅ ࢎࢠ ࡭࡮ ൌ ൤ ൨ ;࡭࡯ ൌ ൤ ൨ ; ࢇ࢟ ൅ ࢈࢝ ࢉ࢟ ൅ ࢊ࢝ ࢋ࢟ ൅ ࢌ࢝ ࢍ࢟ ൅ ࢎ࢝ ሺࢇ൅ࢋሻ࢞൅ሺ࢈൅ࢌሻࢠ ሺࢉ൅ࢍሻ࢞൅ሺࢊ൅ࢎሻࢠ ࡭ሺ࡮൅࡯ሻ ൌ൤ ൨ ሺࢇ൅ࢋሻ࢟൅ሺ࢈൅ࢌሻ࢝ ሺࢉ൅ࢍሻ࢟൅ሺࢊ൅ࢎሻ࢝

b. Verify that ࡭ሺ࡮൅࡯ሻ ൌ࡭࡮൅࡭࡯. ሺࢇ൅ࢋሻ࢞൅ሺ࢈൅ࢌሻࢠ ሺࢉ൅ࢍሻ࢞൅ሺࢊ൅ࢎሻࢠ ࡭ሺ࡮൅࡯ሻ ൌ൤ ൨ ሺࢇ൅ࢋሻ࢟൅ሺ࢈൅ࢌሻ࢝ ሺࢉ൅ࢍሻ࢟൅ሺࢊ൅ࢎሻ࢝ ࢇ࢞൅ࢋ࢞൅࢈ࢠ൅ࢌࢠ ࢉ࢞൅ࢍ࢞൅ࢊࢠ൅ࢎࢠ ൌ൤ ൨ ࢇ࢟൅ࢋ࢟൅࢈࢝൅ࢌ࢝ ࢉ࢟൅ࢍ࢟൅ࢊ࢝൅ࢎ࢝

ࢇ࢞ ൅ ࢈ࢠ ࢉ࢞ ൅ ࢊࢠ ࢋ࢞ ൅ ࢌࢠ ࢍ࢞ ൅ ࢎࢠ ࡭࡮ ൅ ࡭࡯ ൌ ൤ ൨൅൤ ൨ ࢇ࢟ ൅ ࢈࢝ ࢉ࢟ ൅ ࢊ࢝ ࢋ࢟ ൅ ࢌ࢝ ࢍ࢟ ൅ ࢎ࢝ ࢇ࢞൅࢈ࢠ൅ࢋ࢞൅ࢌࢠ ࢉ࢞൅ࢊࢠ൅ࢍ࢞൅ࢎࢠ ൌ൤ ൨ ࢇ࢟൅࢈࢝൅ࢋ࢟൅ࢌ࢝ ࢉ࢟൅ࢊ࢝൅ࢍ࢟൅ࢎ࢝

Therefore, ࡭ሺ࡮൅࡯ሻ ൌ࡭࡮൅࡭࡯.

2. Suppose࡭, ࡮, and ࡯ are ૜ൈ૜ matrices and ࢄ is a point in three-dimensional space. a. Explain why the point ൫࡭ሺ࡮࡯ሻ൯ࢄ must be the same point as ൫ሺ࡭࡮ሻ࡯൯ࢄ.

൫࡭ሺ࡮࡯ሻ൯ࢄൌሺ࡭ሻሺ࡮ሻሺ࡯ሻࢄ. Applying ࡮࡯ and then ࡭is the same as applying࡯, then ࡮, and then ࡭.

ሺ࡭ሻሺ࡮ሻሺ࡯ሻࢄൌ൫ሺ࡭࡮ሻ࡯൯ࢄ. Applying ࡯, then࡮, and then࡭ is the same as applying ࡯ and then ࡭࡮.

b. Explain why matrix multiplication must be associative.

Matrix multiplication is associative because performing the transformation ࡮ and then ࡭ on a point ࢄ is the same as applying the product of ࡭࡮to point ࢄ.

c. Verify using the matrices from Exercise 1 that ࡭ሺ࡮࡯ሻ ൌ ሺ࡭࡮ሻ࡯. ࢇࢉࢋࢍ ࢇࢋ ൅ ࢉࢌ ࢇࢍ ൅ ࢉࢎ ࡮࡯ ൌ ቂ ቃቂ ቃൌ൤ ൨ ࢈ࢊࢌࢎ ࢈ࢋ ൅ ࢊࢌ ࢈ࢍ ൅ ࢊࢎ ࢞ࢠࢇࢋ ൅ ࢉࢌ ࢇࢍ ൅ ࢉࢎ ࢞ሺࢇࢋ ൅ ࢉࢌሻ ൅ࢠሺ࢈ࢋ൅ࢊࢌሻ ࢞ሺࢇࢍ ൅ ࢉࢎሻ ൅ࢠሺ࢈ࢍ൅ࢊࢎሻ ࡭ሺ࡮࡯ሻ ൌቂ ቃ൤ ൨ൌ൤ ൨ ࢟࢝࢈ࢋ ൅ ࢊࢌ ࢈ࢍ ൅ ࢊࢎ ࢟ሺࢇࢋ ൅ ࢉࢌሻ ൅ ࢝ሺ࢈ࢋ ൅ ࢊࢌሻ ࢟ሺࢇࢍ ൅ ࢉࢎሻ ൅࢝ሺ࢈ࢍ൅ࢊࢎሻ ࢞ࢠࢇࢉ ࢇ࢞ ൅ ࢈ࢠ ࢉ࢞ ൅ ࢊࢠ ࡭࡮ ൌ ቂ ቃቂ ቃൌ൤ ൨ ࢟࢝࢈ࢊ ࢇ࢟ ൅ ࢈࢝ ࢉ࢟ ൅ ࢊ࢝ ࢇ࢞ ൅ ࢈ࢠ ࢉ࢞ ൅ ࢊࢠ ࢋࢍ ሺࢇ࢞ ൅ ࢈ࢠሻࢋ൅ሺࢉ࢞ ൅ ࢊࢠሻࢌ ሺࢇ࢞ ൅ ࢈ࢠሻࢍ൅ሺࢉ࢞ ൅ ࢊࢠሻࢎ ሺ࡭࡮ሻ࡯ൌ൤ ൨ቂ ቃൌ൤ ൨ ࢇ࢟ ൅ ࢈࢝ ࢉ࢟ ൅ ࢊ࢝ ࢌࢎ ሺࢇ࢟ ൅ ࢈࢝ሻࢋ൅ሺࢉ࢟ ൅ ࢊ࢝ሻࢌ ሺࢇ࢟ ൅ ࢈࢝ሻࢍ ൅ ሺࢉ࢟ ൅ ࢊ࢝ሻࢎ ࢇࢋ࢞ ൅ ࢉࢌ࢞ ൅ ࢈ࢋࢠ ൅ ࢊࢌࢠ ࢇࢍ࢞ ൅ ࢉࢎ࢞ ൅ ࢈ࢍࢠ ൅ ࢊࢎࢠ ࡭ሺ࡮࡯ሻ ൌ൤ ൨ൌሺ࡭࡮ሻ࡯ ࢇࢋ࢟൅ࢊࢌ࢟൅࢈ࢋ࢝൅ࢊࢌ࢝ ࢇࢍ࢟൅ࢉࢎ࢟൅࢈ࢍ࢝൅ࢊࢎ࢝

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Closing (5 minutes) As a class, discuss the results of Exercises 1 and 2, focusing on the properties of matrix multiplication (distributive and associative) that were discovered/confirmed in this lesson and these exercises. Alternately, revisit the properties used in the example and the exercises of matrix multiplication. What matrix multiplication property did you prove in Exercise 1? Explain how you proved it in part (c). à The distributive property. See exercises to ensure students understand properties. In Exercise 2, parts (a) and (b), explain why you knew that matrix multiplication was associative. à Review the steps from the exercises to ensure students understand the properties. Some properties of matrices: à Each matrix represents a linear transformation, so ܣሺܤܺ ൅ ܥܺሻ ൌܣሺܤܺሻ ൅ܣሺܥܺሻ. à The sum of two matrices acts by summing the image points, ሺܤ൅ܥሻܺൌܤܺ൅ܥܺ. à Applying matrix ܤ and then ܣ is the same as applying the matrix product ܣܤ, meaning ܣሺܤܺሻ ൌ ሺܣܤሻܺ.

Exit Ticket (5 minutes)

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Name Date

Lesson 12: Matrix Multiplication Is Distributive and Associative

Exit Ticket

represents a reflection about ܤrepresents a ͳͺͲι rotation about the ݕ-axis, matrix ܣ In three-dimensional space, matrix :represents a reflection about the ݔݕ-plane. Answer the following ܥ the ݔݖ-plane, and matrix a. Write matrices ܣ, ܤ, and ܥ.

ʹ b. If ܺൌ൥ʹ൩, compute ܣሺܤܺ ൅ ܥܺሻ. ʹ

c. What matrix operations are equivalent to ܣሺܤܺ ൅ ܥܺሻ? What property is shown?

d. Would ൫ܣሺܤܥሻ൯ܺ ൌ ൫ሺܣܤሻܥ൯ܺ? Why?

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Exit Ticket Sample Solutions

In three-dimensional space, matrix ࡭ represents a ૚ૡ૙ι rotation about the ࢟-axis, matrix ࡮ represents a reflection about the ࢞ࢠ-plane, and matrix ࡯ represents a reflection about the ࢞࢟-plane. Answer the following:

a. Write matrices ࡭, ࡮, and ࡯. െ૚ ૙ ૙ ૚૙ ૙ ૚૙ ૙ ࡭ൌ൥ ૙૚૙൩ ;࡮ ൌ ൥૙െ૚૙൩ ;࡯ ൌ ൥૙૚ ૙൩ ૙૙െ૚ ૙૙െ૚ ૙૙െ૚

૛ b. If ࢄൌ൥૛൩, compute ࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ. ૛ ૚૙ ૙૛ ૛ ૚૙ ૙ ૛ ૛ ࡮ࢄ ൌ ൥૙െ૚૙൩൥૛൩ ൌ ൥െ૛൩;࡯ࢄ ൌ ൥૙૚ ૙൩൥૛൩ ൌ ൥ ૛ ൩ ૙૙െ૚૛ െ૛ ૙૙െ૚૛ െ૛ ૝ െ૚ ૙ ૙ ૝ െ૝ ࡮ࢄ ൅ ࡯ࢄ ൌ ൥ ૙ ൩;࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ ൌ ൥ ૙૚૙൩൥ ૙ ൩ ൌ ൥ ૙ ൩ െ૝ ૙૙െ૚െ૝ ૝

c. What matrix operations are equivalent to ࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ? What property is shown?

࡭ሺ࡮ࢄ ൅ ࡯ࢄሻ ൌ ሺ࡭࡮ሻࢄ൅ሺ࡭࡯ሻࢄ; matrix multiplication is distributive.

d. Would ൫࡭ሺ࡮࡯ሻ൯ࢄൌ൫ሺ࡭࡮ሻ࡯൯ࢄ? Why?

Yes; matrix multiplication is associative.

Problem Set Sample Solutions

૜െ૛ ૝૝ ૡ૛ 1. Let matrix ࡭ൌቀ ቁ, matrix࡮ ൌ ቀ ቁ, and matrix ࡯ൌቀ ቁ. Calculate the following: െ૚ ૙ ૜ૢ ૠെ૞ a. ࡭࡮ ૟െ૟ ቀ ቁ െ૝ െ૝

b. ࡭࡯ ૚૙ ૚૟ ቀ ቁ െૡ െ૛

c. ࡭ሺ࡮ ൅ ࡯ሻ ૚૛ ૟ ૚૟ ૚૙ ૚૟ ૚૙ ࡮൅࡯ൌቀ ቁ; ࡭ሺ࡮൅࡯ሻ ൌቀ ቁ; ࡭࡮ ൅ ࡭࡯ ൌ ቀ ቁ ૚૙ ૝ െ૚૛ െ૟ െ૚૛ െ૟ We have that ࡭ሺ࡮൅࡯ሻ ൌ࡭࡮൅࡭࡯.

d. ࡭࡮ ൅ ࡭࡯ ૚૟ ૚૙ ቀ ቁ െ૚૛ െ૟

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e. ሺ࡭൅࡮ሻ࡯ ሺ࡭൅࡮ሻ࡯ൌ࡭࡯൅࡮࡯, so ࡮࡯ has not been calculated yet. We get ૟૙ െ૚૛ ࡮࡯ ൌ ቀ ቁ. ૡૠ െ૜ૢ So, ૠ૙ ૝ ሺ࡭൅࡮ሻ࡯ൌቀ ቁ. ૠૢ െ૝૚

f. ࡭ሺ࡮࡯ሻ ૟૙ െ૚૛ ࡭ሺ࡮࡯ሻ ൌ࡭ቀ ቁ ૡૠ െ૜ૢ ૟૝૛ ൌቀ ቁ െ૟૙ ૚૛

૚ െ૜ 2. Apply each of the transformations you found in Problem 1 to the points ࢞ൌቀ ቁ, ࢟ൌቀ ቁ, and ࢞൅࢟. ૚ ૛ ૙ a. ሺ࡭࡮ሻ࢞ൌቀ ቁ െૡ െ૜૙ ሺ࡭࡮ሻ࢟ൌቀ ቁ ૝ െ૜૙ ሺ࡭࡮ሻሺ࢞൅࢟ሻ ൌቀ ቁ െ૝

૛૟ b. ሺ࡭࡯ሻ࢞ൌቀ ቁ െ૚૙ ૛ ሺ࡭࡯ሻ࢟ൌቀ ቁ ૛૙ ૛ૡ ሺ࡭࡯ሻሺ࢞൅࢟ሻ ൌቀ ቁ ૚૙

૛૟ c. ൫࡭ሺ࡮൅࡯ሻ൯࢞ൌቀ ቁ െ૚ૡ െ૛ૡ ൫࡭ሺ࡮൅࡯ሻ൯࢟ൌቀ ቁ ૛૝ െ૛ ൫࡭ሺ࡮൅࡯ሻ൯ሺ࢞൅࢟ሻ ൌቀ ቁ ૟

d. Same as part (c) ૛૟ ሺ࡭࡮ ൅ ࡭࡯ሻ࢞ൌቀ ቁ െ૚ૡ െ૛ૡ ሺ࡭࡮ ൅ ࡭࡯ሻ࢟ൌቀ ቁ ૛૝ െ૛ ሺ࡭࡮ ൅ ࡭࡯ሻሺ࢞൅࢟ሻ ൌቀ ቁ ૟

ૠ૝ e. ൫ሺ࡭൅࡮ሻ࡯൯࢞ൌቀ ቁ ૜ૡ െ૛૙૛ ൫ሺ࡭൅࡮ሻ࡯൯࢟ൌቀ ቁ െ૜૚ૢ െ૚૛ૡ ൫ሺ࡭൅࡮ሻ࡯൯ሺ࢞൅࢟ሻ ൌቀ ቁ െ૛ૡ૚

૝ૡ f. ൫࡭ሺ࡮࡯ሻ൯࢞ൌቀ ቁ െ૝ૡ ૟૟ ൫࡭ሺ࡮࡯ሻ൯࢟ൌቀ ቁ ૛૙૝ ૚૚૝ ൫࡭ሺ࡮࡯ሻ൯ሺ࢞൅࢟ሻ ൌቀ ቁ ૚૞૟

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3. Let࡭, ࡮, ࡯, and ࡰ be any four square matrices of the same dimensions. Use the distributive property to evaluate the following: a. ሺ࡭ ൅ ࡮ሻሺ࡯ ൅ ࡰሻ ሺ࡭൅࡮ሻ࡯൅ሺ࡭൅࡮ሻࡰ ൌ ࡭࡯ ൅ ࡮࡯ ൅ ࡭ࡰ ൅ ࡮ࡰ

b. ሺ࡭ ൅ ࡮ሻሺ࡭ ൅ ࡮ሻ ࡭࡭ ൅ ࡭࡮ ൅ ࡮࡭ ൅ ࡮࡮

c. What conditions need to be true for part (b) to equal ࡭࡭ ൅ ૛࡭࡮ ൅ ࡮࡮?

࡭࡮ ൌ ࡮࡭ needs to be true.

૛૙ ૜૙ 4. Let ࡭ be a ૛ൈ૛ matrix and ࡮, ࡯ be the scalar matrices ࡮ൌቀ ቁ, and ࡯ൌቀ ቁ. Answer the following ૙૛ ૙૜ questions. a. Evaluate the following: i. ࡮࡯ ૟૙ ቀ ቁ ૙૟

ii. ࡯࡮ ૟૙ ቀ ቁ ૙૟

iii. ࡮൅࡯ ૞૙ ቀ ቁ ૙૞

iv. ࡮െ࡯ െ૚ ૙ ቀ ቁ ૙െ૚

b. Are your answers to part (a) what you expected? Why or why not?

Answers may vary. Students should expect that the matrices behave like real numbers since they represent scalars.

࢞࢟ c. Let ࡭ൌቀ ቁ; does ࡭࡮ ൌ ࡮࡭? Does ࡭࡯ ൌ ࡯࡭? ࢠ࢝ ૛࢞ ૛࢟ ૜࢞ ૜࢟ Yes, ࡭࡮ ൌ ቀ ቁൌ࡮࡭, and ࡭࡯ ൌ ቀ ቁൌ࡯࡭. ૛ࢠ ૛࢝ ૜ࢠ ૜࢝

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d. What is ሺ࡭ ൅ ࡮ሻሺ࡭ ൅ ࡯ሻ? Write the matrix ࡭ with the letter and not in matrix form. How does this compare to ሺ࢞ ൅ ૛ሻሺ࢞ ൅ ૜ሻ?

ሺ࡭൅࡮ሻሺ࡭൅࡯ሻ ൌ ࡭࡭ ൅ ࡮࡭ ൅ ࡭࡯ ൅ ࡮࡯ ૟૙ ൌ࡭࡭൅૛࡭൅૜࡭൅ቀ ቁ ૙૟ ૟૙ ൌ࡭࡭൅૞࡭൅ቀ ቁ ૙૟

This is identical to ሺ࢞ ൅ ૛ሻሺ࢞ ൅ ૜ሻ, with ࢞ൌ࡭.

e. With ࡮ and ࡯ given as above, is it possible to factor ࡭࡭െ࡭െ࡮࡯?

Yes. We need factors of െ࡮࡯ that add to െ૚. It looks like ࡮ and െ࡯ work, so we get ሺ࡭൅࡮ሻሺ࡭െ࡯ሻ.

5. Define the sum of any two functions with the same domain to be the function ࢌ൅ࢍ such that for each ࢞ in the domain of ࢌ and ࢍ, ሺࢌ൅ࢍሻሺ࢞ሻ ൌࢌሺ࢞ሻ ൅ࢍሺ࢞ሻ. Define the product of any two functions to be the function ࢌࢍ, such that for each ࢞ in the domain of ࢌ and ࢍ, ሺࢌࢍሻሺ࢞ሻ ൌ ൫ࢌሺ࢞ሻ൯൫ࢍሺ࢞ሻ൯. ૚ ࢌ ࢍ ࢎ ࢌሺ࢞ሻ ൌ૜࢞൅૚ ࢍሺ࢞ሻ ൌെ ࢞൅૛ Let , , and be real-valued functions defined by the equations , ૛ , and ࢎሺ࢞ሻ ൌ࢞૛ െ૝. a. Does ࢌሺࢍ൅ࢎሻ ൌࢌࢍ൅ࢌࢎ?

Yes. If we can show that ൫ࢌሺࢍ൅ࢎሻ൯ሺ࢞ሻ ൌ ሺࢌࢍሻሺ࢞ሻ ൅ ሺࢌࢎሻሺ࢞ሻ, then we will have shown that the functions are equal to each other.

൫ࢌሺࢍ൅ࢎሻ൯ሺ࢞ሻ ൌ ൫ࢌሺ࢞ሻ൯൫ሺࢍ൅ࢎሻሺ࢞ሻ൯ ૚ ൌ ሺ૜࢞ ൅ ૚ሻ ൬െ ࢞൅૛൅࢞૛ െ૝൰ ૛ ૚ ൌ ሺ૜࢞ ൅ ૚ሻ ቆ൬െ ࢞൅૛൰൅ሺ࢞૛ െ૝ሻቇ ૛ ૚ ൌ ሺ૜࢞ ൅ ૚ሻ ൬െ ࢞൅૛൰൅ሺ૜࢞ ൅ ૚ሻሺ࢞૛ െ૝ሻ ૛ ൌ ൫ࢌሺ࢞ሻ൯൫ࢍሺ࢞ሻ൯ ൅ ൫ࢌሺ࢞ሻ൯൫ࢎሺ࢞ሻ൯ ൌ ሺࢌࢍሻሺ࢞ሻ ൅ ሺࢌࢎሻሺ࢞ሻ

b. Show that this is true for any three functions with the same domains.

൫ࢌሺࢍ൅ࢎሻ൯ሺ࢞ሻ ൌ ൫ࢌሺ࢞ሻ൯൫ሺࢍ൅ࢎሻሺ࢞ሻ൯ ൌ ൫ࢌሺ࢞ሻ൯൫ࢍሺ࢞ሻ ൅ࢎሺ࢞ሻ൯ ൌ ൫ࢌሺ࢞ሻ൯൫ࢍሺ࢞ሻ൯ ൅ ൫ࢌሺ࢞ሻ൯൫ࢎሺ࢞ሻ൯ ൌ ሺࢌࢍሻሺ࢞ሻ ൅ ሺࢌࢎሻሺ࢞ሻ

?ࢎ for the functions described aboveלࢍ൅ࢌלሺࢍ൅ࢎሻ ൌࢌלc. Does ࢌ

No ૚ ሺࢍ൅ࢎሻ൯ሺ࢞ሻ ൌࢌ൬െ ࢞൅૛൅࢞૛ െ૝൰ל൫ࢌ ૛ ૚ ൬െ ࢞൅૛൅࢞૛ െ૝൰൅૚ڄൌ૜ ૛ ૚ ሺ࢞૛ െ૝ሻ ൅૚ڄ൬െ ࢞൅૛൰൅૜ڄൌ૜ ૛

The addition by ૚ prevented it from working. If ࢌሺ࢞ሻ would have been a proportion, then the composition would have worked.

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A STORY OF FUNCTIONS Lesson 13 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 13: Using Matrix Operations for Encryption

Student Outcomes Students study and practice the properties of matrix multiplication. Students understand the role of the multiplicative identity matrix.

Lesson Notes Data encryption has become a necessity with the rise of sensitive data being stored and transmitted via computers. The methods included in this section are not secure enough to use for applications such as Internet banking, but they result in codes that are not easy to break and provide a good introduction to the ideas of encryption. Interested students can research RSA public-key encryption, which relies on the fact that factoring extremely large numbers is a very difficult and slow process. Students interested in the history of data encryption can research the Cherokee and Choctaw Code Talkers from World War I and the Navajo Code Talkers from World War II. This lesson reinforces concepts of matrix multiplication, matrix inverses, and the identity matrix in the context of encoding and decoding strings of characters using multiplication by either an encoding matrix or its inverse decoding matrix. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

The activity in Exercise 2 requires that six stations be set up in advance around the classroom as the messages have been encoded four times. At each station, post the specified decoding matrix: Station 1: Station 2: Station 3: െͳ െͳ ͳ ͳ ͳ ܦ ൌ൥ ͳ ൩ Ͳ െ ଵ ͳ ܦ ൌ൦͸ ൪ ܦ ൌ൦ ͵ ͵൪ ʹ ଶ ͳ ଷ ͳ Ͷ Ͳ െ ͵ ͵ ͵ Station 4: Station 5: Station 6: ͳ ͳ െͳ ͳ ʹെͳ ܦ ൌቂ ቃ ܦ ൌቂ ቃ െ ହ Ͳͳ ଺ െ͵ ʹ ܦ ൌ൦ ʹ ʹ൪ ସ ͵ ͷ െ ʹ ʹ Divide students into at least six groups numbered 1–6, assign each group their coded message, and start them at their numbered station. Groups apply the decoding matrix to their messages and then move to the next station. After applying four decoding matrices, the original message is revealed. Each group decodes ʹͲ characters of the original message, combining the results into the full quote from the entire class.

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Classwork Opening (7 minutes)

The phrase “The crow flies at midnight” appears to have first occurred in Ian Fleming’s James Bond novel From Russia with Love. It has since become a coded message in spy movies and television shows.

Opening

A common way to send coded messages is to assign each letter of the alphabet to a number ૚–૛૟ and send the message as a string of integers. For example, if we encode the message “THE CROW FLIES AT MIDNIGHT” according to the chart below, we get the string of numbers

૛૙,ૡ,૞,૙,૜,૚ૡ,૚૞,૛૜,૙,૟,૚૛,ૢ,૞,૚ૢ,૙,૚,૛૙,૙, ૚૜,ૢ,૝, ૚૝,ૢ,ૠ, ૡ,૛૙.

A B C D E F G H I J K L M ૚ ૛ ૜ ૝ ૞ ૟ ૠ ૡ ૢ ૚૙ ૚૚ ૚૛ ૚૜

N O P Q R S T U V W X Y Z SPACE ૚૝ ૚૞ ૚૟ ૚ૠ ૚ૡ ૚ૢ ૛૙ ૛૚ ૛૛ ૛૜ ૛૝ ૛૞ ૛૟ ૙

However, codes such as these are easily broken using an analysis of the frequency of numbers that appear in the coded messages.

We can instead encode a message using matrix multiplication. If a matrix ࡱ has an inverse, then we can encode a message as follows.

First, convert the characters of the message to integers between ૚and ૛૟ using the chart above. If the encoding matrix ࡱ is an ࢔ൈ࢔matrix, then break up the numerical message into ࢔ rows of the same length. If needed, add extra zeros to make the rows the same length. Place the rows into a matrix ࡹ. Compute the product ࡱࡹ to encode the message. The message is sent as the numbers in the rows of the matrix ࡱࡹ.

Example (10 minutes)

A common way to send coded messages is to assign each letter of the alphabet to a number ͳ–ʹ͸ and send the message as a string of integers. For example, if we encode the message “THE CROW FLIES AT MIDNIGHT” according to the chart below, we get the string of numbers ʹͲ, ͺ, ͷ, Ͳ, ͵, ͳͺ, ͳͷ, ʹ͵, Ͳ, ͸, ͳʹ, ͻ, ͷ, ͳͻ, Ͳ, ͳ, ʹͲ, Ͳ, ͳ͵, ͻ, Ͷ, ͳͶ, ͻ, ͹, ͺ, ʹͲ.

A B C D E F G H I J K L M ͳ ʹ ͵ Ͷ ͷ ͸ ͹ ͺ ͻ ͳͲ ͳͳ ͳʹ ͳ͵

N O P Q R S T U V W X Y Z SPACE ͳͶ ͳͷ ͳ͸ ͳ͹ ͳͺ ͳͻ ʹͲ ʹͳ ʹʹ ʹ͵ ʹͶ ʹͷ ʹ͸ Ͳ

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However, codes such as these are easily broken using an analysis of the frequency of numbers that appear in the coded messages. For example, if the coded phrase is ʹͲ, ͺ, ͷ, Ͳ, ͵, ͳͺ, ͳͷ, ʹ͵, Ͳ, ͸, ͳʹ, ͻ, ͷ, ͳͻ, Ͳ, ͳ, ʹͲ, Ͳ, ͳ͵, ͻ, Ͷ, ͳͶ, ͻ, ͹, ͺ, ʹͲ, we can see that there are three of the letters assigned to the integer ʹͲ, so we may want to try putting a common letter in for the number ʹͲ, like S, T, or E. If we assume the word the is used to start the phrase since we have three letters and then a space, that would lead us to think that maybe the number ͷ is E, and so on. We can instead encode a message using matrix multiplication. If a matrix ܧ has an inverse, then we can encode a message as follows. First, convert the characters of the message to integers between ͳ and ʹ͸ using the chart above. If the encoding matrix ܧ is an ݊ൈ݊matrix, then break up the numerical message into ݊ rows of the same length. If needed, add extra zeros to make the rows the same length. For example, if we want to use an encoding matrix that is ʹൈʹ, we would write the message in two rows of equal length, filling in zero for the last number if the number of letters was odd. If the encoding matrix is ͵ൈ͵, the message would be written in three equal rows, adding zeros as necessary. Place the rows into a matrix ܯ. Scaffolding: ܧܯ Compute the product to encode the message. Students who are The message is sent as the numbers in the rows of the matrix ܧܯ. struggling can be given a ʹͳ simpler phrase such as “Be Using the matrixܧ ൌ ቂ ቃ, we encode our message as follows: ͵ͳ Happy” or “Dream Big.” Ͷ, ͳͶ, ͻ, ͹, ͺ, ʹͲ. Have advanced learners find their own phrase of Since ܧ is a ʹൈʹmatrix, we need to break up our message into two rows. ͵Ͳ characters or more and ʹͲ, ͺ, ͷ, Ͳ, ͵, ͳͺ, ͳͷ, ʹ͵, Ͳ, ͸, ͳʹ, ͻ, ͷ, encode using a ͵ൈ͵ ͳͻ, Ͳ, ͳ, ʹͲ, Ͳ, ͳ͵, ͻ, Ͷ, ͳͶ, ͻ, ͹, ͺ, ʹͲ matrix.

Then, we place the rows into a matrix ܯ. ʹͲ ͺ ͷ Ͳ ͵ ͳͺ ͳͷ ʹ͵ Ͳ ͸ ͳʹ ͻ ͷ ܯൌቂ ቃ ͳͻ Ͳ ͳ ʹͲ Ͳ ͳ͵ ͻ Ͷ ͳͶ ͻ ͹ ͺ ʹͲ Explain how matrix ܯ represents “THE CROW FLIES AT MIDNIGHT.” à Each letter and space in the phrase was assigned an integer value, and these numbers represent the letters in the phrase. .ܯڄܧ by multiplying ܥ We encode the message into matrix ʹͳʹͲ ͺ ͷ Ͳ ͵ ͳͺ ͳͷ ʹ͵ Ͳ ͸ ͳʹ ͻ ͷ ൌቂ ቃቂ ቃܯڄܧൌܥ ͵ͳͳͻ Ͳ ͳ ʹͲ Ͳ ͳ͵ ͻ Ͷ ͳͶ ͻ ͹ ͺ ʹͲ ͷͻͳ͸ͳͳʹͲ͸Ͷͻ͵ͻͷͲͳͶʹͳ͵ͳʹ͸͵Ͳ ܥൌቂ ቃ ͹ͻʹͶͳ͸ʹͲͻ͸͹ͷͶ͹͵ͳͶʹ͹Ͷ͵͵ͷ͵ͷ Thus, the coded message that we send is

ͷͻ, ͳ͸, ͳͳ, ʹͲ, ͸, Ͷͻ, ͵ͻ, ͷͲ, ͳͶ, ʹͳ, ͵ͳ, ʹ͸, ͵Ͳ, ͹ͻ, ʹͶ, ͳ͸, ʹͲ, ͻ, ͸͹, ͷͶ, ͹͵, ͳͶ, ʹ͹, Ͷ͵, ͵ͷ, ͵ͷ.

If this coded message is intercepted, then it cannot easily be decoded unless the recipient knows how it was originally encoded. Be sure to work through this discussion, and emphasize that the way to decode a message is to multiply by the inverse of the encoding message.

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Using what you know about how the message was encoded, as well as matrix multiplication, describe how you would decode this message. à We need to know a decoding matrix ܦ. How can we find that matrix? à The decoding matrix is the inverse of the encoding matrix, so ܦൌܧିଵ. What is the decoding matrix? ʹͳିଵ ͳ ͳെͳ െͳ ͳ à ܦൌܧିଵ ൌቂ ቃ ൌ ቂ ቃൌቂ ቃ ͵ͳ ʹെ͵ െ͵ ʹ ͵െʹ Decode this message! െͳ ͳ ͷͻ ͳ͸ ͳͳ ʹͲ ͸ Ͷͻ ͵ͻ ͷͲ ͳͶ ʹͳ ͵ͳ ʹ͸ ͵Ͳ ൌቂ ቃቂ ቃܥڄܦ à ͵െʹ͹ͻ ʹͶ ͳ͸ ʹͲ ͻ ͸͹ ͷͶ ͹͵ ͳͶ ʹ͹ Ͷ͵ ͵ͷ ͵ͷ ʹͲ ͺ ͷ Ͳ ͵ ͳͺ ͳͷ ʹ͵ Ͳ ͸ ͳʹ ͻ ͷ ൌ ቂ ቃ ͳͻ Ͳ ͳ ʹͲ Ͳ ͳ͵ ͻ Ͷ ͳͶ ͻ ͹ ͺ ʹͲ As expected, this is the matrix ܯ that stored our original message “THE CROW FLIES AT MIDNIGHT.” Why does this process work? We then .ܯڄܧൌܥ so ,ܯ and ܧ is the product of matrices ܥ à The coded message stored in matrix decode the message stored in matrix ܥ by multiplying by matrix ܦ. Since matrices ܦ and ܧ are inverses, we have

.ܯൌܯڄܫൌܯڄ ሻܧڄܦሻ ൌ ሺܯڄܧሺڄܦ

So, encoding and then decoding returns the original message in matrix ܯ. Explain to your neighbor what you learned about how to encode and decode messages. Teachers should use this as an informal way to check for understanding.

Exercise 1 (7 minutes) The encoded phrase in this exercise is “ARCHIMEDES.” Archimedes (c. 287–212 B.C.E., Greece) is regarded as the greatest mathematician of his age and one of the greatest of all time. He developed and applied an early form of integral calculus to derive correct formulas for the area of a circle, volume of a sphere, and area under a parabola. He also found accurate approximations of irrational numbers such as ξ͵andߨ. However, during his lifetime he was known more for his inventions such as the Archimedean screw, compound pulleys, and weapons such as the Claw of Archimedes used to protect Syracuse in times of war. ͳͳͺ͵ͺ The original message is stored in matrix ܯൌ൥ͻͳ͵ͷͶ൩, and the matrix used to encode the message is ͷͳͻͲͲ ͳʹ͵ ܧൌ൥Ͳʹͳ൩. This exercise introduces students to using a larger matrix to perform the encoding and decoding and ͳͳͳ requires that students practice matrix multiplication with non-integer matrix entries. Additionally, because students have not learned a method for finding the inverse of a ͵ൈ͵ matrix, they must demonstrate understanding of the meaning of a matrix inverse in order to decode this matrix.

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Exercises

1. You have received an encoded message: ૜૝, ૚૙૚, ૚૜, ૚૟, ૛૜, ૝૞, ૚૙, ૡ, ૚૞, ૞૙, ૡ, ૚૛. ૚૛૜ Scaffolding: You know that the message was encoded using matrix ࡱൌ൥૙૛૚൩. Students may need to be ૚૚૚ reminded that the a. Store your message in a matrix ࡯. What are the dimensions of ࡯? matrices used to encode There are ૚૛ numbers in the coded message, and it was encoded using a and decode a message are ૜ൈ૜ matrix. Thus, the matrix ࡯ needs to have three rows. That means ࡯ hasfour columns, so࡯ is a ૜ൈ૝ matrix. inverses.

Students may also need to b. You have forgotten whether the proper decoding matrix is matrix ࢄ, ࢅ, or ࢆ as shown be reminded of the below. Determine which of these is the correct matrix to use to decode this message. property that defines ૚ ૚ ૝ ૚ ૚ ૝ ૚ ૚ ૝ inverse matrices: ܣ and ܤ ې െ െۍ ې െ െۍ ې െ െۍ are inverse matrices if ۑ ૜ ૜ ૜ ێ ۑ ૜ ૜ ૜ ێ ۑ ૜ ૜ ૜ ێ ૚ ૛ ૚ ૚ ૛ ૚ ૚ ૛ ૚ .ܫൌܤڄܣ ۑ െێ ࢆ ൌ, ۑ െ െێ ࢅ ൌ, ۑ െێࢄൌ ۑ ૜ ૜ ૜ ێ ۑ૜ ૜ ૜ ێ ۑ ૜ ૜ ૜ ێ ۑ૛ ૚ ૛ ێ ۑ ૛ ૚ ૛ ێ ۑ૛ ૚ ૛ ێ െ െ െ െ െ ے૜ ૜ ૜ ۏ ے ૜ ૜ ૜ ۏ ے૜ ૜ ૜ ۏ Matrices used to encode and decode messages must be inverses of each other. Thus, the correct decoding ,ࡱڄࡱ, ࢅڄࡱ ൌ ࡵ. We can find the correct decoding matrix by multiplying ࢄ ڄ matrix is the matrix ࡰso thatࡰ .ࡱڄand ࢆ ૚ ૚ ૝ ې െ െۍ ૜ ૜ ૜ ૚૙૙ ૚૛૜ ۑ ێ ૙૚૙ ۑ ૚ ૛ ૚ ێ ൥૙૛૚൩ ൌ൦ ૝ ૝ ૠ൪ ڄ ࡱൌ െڄࢄ ૚૚૚ െ െ െ ۑ ૜ ૜ ૜ ێ ૜ ૜ ૜ ۑ૛ ૚ ૛ ێ െ െ ے૜ ૜ ૜ ۏ ૚ ૚ ૝ ૚૙૙ ې െ െۍ ૜ ૜ ૜ ې૛ ૚ ૛ ۍ ૚૛૜ ۑ ێ ۑ െ െێ ۑ૚ ૛ ૚ ێ ۑ૜ ૜ ૜ ێ ൥૙૛૚൩ ൌ ڄ ࡱൌ െ െڄࢅ ۑ ૝ ૝ ૠ ێ ૚૚૚ ۑ૜ ૜ ૜ ێ ۑ ૛ ૚ ૛ ێ ے ૜ ૜ ૜ ۏ െ ے ૜ ૜ ૜ ۏ ૚ ૚ ૝ ې െ െۍ ۑ ૜ ૜ ૜ ێ ૚ ૛ ૚ ૚૛૜ ૚૙૙ ൥૙૛૚൩ ൌ ൥૙૚૙൩ ڄ ۑ െێࡱൌڄࢆ ૚૚૚ ૙૙૚ ۑ ૜ ૜ ૜ ێ ۑ૛ ૚ ૛ ێ െ െ ے૜ ૜ ૜ ۏ

.ࡱൌࡵ, we know that ࢆ is the decoding matrix we needڄSince ࢆ

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c. Decode the message.

Using matrix ࢆ to decode, we have ૚ ૚ ૝ ې െ െۍ ۑ ૜ ૜ ૜ ێ ૚ ૛ ૚ ૜૝ ૚૙૚ ૚૜ ૚૟ ൥૛૜ ૝૞ ૚૙ ૡ ൩ ڄ ۑ െێ࡯ൌڄࡹൌࢆ ૚૞ ૞૙ ૡ ૚૛ ۑ ૜ ૜ ૜ ێ ۑ૛ ૚ ૛ ێ െ െ ے૜ ૜ ૜ ۏ ૜૝ ૛૜ ૟૙ ૚૙૚ ૝૞ ૛૙૙ ૚૜ ૚૙ ૜૛ ૚૟ ૡ ૝ૡ ې െ െ ൅ െ െ ൅ െ െ ൅ െ െ ൅ۍ ۑ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ێ ૜૝ ૝૟ ૚૞ ૚૙૚ ૢ૙ ૞૙ ૚૜ ૛૙ ૡ ૚૟ ૚૟ ૚૛ ۑ െ ൅ ൅ െ ൅ ൅ െ ൅ ൅ െ ൅ ൅ێ ൌ ۑ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ێ ۑ ૟ૡ ૛૜ ૜૙ ૛૙૛ ૢ૙ ૚૙૙ ૛૟ ૚૙ ૚૟ ૜૛ ૡ ૛૝ ێ െ െ െ െ െ െ െ െ ے ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ૜ ۏ ૚૚ૡ૜ૡ ൌ ൥ૢ૚૜૞૝൩ . ૞૝૙૙

The decoded message is “ARCHIMEDES.”

Exercises 2–3 (15 minutes) In this exercise, groups of students decode separate parts of a message that have been encoded four times; as groups complete the decoding of their portion of the message, have them record it in a location that all students can see— either on the white board or projected through a document camera, for example. The decoded messages together spell out a famous quote by Albert Einstein: “Do not worry about your difficulties in mathematics. I can assure you that mine are still greater.” Substitute a different quote if preferred, perhaps a school motto. Carefully encode each of six

portions of a quote stored in matrices ܯଵ to ܯ଺ using encoding matrices ܧଵ to ܧ଺ as follows. ܯڄ ܧڄ ܧڄ ܧڄ ܧൌ ܥ ଵ ଵ ଶ ଷ ସ ଵ Scaffolding: ܯڄ ܧڄ ܧڄ ܧڄ ܧൌ ܥ ଶ ଶ ଷ ସ ହ ଶ For struggling students, select a ଷ shorter quote, or encode it inܯڄ ଺ܧڄ ହܧڄ ସܧڄ ଷܧଷ ൌܥ

.ସ fewer than four stepsܯڄ ଵܧڄ ଺ܧڄ ହܧڄ ସܧସ ൌܥ

ହܯڄ ଶܧڄ ଵܧڄ ଺ܧڄ ହܧହ ൌܥ

଺ܯڄ ଷܧڄ ଶܧڄ ଵܧڄ ଺ܧ଺ ൌܥ ͳʹ ͸Ͳ Ͷͳ ͷͳ െͳ ͳ ʹͳ ܧ ൌቂ ቃ; ܧ ൌቂ ቃ; ܧ ൌቂ ቃ; ܧ ൌቂ ቃ; ܧ ൌቂ ቃ; ܧ ൌቂ ቃ ଵ െʹ െʹ ଶ Ͳ͵ ଷ ͳͳ ସ ͵ͳ ହ Ͳͳ ଺ ͵ʹ Divide the class into at least six groups of two or three students, numbered 1–6, assigning multiple groups to the same number as needed. Set up six stations around the room in a circular arrangement. Have each group start at the station with the same number as the group—Group 1 starts at Station 1, Group 2 starts at Station 2, etc. At each station, the groups apply the posted decoding matrix to their encoded messages shown below, and then they progress to the next station, with groups at Station 6 proceeding to Station 1. It requires four decoding steps with different matrices (such as

ܦଶ, ܦଷ, ܦସ and ܦହ) to uncover a group’s portion of the original message.

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At each station, post the matrix listed below. െͳ െͳ Station 1: ܦ ൌቈ ͳ ቉ ଵ ͳ ʹ ͳ Ͳ ܦ ൌ቎͸ ቏ Station 2: ଶ ͳ Ͳ ͵ ͳ ͳ െ ܦ ൌ቎ ͵ ͵቏ Station 3: ଷ ͳ Ͷ െ ͵ ͵ ͳ ͳ െ ܦ ൌ቎ ʹ ʹ቏ Station 4: ସ ͵ ͷ െ ʹ ʹ െͳ ͳ Station 5: ܦ ൌቂ ቃ ହ Ͳͳ ʹെͳ Station 6: ܦ ൌቂ ቃ ଺ െ͵ ʹ

2. You have been assigned a group number. The message your group receives is listed below. This message is TOP SECRET! It is of such importance that it has been encoded four times. Your group’s portion of the coded message is listed below. Group 1: ૚૞૙૙, ૜ૡ૝૙, ૙, ૜૝૝૝, ૜૝૛૙, ૝૜૞૙, ૙, ૝ૡ૛૝, ૜૟ૠ૛, ૜૝ૠ૝, െ૛૞ૢ૛, െ૟૟૟૙, ૙, െ૞ૢૠ૟, െ૞ૢ૝૙, െૠ૞૟૙, ૙, െૡ૜ૡૡ, െ૟૜ૠ૛, െ૟૙૝ૡ

Group 2: ૛૝૛૝, ૜૙૛૝, െ૚૜ૡ, ૜ૢ૟, െ૞૞ૡ, െ૚ૡૢ૙, െ૚ૠ૞૛, ૚૞૚૛, െ૛ૢ૝૟, ૚૝૞ૡ, ૝૜ૡ, ૞૝૙, െ૛૝, ૠ૛, െૢ૙, െ૜૛૝, െ૜૙૙, ૛ૠ૙, െ૞૚૙, ૛ૠ૙

Group 3: ૝ૡૢ, ૚૝૛૙, ૟૙૟, ૜૞૞, ૚૚૞૚, ૜૜, ૚૙૙૛, ૡ૛ૢ, ૢૢ, ૚૚૛૚, ૚ૡ૙, ૞૛૙, ૛૛૛, ૚૜૙, ૝૛૛, ૚૛, ૜૟૟, ૜૙૝, ૜૟, ૝૚૙

Group 4: െ૚ૡ, ૚૙, െ૚ૡ, ૝૝, െ૞૝, ૝૛, െ૟, െૠ૝, െૢૡ, െ૚૛૝, ૙, ૚૙, െ૚૛, ૝૟, െ૛૟, ૝૛, െ૝, െ૜૟, െ૟૙, െૡ૛

Group 5: െ૚૛૙, ૙, െૠૡ, െ૞૝, െૡ૝, െ૜૙, ૙, െ૟, െ૚૙ૡ, െ૜૙, െ૚૛૙, ૚૚૝, ૝૛, ૙, െ૚૛, ૝૛, ૙, ૜૟, ૙, ૙

Group 6: ૚૛૟, ૚૛૙, ૟૙, ૚૟૛, ૡ૝, ૚૛૙, ૚ૢ૛, ૝૛, ૡ૝, ૚ૢ૛, െ૚ૡ, െ૜૟૙, െૢ૙, െ૜૛૝, ૙, െ૚ૡ, െ૛૚૟, െ૜૟, െૢ૙, െ૜૛૝

a. Store your message in a matrix ࡯ with two rows. How many columns does matrix ࡯ have?

(Sample responses are provided for Group 1.) Our message is stored in a matrix with ten columns: ૚૞૙૙ ૜ૡ૝૙ ૙ ૜૝૝૝ ૜૝૛૙ ૝૜૞૙ ૙ ૝ૡ૛૝ ૜૟ૠ૛ ૜૝ૠ૝ ࡯ൌቂ ቃ െ૛૞ૢ૛ െ૟૟૟૙ ૙ െ૞ૢૠ૟ െ૞ૢ૝૙ െૠ૞૟૙ ૙ െૡ૜ૡૡ െ૟૜ૠ૛ െ૟૙૝ૡ

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b. Begin at the station of your group number, and apply the decoding matrix at this first station. െ૚ െ૚ ࡰ ൌ ൥ ૚ ൩ ૚ ૚ ૛ െ૚ െ૚ ૚૞૙૙ ૜ૡ૝૙ ૙ ૜૝૝૝ ૜૝૛૙ ૝૜૞૙ ૙ ૝ૡ૛૝ ૜૟ૠ૛ ૜૝ૠ૝ ቂ ቃڄ ࡯ൌ൥ ૚ ൩ڄ ࡰ ૚ ૚ െ૛૞ૢ૛ െ૟૟૟૙ ૙ െ૞ૢૠ૟ െ૞ૢ૝૙ െૠ૞૟૙ ૙ െૡ૜ૡૡ െ૟૜ૠ૛ െ૟૙૝ૡ ૛ ૚૙ૢ૛ ૛ૡ૛૙ ૙ ૛૞૜૛ ૛૞૛૙ ૜૛૚૙ ૙ ૜૞૟૝ ૛ૠ૙૙ ૛૞ૠ૝ ൌቂ ቃ ૛૙૝ ૞૚૙ ૙ ૝૞૟ ૝૞૙ ૞ૠ૙ ૙ ૟૜૙ ૝ૡ૟ ૝૞૙

c. Proceed to the next station in numerical order; if you are at Station 6, proceed to Station 1. Apply the decoding matrix at this second station. ૚ ૙ ࡰ ൌ൦૟ ൪ ૛ ૚ ૙ ૜ ૚ ૙ ૚૙ૢ૛ ૛ૡ૛૙ ૙ ૛૞૜૛ ૛૞૛૙ ૜૛૚૙ ૙ ૜૞૟૝ ૛ૠ૙૙ ૛૞ૠ૝ ቂ ቃڄ࡯ൌ൦૟ ൪ڄ ࡰڄ ࡰ ૛ ૚ ૚ ૛૙૝ ૞૚૙ ૙ ૝૞૟ ૝૞૙ ૞ૠ૙ ૙ ૟૜૙ ૝ૡ૟ ૝૞૙ ૙ ૜ ૚ૡ૛ ૝ૠ૙ ૙ ૝૛૛ ૝૛૙ ૞૜૞ ૙ ૞ૢ૝ ૝૞૙ ૝૛ૢ ൌቂ ቃ ૟ૡ ૚ૠ૙ ૙ ૚૞૛ ૚૞૙ ૚ૢ૙ ૙ ૛૚૙ ૚૟૛ ૚૞૙

d. Proceed to the next station in numerical order; if you are at Station 6, proceed to Station 1. Apply the decoding matrix at this third station. ૚ ૚ െ ࡰ ൌ൦ ૜ ૜൪ ૜ ૚ ૝ െ ૜ ૜ ૚ ૚ െ ૚ૡ૛ ૝ૠ૙ ૙ ૝૛૛ ૝૛૙ ૞૜૞ ૙ ૞ૢ૝ ૝૞૙ ૝૛ૢ ቂ ቃڄ࡯ൌ൦ ૜ ૜൪ڄ ࡰڄ ࡰڄ ࡰ ૜ ૛ ૚ ૚ ૝ ૟ૡ ૚ૠ૙ ૙ ૚૞૛ ૚૞૙ ૚ૢ૙ ૙ ૛૚૙ ૚૟૛ ૚૞૙ െ ૜ ૜ ૜ૡ ૚૙૙ ૙ ૢ૙ ૢ૙ ૚૚૞ ૙ ૚૛ૡ ૢ૟ ૢ૜ ൌቂ ቃ ૜૙ ૠ૙ ૙ ૟૛ ૟૙ ૠ૞ ૙ ૡ૛ ૟૟ ૞ૠ

e. Proceed to the next station in numerical order; if you are at Station 6, proceed to Station 1. Apply the decoding matrix at this fourth station. ૚ ૚ െ ࡰ ൌ൦ ૛ ૛൪ ૝ ૜ ૞ െ ૛ ૛ ૚ ૚ െ ૜ૡ ૚૙૙ ૙ ૢ૙ ૢ૙ ૚૚૞ ૙ ૚૛ૡ ૢ૟ ૢ૜ ቂ ቃڄ࡯ൌ൦ ૛ ૛൪ڄ ࡰڄ ࡰڄ ࡰڄ ࡰ ૝ ૜ ૛ ૚ ૜ ૞ ૜૙ ૠ૙ ૙ ૟૛ ૟૙ ૠ૞ ૙ ૡ૛ ૟૟ ૞ૠ െ ૛ ૛ ૝ ૚૞ ૙ ૚૝ ૚૞ ૛૙ ૙ ૛૜ ૚૞ ૚ૡ ൌቂ ቃ ૚ૡ ૛૞ ૙ ૛૙ ૚૞ ૚૞ ૙ ૚૜ ૛૚ ૜

f. If we know the original message had ૛૚ characters and the last letter was an H, decode your message.

The numerical message is ૝,૚૞,૙,૚૝,૚૞,૛૙,૙,૛૜,૚૞,૚ૡ,૚ૡ,૛૞,૙,૛૙,૚૞,૚૞,૙,૚૜,૛૚,૜, which represents the characters “DO NOT WORRY TOO MUCH.”

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ࡰ૝ሻand thenڄ ࡰ૜ڄ ࡰ૛ڄ Sydnie was in Group 1 and tried to decode her message by calculating the matrix ሺࡰ૚ .3 ࡯. This produced the matrixڄࡰ૝ሻڄ ࡰ૜ڄ ࡰ૛ڄ multiplyingሺࡰ૚ ૚૙૞૛૟ ૛ૠ૙૛૙ ૛૝૛૝૛ ૜૙૟૞૞ ૙ ૡ૙૜૙ ૙ ૚૚૜૜૟ ૡ૟૚૟ ૡ૚ૠ૚ ࡹ ൌ ൦ ૜ ૜ ૜ ૜ ൪ . ૡ૝ૠ૞ ૟૚ૠૠ െ૚૝૞૞ െ૜ૠ૜૞ ૙ െ૜૜૞૚ െ૜૜૜૙ െ ૙ െ૝ૠ૙૚ െ૜૞ૠ૜ െ ૛ ૛ a. How did she know that she made a mistake?

If Sydnie had properly decoded her message, all entries in the matrix ࡹ would be integers between૙and ૛૟.

b. Matrix ࡯ was encoded using matrices ࡱ૚,ࡱ૛,ࡱ૜, and ࡱ૝, where ࡰ૚ decodes a message encoded by ࡱ૚, ࡰ૛ decodes a message encoded by ࡱ૛, and so on. What is the relationship between matrices ࡱ૚ and ࡰ૚, between ࡱ૛ and ࡰ૛, etc.?

Matrices ࡱ૚ and ࡰ૚ are inverse matrices, as are ࡱ૛ and ࡰ૛, ࡱ૜, and ࡰ૜, and so on.

ࡹ. Explain to Sydnie how theڄ ࡱ૝ڄ ࡱ૜ڄ ࡱ૛ڄ c. The matrix that Sydnie received was encoded by ࡯ൌࡱ૚ decoding process works to recover the original matrix ࡹ, and devise a correct method for decoding using multiplication by a single decoding matrix.

ࡹ, we can recover the original matrix ࡹ by multiplying both sides of thisڄ ࡱ૝ڄ ࡱ૜ڄ ࡱ૛ڄ Since ࡯ൌࡱ૚ .ࡱ૛ ൌࡵ, etcڄ ࡱ૚ ൌࡵ, ࡰ૛ڄ equation by the proper decoding matrix at each step, remembering that ࡰ૚

ࡹڄ ࡱ૝ڄ ࡱ૜ڄ ࡱ૛ڄ ࡯ൌࡱ૚

ࡹሻڄ ࡱ૝ڄ ࡱ૜ڄ ࡱ૛ڄ ሺࡱ૚ڄ ࡯ൌࡰ૚ڄ ࡰ૚

ࡹሻڄ ࡱ૝ڄ ࡱ૜ڄ ሺࡱ૛ ڄ ࡱ૚ሻڄ ൌ ሺࡰ૚

ࡹሻڄ ࡱ૝ڄ ࡱ૜ڄ ሺࡱ૛ڄൌࡵ

ࡹሻڄ ࡱ૝ڄ ࡱ૜ڄ ൌሺࡱ૛

ࡹሻڄ ࡱ૝ڄ ࡱ૜ڄ ሺࡱ૛ڄ ࡯ൌࡰ૛ڄ ࡰ૚ڄ ࡰ૛

ࡹሻڄ ࡱ૝ڄ ሺࡱ૜ڄ ࡱ૛ሻڄ ൌ ሺࡰ૛

ࡹሻڄ ࡱ૝ڄ ሺࡱ૜ڄൌࡵ

ࡹሻڄ ࡱ૝ڄ ൌሺࡱ૜

ࡹሻڄ ࡱ૝ڄ ሺࡱ૜ڄ ࡯ൌࡰ૜ڄ ࡰ૚ڄ ࡰ૛ڄ ࡰ૜

ࡹሻڄ ሺࡱ૝ ڄ ࡱ૜ሻڄ ൌ ሺࡰ૜

ࡹሻڄ ሺࡱ૝ڄൌࡵ

ࡹڄ ൌࡱ૝

ࡹሻڄ ሺࡱ૝ڄ ࡯ൌࡰ૝ڄ ࡰ૚ڄ ࡰ૛ڄ ࡰ૜ڄ ࡰ૝

ࡹڄ ࡱ૝ሻڄ ൌ ሺࡰ૝ ࡹڄൌࡵ ൌࡹ

.࡯ڄ ࡰ૚ሻڄ ࡰ૛ڄ ࡰ૜ڄ Since matrix multiplication is associative, this means that ࡹൌሺࡰ૝

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d. Apply the method you devised in part (c) to your group’s message to verify that it works. ૚ ૚ ૚ ૚ ૚ െ െ ૙ െ૚ െ૚ ൥ ૚ ൩ڄ൦૟ ൪ڄ൦ ૜ ૜൪ڄࡰ ሻ ൌ൦ ૛ ૛൪ڄ ࡰڄ ࡰڄ ሺࡰ ૝ ૜ ૛ ૚ ૜ ૞ ૚ ૝ ૚ ૚ െ െ ૙ ૛ ૛ ૛ ૜ ૜ ૜ ૚ ૞ ૚ ૚ ૚ ૠ െ െ െ െ െ ൦ ૟ ૟൪ൌ൦ ૜ ૜૟൪ڄൌ൦ ૜ ૟൪ ૝ ૛૜ ૚ ૚ ૜ ૜૚ െ ૜ ૟ ૜ ૟ ૛ ૜૟ So,

࡯ڄ ࡰ૚ሻڄ ࡰ૛ڄ ࡰ૜ڄ ሺࡰ૝ ૚ ૠ െ െ ૚૞૙૙ ૜ૡ૝૙ ૙ ૜૝૝૝ ૜૝૛૙ ૝૜૞૙ ૙ ૝ૡ૛૝ ૜૟ૠ૛ ૜૝ૠ૝ ቂ ቃڄൌ ൦ ૜ ૜૟൪ ૜ ૜૚ െ૛૞ૢ૛ െ૟૟૟૙ ૙ െ૞ૢૠ૟ െ૞ૢ૝૙ െૠ૞૟૙ ૙ െૡ૜ૡૡ െ૟૜ૠ૛ െ૟૙૝ૡ ૛ ૜૟ ૝ ૚૞ ૙ ૚૝ ૚૞ ૛૙ ૙ ૛૜ ૚૞ ૚ૡ ൌቂ ቃ ૚ૡ ૛૞ ૙ ૛૙ ૚૞ ૚૞ ૙ ૚૜ ૛૚ ૜ This is the same decoded message that we found in Exercise 2, part (f).

Exercise 4 (Optional, 8 minutes) The encoded phrase in this exercise is “RAMANUJAN.” Srinivasa Ramanujan (1887–1920) was a self-taught mathematician from India who made significant contributions to many branches of mathematics, particularly analysis and number theory, compiling thousands of mathematical results. Although he died young, he is widely considered to be one of the greatest mathematicians of his time. ͳͺ ͳ ͳ͵ The original message is stored in matrix ܯൌ൥ͳͳͶʹͳ൩, and the matrix used to encode the message is ͳͲ ͳ ͳͶ ͳʹͳ ܧൌ൥െͳ Ͳ ʹ ൩. In this optional exercise, students need to reason through the process of encoding and decoding to ͳͳെͳ recover a missing entry in the decoding matrix when the encoding matrix is unknown. Use this exercise as an extension for students who have finished the previous exercises quickly.

૜૙ ૜૙ ૟ૢ 4. You received a coded message in the matrix ࡯ൌ൥ ૛૚૚૞൩. However, the matrix ࡰthat will decode this ૢ૚૝૛૙ message has been corrupted, and you do not know the value of entry ࢊ૚૛. You know that all entries in matrix ࡰ are ૛࢞െ૝ integers. Using ࢞ to represent this unknown entry, the decoding matrix ࡰ is given by ࡰൌ൥െ૚ ૛ ૜ ൩. Decode ૚െ૚െ૛ the message in matrix ࡯.

:࡯ڄDecoding the message requires that we multiply ࡰ ૛࢞െ૝૜૙ ૜૙ ૟ૢ ൥ ૛૚૚૞൩ ڄ ࡯ൌ൥െ૚ ૛ ૜ ൩ڄࡰ ૚െ૚െ૛ ૢ૚૝૛૙ ૛૝ ൅ ૛࢞ ૝ ൅ ࢞ ૞ૡ ൅ ૚૞࢞ ൌ ൥ ૚૚૝૛૚൩ ૚૙ ૚ ૚૝

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Since we know all entries are integers and that the entries represent letters, we know that ૙ ൑ ૛૝ ൅ ૛࢞ ൑ ૛૟ ૙൑૝൅࢞൑૛૟ ૙ ൑ ૞ૡ ൅ ૚૞࢞ ൑ ૛૟.

Solving these inequalities gives െ૚૛ ൑ ࢞ ൑ ૚ െ૝ ൑ ࢞ ൑ ૛૛ ૞ૡ ૜૛ െ ൑࢞൑െ . ૚૞ ૚૞

Because we know that ࢞ is an integer, the third inequality becomes െ૜ ൑ ࢞ ൑ െ૜, so we know that ࢞ൌെ૜. Then, the decoded message is ૛૝ ൅ ૛ሺെ૜ሻ ૝൅ሺെ૜ሻ ૞ૡ ൅ ૚૞ሺെ૜ሻ ࡯ൌ൥ ૚૚૝૛૚൩ Ǣڄࡰ ૚૙ ૚ ૚૝ thus, ૚ૡ ૚ ૚૜ , ࡯ൌ൥ ૚૚૝૛૚൩ڄࡰ ૚૙ ૚ ૚૝ and the decoded message is “RAMANUJAN.”

Closing (3 minutes) Ask students to write a brief answer to the question, “How do matrix inverses make encoding and decoding messages possible?” Then, have students share responses with a partner before sharing responses as a class. How do matrix inverses make encoding and decoding messages possible? à If an ݊ൈ݊ matrix ܧ is invertible, then it can be used to encode a message. We store the message in a To decode the message, we .ܯڄܧ has ݊ rows, and then encode it by multiplying ܯ where ,ܯ matrix .ܯൌܯڄܫൌܯڄ ሻܧڄ ଵିܧሻ ൌ ሺܯڄܧሺ ڄ ଵିܧ multiply

Exit Ticket (4 minutes) The message encoded in the problem in the Exit Ticket is “HYPATIA.” Hypatia (Hy-pay-shuh) of Alexandria (born between 350 and 370 C.E., died 415 C.E.) is one of the earliest known female mathematicians. She was the head of the Neoplatonic School in Alexandria, Egypt, and the head of the Library of Alexandria. She was murdered in a religious conflict. None of her mathematical works have survived.

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Name Date

Lesson 13: Using Matrix Operations for Encryption

Exit Ticket

ͳെʹ Morgan used matrix ܧൌቂ ቃ to encode the name of her favorite mathematician in the message െͳ ͵ െ͵ʹ, ͹, ͳͶ, ͳ, ͷʹ, ʹ, െͳ͵, െͳ.

a. How can you tell whether or not her message can be decoded?

b. Decode the message, or explain why the original message cannot be recovered.

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Exit Ticket Sample Solutions

૚െ૛ Morgan used matrix ࡱൌቂ ቃ to encode the name of her favorite mathematician in the message െ૚ ૜ െ૜૛, ૠ, ૚૝, ૚, ૞૛, ૛, െ૚૜, െ૚.

a. How can you tell whether or not her message can be decoded?

Since the matrixࡱ has determinant ܌܍ܜሺࡱሻ ൌ૜െ૛ൌ૚, we know that ܌܍ܜሺࡱሻ ്૙, so then a decoding matrix ࡰൌࡱି૚ exists.

b. Decode the message, or explain why the original message cannot be recovered. ૜૛ First, we place the coded message into a ૛ൈ૝ matrix ࡯. Using ࡰൌࡱି૚ ൌቂ ቃ, we have ૚૚ ࡯ڄࡹൌࡰ ૜૛ െ૜૛ ૠ ૚૝ ૚ ቂ ቃڄൌቂ ቃ ૚૚ ૞૛ ૛ െ૚૜ െ૚ െૢ૟ ൅ ૚૙૝ ૛૚ ൅ ૝ ૝૛ െ ૛૟ ૜ െ ૛ ൌቂ ቃ െ૜૛ ൅ ૞૛ ૠ ൅ ૛ ૚૝ െ ૚૜ ૚ െ ૚ ૡ૛૞૚૟૚ ൌቂ ቃ . ૛૙ ૢ ૚ ૙

The decoded message is “HYPATIA.”

Problem Set Sample Solutions Problems 1–4 are optional as they are practice on skills previously taught and assessed. Problems 6–9 allow students to practice the use of matrix multiplication for coding and decoding messages.

૚૜ െ૛ ૠ െ૞ ૜ ૙૙ ૚૙ 1. Let ࡭ൌቂ ቃ, ࡮ൌቂ ቃ, ࡯ൌቂ ቃ, ࢆൌቂ ቃ, and ࡵൌቂ ቃ. Evaluate the following. ૛૞ ૜െ૝ ૛െ૚ ૙૙ ૙૚ a. ࡭൅࡮ b. ࡮൅࡭ െ૚ ૚૙ െ૚ ૚૙ ቂ ቃ ቂ ቃ ૞૚ ૞૚

c. ࡭൅ሺ࡮൅࡯ሻ d. ሺ࡭൅࡮ሻ ൅࡯ െ૟ ૚૜ െ૟ ૚૜ ቂ ቃ ቂ ቃ ૠ૙ ૠ૙

e. ࡭൅ࡵ f. ࡭൅ࢆ ૛૜ ૚૜ ቂ ቃ ቂ ቃ ૛૟ ૛૞

࡭ڄࢆ h. ࢆڄg. ࡭ ૙૙ ૙૙ ቂ ቃ ቂ ቃ ૙૙ ૙૙

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࡮ڄ࡭ j. ࡭ڄi. ࡵ ૚૜ ૠെ૞ ቂ ቃ ቂ ቃ ૛૞ ૚૚ െ૟

࡯ڄ࡭ l. ࡭ڄk. ࡮ ૚૛ ૛ૢ ૚૙ ቂ ቃ ቂ ቃ െ૞ െ૚૚ ૙૚

࡯ڄ࡮൅࡭ڄ࡭ n. ࡭ڄm. ࡯ ૚૙ ૡെ૞ ቂ ቃ ቂ ቃ ૙૚ ૚૚ െ૞

࡯ڄ࡮ڄሺ࡮൅࡯ሻ p. ࡭ڄo. ࡭ ૡെ૞ െ૝૞ ૛૟ ቂ ቃ ቂ ቃ ૚૚ െ૞ െ૟ૠ ૜ૢ

࡮ڄ࡯ڄ࡭ r. ࡭ڄ࡮ڄq. ࡯ െૠ૞ െ૚ૠૡ െ૛ ૠ ቂ ቃ ቂ ቃ ૛ૢ ૟ૢ ૜െ૝

s. ܌܍ܜሺ࡭ሻ t. ܌܍ܜሺ࡮ሻ െ૚ െ૚૜

u. ܌܍ܜሺ࡯ሻ v. ܌܍ܜሺࢆሻ െ૚ ૙

࡯ሻ ڄ ࡮ ڄ ሺ࡭ܜ܍܌ .ሺࡵሻ xܜ܍܌ .w ૚ െ૚૜

࡭ሻ ڄ ࡮ ڄ ሺ࡯ܜ܍܌ .y െ૚૜

૙૙ ૙૙ 2. For any ૛ൈ૛ matrix ࡭ and any real number ࢑, show that if ࢑࡭ ൌ ቂ ቃ, then ࢑ൌ૙ or ࡭ൌቂ ቃ. ૙૙ ૙૙ ࢇ࢈ ࢑ࢇ ࢑࢈ ૙૙ Let ࡭ൌቂ ቃ; then ࢑࡭ ൌ ቂ ቃ. Suppose that ࢑࡭ ൌ ቂ ቃ. ࢉࢊ ࢑ࢉ ࢑ࢊ ૙૙ Case 1: Suppose ࢑്૙. Then, ࢑ࢇ ൌ ૙, ࢑࢈ ൌ ૙, ࢑ࢉ ൌ ૙, and ࢑ࢊ ൌ ૙; all imply that ࢇൌ࢈ൌࢉൌࢊൌ૙. Thus, if ૙૙ ࢑്૙, then ࡭ൌቂ ቃ. ૙૙ ૙૙ Case 2: Suppose that ࡭്ቂ ቃ. Then, at least one of ࢇ, ࢈, ࢉ, and ࢊ is not zero, so ࢑ࢇ ൌ ૙, ࢑࢈ ൌ ૙, ࢑ࢉ ൌ ૙, and ૙૙ ࢑ࢊ ൌ ૙ imply that ࢑ൌ૙. ૙૙ ૙૙ Thus, if ࢑࡭ ൌ ቂ ቃ, then either ࢑ൌ૙ or ࡭ൌቂ ቃ. ૙૙ ૙૙

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െ૜ ૛ െ૜ ૛ 3. Claire claims that she multiplied ࡭ൌቂ ቃ by another matrix ࢄ and obtained ቂ ቃ as her result. What ૙૝ ૙૝ matrix did she multiply by? How do you know? ૚૙ She multiplied࡭ by the multiplicative identity matrixࡵ ൌ ቂ ቃ. Since the product is a ૛ൈ૛ matrix, we know that ૙૚ ࢇ࢈ െ૜ࢇ ൅ ૛ࢉ െ૜࢈ ൅ ૛ࢊ ࢄ ൌ ቂ ቃ. Since ڄ ࢄ gives࡭ ڄ ࢄ is a ૛ൈ૛ matrix of the form ൌቂ ቃ . Multiplying࡭ ࢉࢊ ૙ࢇ ൅ ૝ࢉ ૙࢈ ൅ ૝ࢊ :ࢄൌ࡭, we have the following system of equationsڄ࡭ െ૜ࢇ ൅ ૛ࢉ ൌ െ૜ െ૜࢈ ൅ ૛ࢊ ൌ ૛ ૙ࢇ ൅ ૝ࢉ ൌ ૙ ૙࢈ ൅ ૝ࢊ ൌ ૝.

The third and fourth equations give ࢉൌ૙ and ࢊൌ૚, respectively, and substituting into the first two equations gives െ૜ࢇ ൌ െ૜ and െ૜࢈ ൅ ૛ ൌ ૛. Thus, ࢇൌ૚ and ࢈ൌ૙, and the matrix ࢄ must be ࢄൌࡵ.

4. Show that the only matrix ࡮ such that ࡭൅࡮ൌ࡭ is the zero matrix. ࢇ࢈ ࢞࢟ ࢇ൅࢞ ࢈൅࢟ Let ࡭ൌቂ ቃ and ࡮ൌቂ ቃ; then, we have ࡭൅࡮ൌቂ ቃ, ࢇ൅࢞ൌࢇ, ࢈൅࢟ൌ࢈, ࢉ൅ࢠൌࢉ, and ࢉࢊ ࢠ࢝ ࢉ൅ࢠ ࢊ൅࢝ ૙૙ ࢊ൅࢝ൌࢊ. In each case, solving for the elements of ࡮, we find that ࡮ൌቂ ቃ. ૙૙

ࢇ૙ 5. A ૛ൈ૛ matrix of the form ቂ ቃ is a diagonal matrix. Daniel calculated ૙࢈

૛૙ ૛૜ ૝૟ ቂ ቃൌቂ ቃڄቂ ቃ ૙૛ ૞െ૜ ૚૙ െ૟ ૛૜ ૛૙ ૝૟ ቂ ቃൌቂ ቃڄቂ ቃ ૞െ૜ ૙૛ ૚૙ െ૟

.ࢄڄ࡭ൌ࡭ڄand concluded that if ࢄ is a diagonal matrix and ࡭ is any other matrix, then ࢄ a. Is there anything wrong with Daniel’s reasoning? Prove or disprove that if ࢄis adiagonal ૛ൈ૛ matrix, then .ࢄ for any other matrix ࡭ڄ࡭ൌ࡭ڄࢄ

Yes, there is something wrong with Daniel’s reasoning. A single example does not establish that a statement is true, and the example he calculated used a special case of a diagonal matrix in which the entries on the main diagonal are equal. ૚૛ ૛૙ ૛૝ ૛૟ ࢄൌቂ ቃ. Thus, it is not true thatڄ࡭ ൌ ቂ ቃ and ࡭ ڄ If࡭ ൌ ቂ ቃ and ࢄൌቂ ቃ, thenࢄ ૜૝ ૙૜ ૢ૚૛ ૟૚૛ .ࢄ for all diagonal matrices ࢄ and all other matrices ࡭ڄ࡭ൌ࡭ڄࢄ

ࢉ૙૙ ࢄڄ࡭ൌ࡭ڄb. For ૜ൈ૜ matrices, Elda claims that only diagonal matrices of the form ࢄൌ൥૙ࢉ૙൩ satisfy ࢄ ૙૙ࢉ for any other ૜ൈ૜ matrix ࡭. Is her claim correct? ࢉ૙૙ Elda is correct since ࢄൌ൥૙ࢉ૙൩ ൌࢉࡵ. Then, ૙૙ࢉ ࢄڄሺࢉࡵሻ ൌ࡭ڄࡵሻࢉൌ࡭ڄ࡭ሻ ൌࢉ࡭ൌ࡭ࢉൌሺ࡭ڄ࡭ൌࢉሺࡵڄ࡭ൌࢉࡵڄࢄ for all matrices ࡭.

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૛૛ 6. Calvin encoded a message using ࡱൌቂ ቃ,giving the coded message ૝, ૛ૡ, ૝૛, ૞૟, ૛, െ૟, െ૚, ૞૛. Decode the െ૚ ૜ message, or explain why the original message cannot be recovered. ૝૛ૡ૝૛૞૟ Putting the message in a ૛ൈ૝ matrix, we have ࡯ൌቂ ቃ. We can decode the message with ૛െ૟െ૚૞૛ ૜ ૛ ૚ ૜െ૛ െ ࡰൌࡱି૚ ൌ ቂ ቃൌ቎ૡ ૡ቏. Then, the original message is found in message ࡹ: ૟െ൫െ૛൯ ૚૛ ૚ ૛ ૡ ૡ ૜ ૛ െ ૝૛ૡ૝૛૞૟ ૚૚૛૚૟ૡ . ቂ ቃൌቂ ቃڄ࡯ൌ൦ૡ ૡ൪ڄࡹൌࡰ ૚ ૛ ૛െ૟െ૚૞૛ ૚૛ ૞૛૙ ૡ ૡ The original message is “ALPHABET.”

૚૚െ૚ 7. Decode the message below using the matrix ࡰൌ൥െ૚ ૙ ૛ ൩: ૚૛૚ ૛૛, ૚ૠ, ૛૝, ૢ, െ૚, ૚૝, െૢ, ૜૝, ૝૝, ૟૝, ૝ૠ, ૠૠ. െ૝ െ૜ ૛ ૛૛ ૚ૠ ૛૝ ૢ ૜૚ૡ૛૞૚૟ ,൥െ૚ ૚૝ െૢ ૜૝൩ ൌ ൥૛૙ ૚૞ ૠ ૚ૡ൩. Then ڄ The decoded message is found by multiplying ൥ ૜૛െ૚൩ െ૛ െ૚ ૚ ૝૝ ૟૝ ૝ૠ ૠૠ ૚૚૟ૡ૛૞ the message is “CRYPTOGRAPHY.”

૚૚ ૟૜૜૚૞૚૝ 8. Brandon encoded his name with the matrix ࡱൌቂ ቃ, producing the matrix ࡯ൌቂ ቃ. Decode the ૛૛ ૚૛ ૟૟ ૜૙ ૛ૡ message, or explain why the original message cannot be recovered.

Brandon used a matrix that is not invertible. The original matrix cannot be recovered.

૚૛ 9. Janelle used the encoding matrix ࡱൌቂ ቃ to encode the message “FROG” by multiplying ૚െ૚ ૟૚ૡ ૚૛ ૛૝ ૜૙ ቂ ቃൌቂ ቃ. When Taylor decoded it, she computedڄ࡯ൌቂ ቃ ૚૞ ૠ ૚െ૚ ૛૛ ૜ૠ െ૚ ૛ ૛૝ ૜૙ ૛૙ ૝૝ ?ቂ ቃൌቂ ቃ. What went wrongڄࡹൌቂ ቃ ૚૚૛૛ ૜ૠ ૛െૠ ૛૝ ૜૙ Janelle multiplied her matrices in the wrong order. When Janelle tried to decode the matrix ࡯ൌቂ ቃ using the ૛૛ ૜ૠ ૚૛ି૚ decoding matrix ࡰൌቂ ቃ , she ended up calculating ૚െ૚ ࡱڄࡹڄ࡯ൌࡰڄࡰ .ࡱڄࡹڄ ൌࡱି૚

ࡱ്ࡹ, Taylor was unable to recover the originalڄࡹڄ Because matrix multiplication is not commutative, ࡱି૚ message.

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Name Date

1. Kyle wishes to expand his business and is entertaining four possible options. If he builds a new store, he expects to make a profit of ͻ million dollars if the market remains strong; however, if market growth declines, he could incur a loss of ͷ million dollars. If Kyle invests in a franchise, he could profit Ͷ million dollars in a strong market but lose ͵ million dollars in a declining market. If he modernizes his current facilities, he could profit Ͷ million dollars in a strong market but lose ʹ million dollars in a declining one. If he sells his business, he will make a profit of ʹ million dollars irrespective of the state of the market.

a. Write down a Ͷൈʹ payoff matrix ܲ summarizing the profits and losses Kyle could expect to see with all possible scenarios. (Record a loss as a profit in a negative amount.) Explain how to interpret your matrix.

b. Kyle realizes that all his figures need to be adjusted by ͳͲΨ in magnitude due to inflation costs. What is the appropriate value of a real number ߣ so that the matrix ߣܲ represents a correctly adjusted payoff matrix? Explain your reasoning. Write down the new payoff matrix ߣܲ.

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c. Kyle hopes to receive a cash donation of ͳ million dollars. If he does, all the figures in his payoff matrix will increase by ͳ million dollars.

Write down a matrix ܳ so that if Kyle does receive this donation, his new payoff matrix is given by ܳ൅ߣܲ. Explain your thinking.

2. The following diagram shows a map of three land masses, numbered region 1, region 2, and region 3, connected via bridges over water. Each bridge can be traversed in either direction.

a. Write down a ͵ൈ͵ matrix ܣ with ܽ௜௝, for ݅ൌͳ, ʹ, or ͵ and ݆ൌͳ, ʹ, or ͵, equal to the number of ways to walk from region ݅ to region ݆ by crossing exactly one bridge. Notice that there are no paths that start and end in the same region crossing exactly one bridge.

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b. Compute the matrix product ܣଶ.

c. Show that there are ͳͲ walking routes that start and end in region 2, crossing over water exactly twice. Assume each bridge, when crossed, is fully traversed to the next land mass.

d. How many walking routes are there from region 3 to region 2 that cross over water exactly three times? Again, assume each bridge is fully traversed to the next land mass.

e. If the number of bridges between each pair of land masses is doubled, how does the answer to part (d) change? That is, what would be the new count of routes from region 3 to region 2 that cross over water exactly three times?

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͵െͷ െʹ ͳ 3. Let ܲൌቂ ቃ and ܳൌቂ ቃ. ͷ͵ െͳ െʹ

a. Show the work needed and compute ʹܲ െ ͵ܳ.

b. Show the work needed and compute ܲܳ.

c. Show that ܲଶܳൌܲܳܲ.

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a. Show that if the matrix equation ሺܣ൅ܤሻଶ ൌܣଶ ൅ʹܣܤ൅ܤଶ holds for two square matrices ܣ and ܤ of the same dimension, then these two matrices commute under multiplication.

b. Give an example of a pair of ʹൈʹ matrices ܣ and ܤ for which ሺܣ൅ܤሻଶ ്ܣଶ ൅ʹܣܤ൅ܤଶ.

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c. In general, does ܣܤ ൌ ܤܣ? Explain.

d. In general, does ܣሺܤ൅ܥሻ ൌܣܤ൅ܣܥ? Explain.

e. In general, does ܣሺܤܥሻ ൌ ሺܣܤሻܥ? Explain.

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ݔ the ͵ൈ͵ zero matrix. Let the ͵ൈͳ column ݔൌቈݕ቉ represent a ܣ be the ͵ൈ͵ identity matrix and ܫ Let .5 ݖ ʹͲͷ point in three-dimensional space. Also, set ܲൌ൥ͲͲͲ൩. ͲͲͶ

a. Use examples to illustrate how matrix ܣ plays the same role in matrix addition that the number Ͳ plays in real number addition. Include an explanation of this role in your response.

b. Use examples to illustrate how matrix ܫ plays the same role in matrix multiplication that the number ͳ plays in real number multiplication. Include an explanation of this role in your response.

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c. What is the row 3, column 3 entry of ሺܣܲ ൅ ܫሻଶ? Explain how you obtained your answer.

d. Show that ሺܲ െ ͳሻሺܲ ൅ ͳሻequals ܲଶ െܫ.

.e. Show that ܲݔ is sure to be a point in the ݔݖ-plane in three-dimensional space

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f. Is there a ͵ൈ͵ matrix ܳ, not necessarily the matrix inverse for ܲ, for which ܳܲݔ ൌ ݔ for every .ൈͳ column ݔ representing a point? Explain your answer͵

g. Does the matrix ܲ have a matrix inverse? Explain your answer.

h. What is the determinant of the matrix ܲ?

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ͳ 6. What is the image of the point given by the ͵ൈͳ column matrix ൥ ʹ ൩ when it is rotated Ͷͷι about the െͳ ݖ-axis in the counterclockwise direction (as according to the orientation of the ݔݕ-plane) and then ͳͺͲι ?about the ݕ-axis

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A Progression Toward Mastery STEP 1 STEP 2 STEP 3 STEP 4 Assessment Missing or Missing or A correct answer A correct answer Task Item incorrect answer incorrect answer with some supported by and little evidence but evidence of evidence of substantial of reasoning or some reasoning or reasoning or evidence of solid application of application of application of reasoning or mathematics to mathematics to mathematics to application of solve the problem. solve the problem. solve the problem, mathematics to OR an incorrect solve the problem. answer with substantial evidence of solid reasoning or application of mathematics to solve the problem. 1 a Student gives an Student gives a correct Student gives a correct Student gives a correct incorrect answer and answer but not as a matrix with no matrix with a clear not as a matrix. matrix. explanation of how to explanation of how to interpret it. interpret it.

b Student gives an Student gives a correct Student gives a correct Student gives a correct incorrect answer and answer but not as a matrix with no matrix with a clear not as a matrix. matrix. explanation of explanation of reasoning supporting reasoning supporting the calculation. the calculations. c Student gives an Student gives a correct Student gives a correct Student gives a correct incorrect answer, not as answer but not as a matrix with no matrix with clear a matrix, and with no matrix. explanation of explanation of explanation. reasoning. reasoning.

2 a Student shows little or Student writes a matrix, Student writes a ͵ൈ͵ Student writes a correct no evidence of writing a but it is not a ͵ൈ͵ matrix, but one or two ͵ൈ͵ matrix. matrix. matrix. entries are incorrect. OR Student writes a ͵ൈ͵matrix with three or more mistakes. b Student shows little or Student attempts Student multiplies the Student multiplies the no evidence of matrix multiplication matrices but makes a matrices correctly. understanding matrix but has major errors. small computational multiplication. error.

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c Student shows little or Student answers the Student answers Student answers no evidence of question using matrix correctly but does not correctly and fully interpreting matrix but uses the wrong fully explain the answer. explains the answer. entries. entry.

d Student shows little or Student multiplies to Student multiplies to Student multiplies to ଷ ଷ no evidence of matrix find ͵ but does not find ܣ but makes find ܣ , identifies the operations needed to explain the answer or mistakes multiplying. correct answer, and answer the question. explains the wrong The answer is incorrect, justifies it correctly. entry as the answer. but the reasoning and justification are correct. e Student shows little or Student multiplies to Student multiplies to Student multiplies to ଷ ଷ ଷ no evidence of matrix find ͺܣ but does not find ͺܣ and explains find ͺܣ and explains operations needed to explain the answer or the answer but not the answer completely. answer the question. explains the wrong completely. entry as the answer.

3 a Student shows little or Student shows some Student shows Student shows no evidence of matrix knowledge of matrix knowledge of matrix knowledge of matrix operations. operations but makes operations but has one operations arriving at mistakes on two or entry wrong in the final the correct final matrix. more entries in the final matrix. matrix. b Student shows little or Student shows some Student shows Student shows no evidence of matrix knowledge of matrix knowledge of matrix knowledge of matrix multiplication. multiplication but multiplication but has multiplication arriving makes mistakes on two one entry wrong in the at the correct final or more entries in the final matrix. matrix. final matrix. c Student shows little or Student shows some Student shows Student shows no evidence of matrix knowledge of matrix knowledge of matrix knowledge of matrix multiplication. multiplication but multiplication, finding multiplication, finding makes mistakes leading the correct matrices but correct matrices and to an incorrect answer. not explaining that they explaining that they are are equal. equal. 4 a Student makes little or Student expands the Student expands the Student expands the ଶ ଶ ଶ no attempt to answer binomial ሺܣ൅ܤሻ but binomial ሺܣ൅ܤሻ and binomial ሺܣ൅ܤሻ , the question. does not continue with shows that shows that ܤܣ ൌ ܣܤ, proof, or the steps are ܤܣ ൌ ܣܤ but does not and explains the not correct. explain the reasoning reasoning that these for these matrices being matrices are commutative under commutative under multiplication. multiplication.

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b Student makes little or Student lists two Student lists two Student lists two no attempt to find the ʹൈʹ matrices but does ʹൈʹ matrices but ʹൈʹ matrices and matrices. not support or prove makes mistakes in the shows supporting the answer. calculations or evidence to verify the reasoning to support answer. the answer.

c Student states that Student states that Student states that Student states ܣܤ ൌ ܤܣ is true for ܣܤ ് ܤܣbut does not ܣܤ ് ܤܣand attempts thatܣܤ ് ܤܣ, explains matrices. support the answer to explain reasoning but the reasoning, and with reasoning. does not use the term states that matrix commute or multiplication is not commutative. generally commutative. d Student states that Student states that Student states that Student states that ܣሺܤ ൅ ܥሻ ് ܣܤ ൅ ܤܥ. ܣሺܤ ൅ ܥሻ ൌ ܣܤ ൅ ܤܥ ܣሺܤ ൅ ܥሻ ൌ ܣܤ ൅ ܣܥ ܣሺܤ ൅ ܥሻ ൌ ܣܤ ൅ ܣܥ but does not support and attempts to explain and explains reasoning the answer with but makes minor errors correctly. reasoning. in reasoning. e Student states that Student states that Student states that Student states that ܣሺܤܥሻ ് ሺܣܤሻܥ. ܣሺܤܥሻ ൌ ሺܣܤሻܥ but ܣሺܤܥሻ ൌ ሺܣܤሻܥ and ܣሺܤܥሻ ൌ ሺܣܤሻܥ and does not support the attempts to explain but explains reasoning answer with reasoning. makes minor errors in correctly. reasoning. 5 a Student shows little or Student writes the Student writes the Student writes the no understanding of the ͵ൈ͵ zero matrix but ͵ൈ͵ zero matrix, ͵ൈ͵ zero matrix, ͵ൈ͵ zero matrix. does not explain its role showing an example of shows an example of its in matrix addition. its role in matrix role in matrix addition, addition, but does not and explains the explain the connection connection to the to the number Ͳ in real number Ͳ in real number addition. number addition. b Student shows little or Student writes the Student writes the Student writes the no understanding of the ͵ൈ͵ identity matrix ͵ൈ͵ identity matrix, ͵ൈ͵ identity matrix, ͵ൈ͵ identity matrix. but does not explain its showing an example of shows an example of its role in matrix its role in matrix role in matrix multiplication. multiplication, but does multiplication, and not explain the explains the connection connection to the to the number ͳ in real number ͳ in real number multiplication. number multiplication. c Student shows little or Student finds ሺܣܲ ൅ ܫሻଶ Student finds ሺܣܲ ൅ ܫሻଶ Student finds ଶ no understanding of but does not identify and identifies the entry ሺܣܲ ൅ ܫሻ , identifies matrix operations. the entry in row 3 in row 3 column 3 but the entry in row 3 column 3. does not explain the column 3, and explains answer. the answer.

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d Student shows little or Student calculates two Student calculates Student calculates no understanding of of ܲെͳ, ܲ൅ͳ, or ሺܲ െ ͳሻሺܲ ൅ ͳሻ and ሺܲ െ ͳሻሺܲ ൅ ͳሻ and matrix operations. ܲଶ െܫ correctly. ܲଶ െܫ but does not ܲଶ െܫ and explains why explain why the the expressions are expressions are equal. equal.

e Student shows little or Student sets up ܲݔ but Student sets up and Students sets up and no understanding of does not find the matrix finds the matrix finds the matrix matrix operations. representing the representing ܲݔ but representing ܲݔ and product. does not explain the explains the meaning of meaning of the point in the point in 3-dimensional space. 3-dimensional space.

f Student shows little or Student finds ܲݔ but Student finds ܳܲݔ and Student finds ܳܲݔ and no understanding of does not find ܳܲݔ or attempts to explain, but clearly shows that matrix operations. explain reasoning. not clearly, why matrix matrix ܳ cannot exist. ܳ cannot exist.

g Student shows little or Student shows some Student says that the Student clearly explains no understanding of understanding of inverse does not exist why the inverse matrix inverse matrices. inverse matrices but and attempts to does not exist. cannot answer or explain, but the explain the question. explanation has minor mistakes. h Student shows little or Student incorrectly Student states that the Student explains clearly no understanding of the attempts to find the determinant is zero but why the determinant is determinant of matrix determinant. with no explanation. zero. ܲ.

6 Student shows little or Student attempts to Student writes the Student writes the no understanding of write the matrices correct matrices correct matrices matrices producing producing rotations but producing the rotations producing the rotations rotations. with errors or with only required but makes required and calculates one correct. calculation errors the correct final image leading to an incorrect point. final answer.

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Name Date

9 െ5 4 െ3 We have P ൌ൦ ൪. 4 െ2 22 Here the four rows correspond to, in turn, the options of building a new store, investing in a franchise, modernizing, and selling. The first column gives the payoffs in a strong market, and the second column gives the payoffs in a declining market.

All entries are in units of millions of dollars.

Note: Other presentations for the matrix PǤare possible.

Each entry in the matrix needs to increase 10% in magnitude. This can be accomplished by multiplying each entry by 1.10. If we set ž ൌ 1.1, then 9.9 െ5.5 4.4 െ3.3 žP ൌ൦ ൪ is the appropriate new payoff matrix. 4.4 െ2.2 2.2 2.2

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c. Kyle hopes to receive a cash donation of ͳ million dollars. If he does, all the figures in his payoff matrix will increase by ͳ million dollars.

Write down a matrix ܳ so that if Kyle does receive this donation, his new payoff matrix is given by ܳ൅ߣܲ. Explain your thinking.

11 11 Set Q ൌ൦ ൪. Then Q+žP is the matrix žP with 1 added to each entry. This is the 11 11 effect we seek, increasing each expected payoff by 1 million dollars.

2. The following diagram shows a map of three land masses, numbered region 1, region 2, and region 3, connected via bridges over water. Each bridge can be traversed in either direction.

011 Aൌ ൥103൩ 130

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b. Compute the matrix product ܣଶ.

23 3 2 A ൌ൥3101൩ 3110

c. Show that there are ͳͲ walking routes that start and end in region 2, crossing over water exactly twice. Assume each bridge, when crossed, is fully traversed to the next land mass.

The entries of A2 give the number of paths via two bridges between land regions. As the row 2, column 2 entry of A2 is 10, this is the count of two-bridge journeys that start and end in region 2.

d. How many walking routes are there from region 3 to region 2 that cross over water exactly three times? Again, assume each bridge is fully traversed to the next land mass.

The entries of A3 give the counts of three-bridge journeys between land masses. We seek the row 3, column 2 entry of the product.

23 3011 ൥3101൩൥103൩ 3110130 .3ሻ=33. Therefore, there are 33 such routesڄ0ሻ+ሺ10ڄ1ሻ+ሺ1ڄThis entry is ሺ3

We are now working with the matrix 2A. The number of routes from region 3 to region 2 via three bridges is the row 3, column 2 entry of ሺ2Aሻ3=8A3. As all the entries are multiplied by eight, there are 8×33=264 routes of the particular type we seek.

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͵െͷ െʹ ͳ 3. Let ܲൌቂ ቃ and ܳൌቂ ቃ. ͷ͵ െͳ െʹ

a. Show the work needed and compute ʹܲ െ ͵ܳ.

6–10 –6 3 12 െ13 2P െ 3Q ൌቂ ቃെቂ ቃൌቂ ቃ 10 6 –3 –6 13 12

b. Show the work needed and compute ܲܳ.

–6 ൅ 53൅ 10 –1 13 PQ ൌ ቂ ቃൌቂ ቃ –10 െ 35െ 6 –13 –1

c. Show that ܲଶܳൌܲܳܲ.

–1 13 PQ = ቂ ቃ –13 –1 62 44 P2Q ൌ PሺPQሻ ൌቈ ቉ –44 62 –1 13 3–5 62 44 PQP ൌ ሺPQሻP ൌቂ ቃቂ ቃൌቂ ቃ –13 –1 53 –44 62 These are identical matrices.

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A STORY OF FUNCTIONS Mid-Module Assessment Task M2 PRECALCULUS AND ADVANCED TOPICS

a. Show that if the matrix equation ሺܣ൅ܤሻଶ ൌܣଶ ൅ʹܣܤ൅ܤଶ holds for two square matrices ܣ and ܤ of the same dimension, then these two matrices commute under multiplication.

We have ሺA+Bሻ2 ൌ (A+B)(A+B).

By the distributive rule, which does hold for matrices, this equals AሺA+Bሻ+B(A+B), which, again by the distributive rule, equals A2+AB+BA+B2.

On the other hand, A2+2AB+B2 equals A2+AB+A+B2.

So, if ሺA+Bሻ2 ൌ A2+2AB+B2, then we have A2+AB+BA+B2 ൌ A2+AB+A+B2.

Adding െA2 and െAB and െB2 to each side of this equation gives BA ൌ AB.

This shows that A and B commute under multiplication in this special case when 2 (A+B) =A2 + 2AB + B2, but in general, matrix multiplication is not commutative.

b. Give an example of a pair of ʹൈʹ matrices ܣ and ܤ for which ሺܣ൅ܤሻଶ ്ܣଶ ൅ʹܣܤ൅ܤଶ.

10 A pair of matrices that do not commute under multiplication, such asA ൌቂ ቃ and 11 11 B ൌቂ ቃ, should do the trick. 01 212 54 To check: ሺA+Bሻ2 ൌቂ ቃ ൌቂ ቃ 12 45 2 2 2 2 10 1011 11 A +2AB+B ൌቂ ቃ +2 ቂ ቃቂ ቃ+ቂ ቃ 11 1101 01 10 11 12 ൌቂ ቃ+2 ቂ ቃ+ቂ ቃ 21 12 01 44 ൌቂ ቃ 46

These are indeed different.

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c. In general, does ܣܤ ൌ ܤܣ? Explain.

No. Since matrix multiplication can represent linear transformations, we know that they will not always commute since linear transformations do not always commute.

d. In general, does ܣሺܤ൅ܥሻ ൌܣܤ൅ܣܥ? Explain.

Yes. Consider the effect on the point x made by both sides of the equation. On the left- hand side, the transformation B+C is applied to the point x, but we know that this is the same as Bx+Cxfrom our work with linear transformations. Applying the transformation represented by A to either Bx+Cx or ሺB+Cሻx now is ABx+ACx because they work like linear maps.

e. In general, does ܣሺܤܥሻ ൌ ሺܣܤሻܥ? Explain.

Yes. If we consider the effect the matrices on both sides make on a point x, the matrices are applied in the exact same order, C, then B, then A, regardless of whether AB is computed first or BC is computed first.

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A STORY OF FUNCTIONS Mid-Module Assessment Task M2 PRECALCULUS AND ADVANCED TOPICS

a. Use examples to illustrate how matrix ܣ plays the same role in matrix addition that the number Ͳ plays in real number addition. Include an explanation of this role in your response.

The sum of two 3×3 matrices is determined by adding entries in corresponding positions of the two matrices to produce a new 3×3 matrix. Each and every entry of matrix A is zero, so a sum of the form A + P, where P is another 3×3 matrix, is given by adding zero to each entry of P. Thus, A + P = P. For example: 000 147 0+10+40+7 147 ൥000൩൅൥258൩ൌ൥0+20+50+8൩ൌ൥258൩. 000 369 0+30+60+9 369 This is analogous to the role of zero in the real number system: 0+p=p for every real number p.

In the same way, P+A=P for all 3×3 matrices P, analogous to p+0=p for all real numbers p.

b. Use examples to illustrate how matrix ܫ plays the same role in matrix multiplication that the number ͳ plays in real number multiplication. Include an explanation of this role in your response.

100 We have I ൌ൥010൩. By the definition of matrix multiplication, we see, for example: 001 9ሻڄ8ሻ+ሺ0ڄ7ሻ+ሺ0ڄ6ሻሺ1ڄ5ሻ+ሺ0ڄ4ሻ+ሺ0ڄ3ሻሺ1ڄ2ሻ+ሺ0ڄ1ሻ+ሺ0ڄሺ1 147 100 9ሻ቏ڄ8ሻ+ሺ0ڄ7ሻ+ሺ1ڄ6ሻሺ0ڄ5ሻ+ሺ0ڄ4ሻ+ሺ1ڄ3ሻሺ0ڄ2ሻ+ሺ0ڄ1ሻ+ሺ1ڄ൥010൩ൈ൥258൩ൌ቎ሺ0 9ሻڄ8ሻ+ሺ1ڄ7ሻ+ሺ0ڄ6ሻሺ0ڄ5ሻ+ሺ1ڄ4ሻ+ሺ0ڄ3ሻሺ0ڄ2ሻ+ሺ1ڄ1ሻ+ሺ0ڄሺ0 369 001 147 ൌ൥258൩ 369

We have, in general, that I×P=P for every 3×3 matrix P. We also have P×I=P.

Thus, the matrix I plays the role of the number I in real number arithmetic where 1×p=p and p×1=p for each real number p.

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A STORY OF FUNCTIONS Mid-Module Assessment Task M2 PRECALCULUS AND ADVANCED TOPICS

c. What is the row 3, column 3 entry of ሺܣܲ ൅ ܫሻଶ? Explain how you obtained your answer.

Since A is the zero matrix, AP equals the zero matrix. That is, AP = A.

Thus, ሺAP + Iሻ2 = ሺA + Iሻ2 = I2 = I.

So, ሺAP+Iሻ2 is just the 3×3 identity matrix. The row 3, column 3 entry is thus 1.

d. Show that ሺܲ െ ͳሻሺܲ ൅ ͳሻequals ܲଶ െܫ.

We have ሺP െ IሻሺP+IሻൌP2 െ IP+PI െ I2 (using the distributive property, which holds for matrices). Since PI=P, IP=P, and I2=I, this equals P2 െ P+P െ IൌP2 െ I.

.e. Show that ܲݔ is sure to be a point in the ݔݖ-plane in three-dimensional space

We have 205x 2x+5z Px = ൥000൩ቈy቉ = ൥ 0 ൩Ǥ 004z 4z The image is a point with y-coordinate zero, and so it is a point in the xz-plane in three-dimensional space.

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f. Is there a ͵ൈ͵ matrix ܳ, not necessarily the matrix inverse for ܲ, for which ܳܲݔ ൌ ݔ for every .ൈͳ column ݔ representing a point? Explain your answer͵

0 2050 0 If there were such a matrix Q, then QPx = x for x = ൥1൩. But Px = ൥000൩൥1൩ = ൥0൩, 0 0040 0 0 0 0 and so QPx = Q ൥0൩ = ൥0൩, which is not x = ൥1൩ after all. There can be no such matrix Q. 0 0 0

g. Does the matrix ܲ have a matrix inverse? Explain your answer.

ି ି If P had a matrix inverse P 1, then we would have P 1P=I and so P-1Px=x for all 3×1 columns x representing a point. By part (d), there is no such matrix.

By part (c), P takes all points in the three-dimensional space and collapses them to a plane. So there are points that are taken to the same image point by P. Thus, no ି inverse transformation, P 1, can exist.

h. What is the determinant of the matrix ܲ?

The unit cube is mapped onto a plane, and so the image of the unit cube under P has zero volume. The determinant of P is thus 0.

By part (e), P has no multiplicative inverse, and so its determinant must be 0.

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A STORY OF FUNCTIONS Mid-Module Assessment Task M2 PRECALCULUS AND ADVANCED TOPICS

A rotation about the z-axis of 45° is effected by multiplication by the matrix: 1 1 ې െ 0 ۍ ۑ ξ2 ξ2ێ cosሺ45°ሻ െ sinሺ45°ሻ 0 ൥ ൩ Ǥ ۑ 1 1 ێ = R1 = sinሺ45°ሻ cosሺ45°ሻ 0 0 ۑ ξ2 ξ2ێ001 ے001 ۏ So, the image of the point under the first rotation is 1 ې –ۍ ۑξ2 ێ 1 . ۑ 3 ێ = R1 ൥ 2 ൩ ۑ ξ2 ێ 1– ے 1- ۏ A rotation of 180° about the y-axis is effected by multiplication by

cosሺ180°ሻ 0 െ sinሺ180°ሻ –1 0 0 R2 ൌ൥ 01 0൩ൌ൥ 010൩ . sinሺ180°ሻ 0cosሺ180°ሻ 00–1

The final image point we seek is thus 1 1 ې ۍ ې –ۍ ۑξ2ێ ۑξ2 ێ Ǥۑ 3 ێ ൌ ۑ 3 ێ R2 ۑξ2ێ ۑ ξ2 ێ ے 1 ۏ ے 1– ۏ

238 Module 2: Vectors and Matrices

^dKZzK&&hEd/KE^

DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ

PRECALCULUS AND ADVANCED TOPICS ͻDKh>2

Topic C ^ǇƐƚĞŵƐŽĨ>ŝŶĞĂƌƋƵĂƚŝŽŶƐ

&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of Ͳ and ͳ in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

(+) Represent a system of linear equation as a single matrix equation in a vector variable.

(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension ͵ൈ͵ or greater). /ŶƐƚƌƵĐƚŝŽŶĂůĂǇƐ͗ 3 >ĞƐƐŽŶϭϰ: Solving Equations Involving Linear Transformations of the Coordinate Plane (P)1 >ĞƐƐŽŶϭϱ: Solving Equations Involving Linear Transformations of the Coordinate Space (P) >ĞƐƐŽŶϭϲ: Solving General Systems of Linear Equations (P)

Topic C provides a third context for the appearance of matrices via the study of systems of linear equations. Students see that a system of linear equations can be represented as a single matrix equation in a vector variable and can be solved with the aid of the multiplicative inverse to a matrix, if it exists.

In Lesson 14, students explore the relationship between linear transformations of points in two-dimensional space and systems of equations. Students represent systems of equations as linear transformations represented by matrix equations and apply inverse matrix multiplication to find the solutions to systems of equations, establishing a foundation for solving systems of three or more equations using inverse matrix operations. This work is expanded in Lesson 15 to more complicated systems of equations as students use software to calculate inverse matrices for systems of degree 3 and apply the inverse of the coefficient matrix to the linear transformation equation to solve systems. Topic C concludes with Lesson 16 as students discover that, while it is difficult to geometrically describe linear transformations in four- or higher- dimensional space, the mathematics behind representing systems of equations as a linear transformation using matrices is valid for higher-degree space.

1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

dŽƉŝĐ: Systems of Linear Equations 239

A STORY OF FUNCTIONS dŽƉŝĐ DϮ PRECALCULUS AND ADVANCED TOPICS

They apply this reasoning to represent complicated systems of equations using matrices and then use technology to solve the systems. Throughout Topic C, students use calculators to perform matrix operations, find the inverse of matrices, and solve systems of equations with matrices. Students also see that the structure of solving systems does not

change as systems grow in complexity or in size.

240 dŽƉŝĐ: Systems of Linear Equations

A STORY OF FUNCTIONS Lesson 14 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 14: Solving Equations Involving Linear Transformations of the Coordinate Plane

Student Outcomes .ݔ ൌ ܾ, using matrix notationܮ Students represent systems of equations as linear transformations of the form ݔ ൌ ܾ can be solved by computingܮ Students discover that the systems of equations written in the form and they apply this process to find solutions to systems of two linear ,ܮ ,ଵܾ for all invertible matricesିܮݔൌ equations in two variables.

Lesson Notes In this lesson, students explore the relationship between linear transformations of points in two-dimensional space and systems of equations. They represent systems of equations as linear transformations represented by matrix equations and apply inverse matrix multiplication to find the solutions to systems of equations, which establishes a foundation for solving systems of three or more equations using inverse matrix operations.

Classwork Opening Exercise (3 minutes) Students briefly review how to represent linear transformations of points in the coordinate plane as matrices in preparation for representing systems of equations as linear transformations. Students analyze and critique the reasoning in the given problem, and they can perform the matrix multiplication for each problem to verify that the matrices correctly represent the transformations. Debrief with the class after students work individually.

Opening Exercise െ૚ ૙ ૝ ૚ Ahmad says the matrix ቂ ቃ applied to the point ቂ ቃ will reflect the point to ቂ ቃ. Randelle says that applying the ૙െ૚ ૚ ૝ matrix to the given point will produce a rotation of ૚ૡ૙ι about the origin. Who is correct? Explain your answer and verify the result. െ૚ ૙ Randelle is correct. Applying the matrix ቂ ቃ to the given point produces a rotation of ૚ૡ૙ι about the origin of the ૙െ૚ ૝ െ૝ ૙૚ point ቂ ቃ to the image point ቂ ቃ. Applying the matrix ቂ ቃto the given point would produce a reflection to the image ૚ െ૚ ૚૙ ૚ point ቂ ቃ. ૝

Lesson 14: Solving Equations Involving Linear Transformations of the Coordinate 241 Plane

A STORY OF FUNCTIONS Lesson 14 M2 PRECALCULUS AND ADVANCED TOPICS

Example (20 minutes)

Students should complete part (a) with a partner. After a few minutes, each pair can share its example with another pair. Students should complete part (b) in small groups, such as with the pairs that exchanged examples for part (a). After a few minutes, a different group should describe each example from part (b) and justify why the linear transformation does not produce an image point of ൤Ͷ൨. Students should be allowed to question the reasoning of other ͳ groups. Students should recognize that not all linear transformations produce a desired image point. In particular, linear ݔ transformations represented by matrices with a determinant of Ͳ do not transform a point ൤ݕ൨ to any desired image point. Part (c) should be completed as a teacher-led discussion. Students discover that systems of equations in two- ݔ ൌ ܾ, whereܮ dimensional space can be represented as a linear transformation represented by the equation represents the linear transformation of point ݔ resulting in image point ܾ. The coordinates of the pre-image can be ܮ ଵܾିܮݔ ൌ ܾ, resulting in the equation ݔൌܮ found by applying the reverse of the linear transformation to the equation for all invertible matrices ܮ. How can you verify that the transformation matrix you found in part (a) accurately represents the transformation you described? Ͷ à Multiply the pre-image by the matrix to verify that the result is the column matrix ቂ ቃ. ͳ If we calculate the determinants of the transformation matrices you found for part (a), what do they have in common? à They are all nonzero. Scaffolding: ͲͲ Ͷ Why can’t the transformation matrix ቂ ቃ produce an image point of ቂ ቃ? Have advanced students ͲͲ ͳ find additional examples à It collapses the points to the origin. of linear transformations ͳͳ Ͷ Why can’t the transformation matrix ቂ ቃ produce an image point of ቂ ቃ? that are not invertible. ͳͳ ͳ Consider asking advanced à ݔ ݕ It produces only ordered pairs that have the same - and -coordinates. students to make ͲͲ Ͷ Why can’t the transformation matrix ቂ ቃ produce an image point of ቂ ቃ? conjectures about the Ͳͳ ͳ properties of linear à It transforms all points to the ݕ-axis. transformations that can What do you notice about the determinants of the examples that cannot (or cannot) produce the Ͷ produce the image point of ቂ ቃ? image point without being ͳ prompted about the à They all have determinants of Ͳ. determinant. What have we learned about matrices that have a determinant of Ͳ? When reviewing Example à They do not have inverses. 1, select several pre-image If matrix ܮ represents a linear transformation, what does ܮିଵ represent points and multiply them geometrically? by the transformation matrices. Ask students if à It represents undoing the transformation. they notice any patterns in So if ܮ represents a rotation ͻͲι clockwise about the origin, what would the image points and to ିଵ ܮ represent? make a conjecture about à Rotation of ͻͲι counterclockwise or ʹ͹Ͳι clockwise about the origin. whether they could produce the image Ͷ point ቂ ቃ. ͳ

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Then if ܮ does not have an inverse, what does that suggest geometrically? à The transformation cannot be undone using another single transformation. ͲͲ How would this apply to the matrix ቂ ቃ? ͲͲ à You cannot perform a single transformation to reverse the transformation. ݔ Ͷ Looking at part (c), what does the equation ܮቂ ቃൌቂ ቃ represent geometrically? ݕ ͳ ݔ Ͷ à A linear transformation is applied to the point ቂ ቃ to produce the image point ቂ ቃ. ݕ ͳ ݔ ݔ Ͷ Explain geometrically how we could find the coordinates of ቂ ቃ given the equation ܮቂ ቃൌቂ ቃ. ݕ ݕ ͳ à You can undo the linear transformation represented by matrix ܮ. What would this look like algebraically? ݔ Ͷ à ܮିଵܮቂ ቃൌܮିଵ ቂ ቃ. ݕ ͳ And this simplifies to…? Explain. ݔ Ͷ à ቂ ቃൌܮିଵ ቂ ቃbecause ܮିଵܮൌܫ; it represents applying and then undoing the linear transformation. ݕ ͳ Does ܮ have an inverse? How can you tell? à Yes. It has a determinant of ͷ, and all square matrices with nonzero determinants are invertible. ܾܽ ͳ ݀െܾ Recall the formula that if ܣൌቂ ቃ where ܽ݀ െ ܾܿ ് Ͳ, then ܣିଵ ൌ ቂ ቃ. Given this, find the ܿ݀ ܽ݀െܾܿ െܿ ܽ ݔ .pre-image point ቂݕቃ Ͷ ͳ ͳͳͶ ͳ ͷ ͳ à ܮିଵ ቂ ቃൌ ቂ ቃ ቂ ቃൌ ቂ ቃൌቂ ቃ ͳ ʹሺͳሻെሺെͳሻሺ͵ሻ െ͵ ʹ ͳ ͷ െͳͲ െʹ ݔ ?How else could you have found the coordinates of ቂݕቃwithout using inverse matrices െͳݔ Ͷʹ à Multiply the matrices on the left side of the equation ቂ ቃቂ ቃൌቂ ቃ to create the system of ͳݕ ͳ͵ equations: ݔ െ ݕ ൌ Ͷʹ ݔ ൅ ݕ ൌ ͳ͵ How can we verify that the solution we found using inverse matrices is correct? ͳ à Substitute the ordered pair ቂ ቃ into both equations, and verify that the resulting number sentences െʹ are true. What does our solution mean geometrically? ʹെͳ ͳ à If you apply the transformation represented by the matrix ቂ ቃ to the point ቂ ቃ, the image point ͵ͳ െʹ Ͷ has coordinates of ቂ ቃ. ͳ What do you notice about the relationship between the system of equations and the transformation matrix ܮ? Explain your observations to your partner. à The entries in the matrix represent the coefficients of the system when both equations are written in standard form.

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A STORY OF FUNCTIONS Lesson 14 M2 PRECALCULUS AND ADVANCED TOPICS

Why might it be useful to learn the technique of using inverse matrix operations to solve systems rather than using algebraic methods like substitution or elimination? Discuss your ideas with your partner. à Algebraic methods can become messy and cumbersome with increasing numbers of equations and variables.

Example ૝ a. Describe a transformation not already discussed that results in an image point of ቂ ቃ, and represent the ૚ transformation using a ૛ൈ૛. െ૚ Answers will vary. An example of an appropriate response is as follows: ࡭ rotation of the point ቂ ቃૢ૙ι ૝ ૝ ૙૚ to the point ቂ ቃcan be represented with the matrix ቂ ቃ. ૚ െ૚ ૙

b. Determine whether any of the matrices listed represent linear transformations that can produce the image ૝ point ቂ ቃ. Justify your answers by describing the transformations represented by the matrices. ૚ ૙૙ i. ቂ ቃ ૙૙ ૝ This matrix represents a collapse to the origin, so it cannot produce the image point ቂ ቃ. ૚

૚૚ ii. ቂ ቃ ૚૚ This matrix represents a transformation to the diagonal defined by ࢟ൌ࢞, so it cannot produce the ૝ image point ቂ ቃ. ૚

૙૙ iii. ቂ ቃ ૙૚ ૝ This matrix represents a transformation to the ࢟-axis, so it cannot produce the image point ቂ ቃ. ૚

૛െ૚ c. Suppose a linear transformation ࡸ is represented by the matrix ቂ ቃ. ૜૚ ࢞ ࢞ ૝ Find a point ࡸቂ ቃ so that ࡸቂ ቃൌቂ ቃ. ࢟ ࢟ ૚ ࢞ ૚ ቂ ቃൌቂ ቃ ࢟ െ૛

Exercises (11 minutes) Allow students to complete the work in pairs. They should both complete the work and then compare answers. Some pairs may need more help or additional instruction. After a few minutes, discuss Exercise 1 to ensure that students are able to represent the systems of equations as linear transformations using matrix notation. Exercise 3 is discussed in detail to summarize the main points of the lesson. Exercise 4 is a challenge exercise.

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Exercises

1. Given the system of equations ૛࢞ ൅ ૞࢟ ൌ ૝ ૜࢞ െ ૡ࢟ ൌ െ૛૞ a. Show how this system can be written as a statement about a linear transformation of the form ࡸ࢞ ൌ ࢈, ࢞ ૝ with ࢞ൌቂ ቃ and ࢈ൌቂ ቃ. ࢟ െ૛૞ ૛૞࢞ ૝ ቂ ቃቂ ቃൌቂ ቃ ૜െૡ࢟ െ૛૞

b. Determine whether ࡸ has an inverse. If it does, compute ࡸି૚࢈,and verify that the coordinates represent the solution to the system of equations.

૚ െૡ െ૞ ૝ ࡸି૚࢈ൌ ቂ ቃቂ ቃ ૛ሺെૡሻ െ ሺ૞ሻሺ૜ሻ െ૜ ૛ െ૛૞ ૚ െૡ െ૞ ૝ ૚ ૢ૜ െ૜ ࡸି૚࢈ൌ ቂ ቃቂ ቃൌ ቂ ቃൌቂ ቃ െ૜૚ െ૜ ૛ െ૛૞ െ૜૚ െ૟૛ ૛

Verification using back substitution:

૛ሺെ૜ሻ ൅૞ሺ૛ሻ ൌ૝ ૜ሺെ૜ሻ െૡሺ૛ሻ ൌ െ૛૞

2. The path of a piece of paper carried by the wind into a tree can be modeled with a linear transformation, ૜െ૝ ૟ where ࡸൌቂ ቃ and ࢈ൌቂ ቃ. ૞૜ ૚૙ a. Write an equation that represents the linear transformation of the piece of paper. ૜െ૝࢞ ૟ ቂ ቃቂ ቃൌቂ ቃ ૞૜࢟ ૚૙

b. Solve the equation from part (a).

࢞ ૚ ૜૝૟ ૚ ૞ૡ ૛ ቂ ቃൌ ቂ ቃቂ ቃൌ ቂ ቃൌቂ ቃ ࢟ ૛ૢ െ૞ ૜ ૚૙ ૛ૢ ૙ ૙

c. Use your solution to provide a reasonable interpretation of the path of the piece of paper under the transformation by the wind.

Answers will vary. An example of an appropriate response would be that the piece of paper started on the ground ૛ feet to the right of the location defined as the origin, and it was moved by the wind to a spot ૟ feet to the right of the origin and ૚૙ feet above the ground (in the tree).

3. For each system of equations, write the system as a linear transformation represented by a matrix and apply inverse matrix operations to find the solution, or explain why this procedure cannot be performed. a. ૟࢞ ൅ ૛࢟ ൌ ૚ ࢟ൌ૜࢞൅૚ ૟૛࢞ ૚ ቂ ቃቂ ቃൌቂ ቃ െ૜ ૚ ࢟ ૚ െ૚ ࢞ ૚ ૚െ૛૚ ૚ െ૚ ቂ ቃൌ ቂ ቃቂ ቃൌ ቂ ቃൌ൦૚૛൪ ࢟ ૚૛ ૜૟૚ ૚૛ ૢ ૜ ૝

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b. ૝࢞ െ ૟࢟ ൌ ૚૙ ૛࢞ െ ૜࢟ ൌ ૚

This system cannot be represented as a linear transformation because the transformation matrix ࡸ has a determinant of ૙. The system represents parallel lines, so there is no solution.

4. In a two-dimensional plane, ࡭ represents a rotation of ૜૙ι counterclockwise about the origin, ࡮ represents a reflection over the line ࢟ൌ࢞, and ࡯ represents a rotation of ૟૙ι counterclockwise about the origin. a. Write matrices ࡭, ࡮, and ࡯.

ξ૜ ૚ ૚ ξ૜ ې െ ۍ ې െ ۍ ۑ ێ ૙૚ ۑ ێ ࡭ൌ ૛ ૛ ࡮ൌቂ ቃ࡯ൌ ૛ ૛ ۑ ێ ૚૙ ۑ ێ ۑ ξ૜ ૚ێ ۑ૚ ξ૜ ێ ے ૛ ૛ ۏ ے ૛ ૛ ۏ

࢞ ૚൅૛ ૜ b. Transformations ࡭,࡮, and ࡯ are applied to pointቂ ቃ successively and produce the image point ൤ ξ ൨. ࢟ ૛െξ૜ ࢞ Use inverse matrix operations to find ቂ࢟ቃ. We must apply the inverse transformations in the reverse order. The inverse of matrix ࡯ is

ې૚ ξ૜ ۍ ۑ ૛ ૛ ێ ૚ି . ۑ ێ࡯ ൌ૚ ξ૜ ૚ ۑ െێ ے ૛ ૛ ۏ Applied to the image

ې૚ ξ૜ ۍ ૚൅૛ξ૜ െ૚ ൅ ૛ξ૜ ۑ ૛ ૛ ێ ൤ ൨ൌ൤ ൨ ۑ ێ ξ૜ ૚ ૛െξ૜ െ૛ െ ξ૜ ۑ െێ ے ૛ ૛ ۏ ૙െ૚ ૙૚െ૚ ൅ ૛ ૜ െ૛ െ ૜ The inverse of matrix ࡮ is ࡮ି૚ ൌെ૚ቂ ቃ. Applied to the image ቂ ቃ൤ ξ ൨ൌ൤ ξ ൨ െ૚ ૙ ૚૙ െ૛ െ ξ૜ െ૚ ൅ ૛ξ૜ The inverse of matrix ࡭ is

ې ξ૜ ૚ۍ ۑ ૛ ૛ ێ ૚ି . ۑ ێ࡭ ൌ૚ ૚ ξ૜ ۑ െێ ے ૛ ૛ ۏ Applied to the image

ې ξ૜ ૚ۍ െ૛ െ ξ૜ െ૛ ۑ ૛ ૛ ێ ൤ ൨ൌቂ ቃ ۑ ێ ૚ ξ૜ െ૚ ൅ ૛ξ૜ ૝ ۑ െێ ے ૛ ૛ ۏ

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Closing (6 minutes) As a class, discuss the results of Exercises 1–3, focusing on how to represent systems of two linear equations as a linear transformation represented by matrix multiplication. How does the format of a linear system affect how it can be written as a linear transformation? à The equations need to be written in standard form so the coefficients are represented accurately in the transformation matrix ܮ. What is the geometric interpretation of your conclusion to Exercise 3, part (b)? à There are no points in the coordinate plane that, when the transformation represented by the matrix Ͷെ͸ ͳͲ ቂ ቃ is applied, result in an image point of ቂ ቃ. ʹെ͵ ͳ How are problems involving linear transformations solved in the coordinate plane? à Systems of equations with a single solution can be represented as a linear transformation in the form of ݔ is the transformation matrix, ݔൌቂݕቃ and ܾൌthe coordinates of the ܮ ݔ ൌ ܾ, whereܮ the equation image. à The coordinates of the pre-image can be found by multiplying both sides of the transformation equation byܮିଵ, which effectively undoes the transformation. à When the determinant of ܮ is Ͳ, the linear transformation matrix does not have an inverse, and there is no solution to the system of equations whose coefficients are represented by ܮ.

Exit Ticket (5 minutes)

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Name Date _

Lesson 14: Solving Equations Involving Linear Transformations of the Coordinate Plane

Exit Ticket

͵ .In two-dimensional space, point ݔ is rotated ͳͺͲι to the point ቂ ቃ Ͷ .ܾ ݔ ൌܮ a. Represent the transformation of point ݔ using an equation in the format

.b. Use inverse matrix operations to find the coordinates of ݔ

c. Verify that this solution makes sense geometrically.

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Exit Ticket Sample Solutions

૜ In two-dimensional space, point ࢞ is rotated ૚ૡ૙ιto the point ቂ ቃ. ૝ a. Represent the transformation of point ࢞ using an equation in the format ࡸ࢞ ൌ ࢈. െ૚ ૙ ࢞ ૜ ቂ ቃቂ ቃൌቂ ቃ ૙െ૚࢟ ૝

b. Use inverse matrix operations to find the coordinates of ࢞.

࢞ ૚ െ૚ ૙ ૜ െ૜ ቂ ቃൌ ቂ ቃቂ ቃൌቂ ቃ ࢟ ૚ ૙െ૚૝ െ૝ c. Verify that this solution makes sense geometrically.

When a point is rotated ૚ૡ૙ι about the origin in the coordinate plane, the ࢞- and ࢟-coordinates of the image point are the opposite of those of the pre-image point.

Problem Set Sample Solutions

૚૞ 1. In a two-dimensional plane, a transformation represented by ࡸൌቂ ቃ is applied to point ࢞, resulting in an ૛െ૝ ૙ image point ቂ ቃ. Find the location of the point before it was transformed. ૞ a. Write an equation to represent the linear transformation of point ࢞. ૚૞࢞ ૙ ቂ ቃቂ ቃൌቂ ቃ ૛െ૝࢟ ૞

b. Solve the equation to find the coordinates of the pre-image point.

࢞ െ૚ െ૝ െ૞ ૙ ቂ ቃൌ ቂ ቃቂ ቃ ࢟ ૚૝ െ૛ ૚ ૞ ૛૞ ࢞ െ૚ െ૛૞ ቂ ቃൌ ቂ ቃൌ൦૚૝൪ ࢟ ૚૝ ૞ െ૞ ૚૝

࢞ 2. Find the location of the point ቂ࢟ቃ before it was transformed when given: ૜૞ ૚ a. The transformation ࡸൌቂ ቃ, and the resultant is ቂ ቃ. Verify your answer. ૚૛ ૛ ૜૞࢞ ૚ ቂ ቃቂ ቃൌቂ ቃ ૚૛࢟ ૛ ࢞ ૚ ૛െ૞૚ ቂ ቃൌ ቂ ቃቂ ቃ ࢟ ૚ െ૚ ૜ ૛ ࢞ െૡ ቂ ቃൌቂ ቃ ࢟ ૞ ૜૞െૡ ૚ ቂ ቃቂ ቃൌቂ ቃ ૚૛ ૞ ૛

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૝ૠ ૛ b. The transformation ࡸൌቂ ቃ, and the resultant is ቂ ቃ. Verify your answer. െ૚ െ૛ െ૚ ૝ૠ࢞ ૛ ቂ ቃቂ ቃൌቂ ቃ െ૚ െ૛ ࢟ െ૚ ࢞ ૚ െ૛ െૠ ૛ ቂ ቃൌ ቂ ቃቂ ቃ ࢟ െ૚ ૚૝െ૚ ࢞ ૜ െ૜ ቂ ቃൌെ૚ቂ ቃൌቂ ቃ ࢟ െ૛ ૛ ૝ૠെ૜ ૛ ቂ ቃቂ ቃൌቂ ቃ െ૚ െ૛ ૛ െ૚

૙െ૚ ૚ c. The transformation ࡸൌቂ ቃ, and the resultant is ቂ ቃ. Verify your answer. ૛૚ ૜ ૙െ૚࢞ ૚ ቂ ቃቂ ቃൌቂ ቃ ૛૚࢟ ૜ ࢞ ૚ ૚૚૚ ቂ ቃൌ ቂ ቃቂ ቃ ࢟ ૛ െ૛ ૙ ૜ ࢞ ૚ ૝ ૛ ቂ ቃൌ ቂ ቃൌቂ ቃ ࢟ ૛ െ૛ െ૚ ૙െ૚ ૛ ૚ ቂ ቃቂ ቃൌቂ ቃ ૛૚െ૚ ૜

૛૜ ૜ d. The transformation ࡸൌቂ ቃ, and the resultant is ቂ ቃ. Verify your answer. ૙െ૚ ૙ ૛૜࢞ ૜ ቂ ቃቂ ቃൌቂ ቃ ૙െ૚࢟ ૙ ࢞ ૚ െ૚ െ૜ ૜ ቂ ቃൌ ቂ ቃቂ ቃ ࢟ െ૛ ૙૛૙ ૜ ࢞ ૚ െ૜ ቂ ቃൌ ቂ ቃൌ൥ ൩ ࢟ െ૛ ૙ ૛ ૙ ૜ ૛૜ ૜ ቂ ቃ ൥ ൩ ൌቂ ቃ ૙െ૚૛ ૙ ૙

૛െ૚ ૜ e. The transformation ࡸൌቂ ቃ, and the resultant is ቂ ቃ. Verify your answer. ૚૛ ૛ ૛െ૚࢞ ૜ ቂ ቃቂ ቃൌቂ ቃ ૚૛࢟ ૛ ࢞ ૚ ૛૚૜ ቂ ቃൌ ቂ ቃቂ ቃ ࢟ ૞ െ૚ ૛ ૛ ૡ ࢞ ૚ ૡ ቂ ቃൌ ቂ ቃൌ൦૞൪ ࢟ ૞ ૚ ૚ ૞ ૡ ૛െ૚ ૜ ቂ ቃ൦૞൪ൌቂ ቃ ૚૛૚ ૛ ૞

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3. On a computer assembly line, a robot is placing a CPU onto a motherboard. The robot’s arm is carried out by the ૛૜ transformation ࡸൌቂ ቃ. ૚૛ െ૛ a. If the CPU is attached to the motherboard at point ቂ ቃ, at what location does the robot pick up the CPU? ૜ ૛૜࢞ െ૛ ቂ ቃቂ ቃൌቂ ቃ ૚૛࢟ ૜ ࢞ ૚ ૛െ૜െ૛ െ૚૜ ቂ ቃൌ ቂ ቃቂ ቃൌቂ ቃ ࢟ ૚ െ૚ ૛ ૜ ૡ

૜ b. If the CPU is attached to the motherboard at point ቂ ቃ, at what location does the robot pick up the CPU? ૛ ૛૜࢞ ૜ ቂ ቃቂ ቃൌቂ ቃ ૚૛࢟ ૛ ࢞ ૚ ૛െ૜૜ ૙ ቂ ቃൌ ቂ ቃቂ ቃൌቂ ቃ ࢟ ૚ െ૚ ૛ ૛ ૚

െ૚ ࢉ ૛ c. Find the transformation ࡸൌቂ ቃ that places the CPU starting at ቂ ቃ onto the motherboard at the ࢈૜ െ૜ െૡ location ቂ ቃ. ૜ െ૚ ࢉ ૛ െૡ ቂ ቃቂ ቃൌቂ ቃ ࢈૜െ૜ ૜

െ૛ െ ૜ࢉ ൌ െૡ,ࢉ ൌ ૛

૛࢈ െ ૢ ൌ ૜,࢈ ൌ ૟

െ૚ ૛ ቂ ቃ ૟૜

4. On a construction site, a crane is moving steel beams from a truck bed to workers. The crane is programmed to ૚૚ perform the transformation ࡸൌቂ ቃ. ૛૜ ૛ a. If the workers are at location ቂ ቃ, where does the truck driver need to unload the steel beams so that the ૞ crane can pick them up and bring them to the workers? ૚૚࢞ ૛ ቂ ቃቂ ቃൌቂ ቃ ૛૜࢟ ૞ ࢞ ૚ ૜െ૚૛ ૚ ቂ ቃൌ ቂ ቃቂ ቃൌቂ ቃ ࢟ ૚ െ૛ ૚ ૞ ૚

െ૜ b. If the workers move to another location ቂ ቃ,where does the truck driver need to unload the steel beams so ૚ that the crane can pick them up and bring them to the workers? ૚૚࢞ െ૜ ቂ ቃቂ ቃൌቂ ቃ ૛૜࢟ ૚ ࢞ ૚ ૜െ૚െ૜ െ૚૙ ቂ ቃൌ ቂ ቃቂ ቃൌቂ ቃ ࢟ ૚ െ૛ ૚ ૚ ૠ

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૙ 5. A video game soccer player is positioned at ቂ ቃ, where he kicks the ball. The ball goes into the goal, which is at point ૛ ૚૙ ૚ ૚૙ ቂ ቃ. When the player moves to point ቂ ቃand kicks the ball, he misses the goal. The ball lands at point ቂ ቃ. What ૙ ૚ െ૚ ࢇࢉ is the program/transformation ࡸൌቂ ቃ that this video soccer player uses? ࢈ࢊ ࢇࢉ૙ ૚૙ ቂ ቃቂ ቃൌቂ ቃ ,૛ࢉ ൌ ૚૙,ࢉ ൌ ૞.૛ࢊ ൌ ૙,ࢊ ൌ ૙ ࢈ࢊ૛ ૙ ࢇ૞૚ ૚૙ ቂ ቃቂ ቃൌቂ ቃ ,ࢇ ൅ ૞ ൌ ૚૙,ࢇ ൌ ૞.࢈ ൌ െ૚ ࢈૙૚ െ૚ ૞૞ ቂ ቃ െ૚ ૙

6. Tim bought ૞ shirts and ૜ pairs of pants, and it cost him ̈́૛૞૙. Scott bought ૜ shirts and ૛ pairs of pants, and it cost him ̈́૚૟૙. All the shirts have the same cost, and all the pants have the same cost. a. Write a system of linear equations to find the cost of the shirts and pants. ૞ࡿ ൅ ૜ࡼ ൌ ૛૞૙ ቄ ૜ࡿ ൅ ૛ࡼ ൌ ૚૟૙

b. Show how this system can be written as a statement about a linear transformation of the form ࡸ࢞ ൌ ࢈ with ࡿ ૛૞૙ ࢞ൌቂ ቃ and ࢈ൌቂ ቃ. ࡼ ૚૟૙ ૞૜ࡿ ૛૞૙ ቂ ቃቂ ቃൌቂ ቃ ૜૛ࡼ ૚૟૙

c. Determine whether ࡸ has an inverse. If it does, compute ࡸି૚࢈, and verify your answer to the system of equations. ૞૜ ૚ ૛െ૜ ૛െ૜ The determinant of ቂ ቃ is ૚. ࡸି૚ ൌ ቂ ቃൌቂ ቃ ૜૛ ૚ െ૜ ૞ െ૜ ૞ ࡿ ૛െ૜૛૞૙ ૛૙ ቂ ቃൌቂ ቃቂ ቃൌቂ ቃ ࡼ െ૜ ૞ ૚૟૙ ૞૙ Verification using back substitution:૞ሺ૛૙ሻ ൅૜ሺ૞૙ሻ ൌ ૛૞૙,૜ሺ૛૙ሻ ൅૛ሺ૞૙ሻ ൌ ૚૟૙

7. In a two-dimensional plane, ࡭ represents a reflection over the ࢞-axis, ࡮ represents a reflection over the ࢟-axis, and ࡯ represents a reflection over the line ࢟ൌ࢞. a. Write matrices ࡭, ࡮, and ࡯. ૚૙ െ૚ ૙ ૙૚ ࡭ൌቂ ቃ࡮ൌቂ ቃ࡯ൌቂ ቃ ૙െ૚ ૙૚ ૚૙

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െ૛ b. Write an equation for each linear transformation, assuming that each one produces an image point of ቂ ቃ. െ૜ For transformation ࡭, ૚૙࢞ െ૛ ቂ ቃቂ ቃൌቂ ቃ ૙െ૚࢟ െ૜ For transformation ࡮, െ૚ ૙ ࢞ െ૛ ቂ ቃቂ ቃൌቂ ቃ ૙૚࢟ െ૜ For transformation ࡯, ૙૚࢞ െ૛ ቂ ቃቂ ቃൌቂ ቃ ૚૙࢟ െ૜

c. Use inverse matrix operations to find the pre-image point for each equation. Explain how your solutions make sense based on your understanding of the effect of each geometric transformation on the coordinates of the pre-image points. ࢞ െ૚ ૙ െ૛ െ૛ For transformation ࡭, ቂ ቃൌെ૚ቂ ቃቂ ቃൌቂ ቃ. When a point is reflected over the ࢞-axis, the ࢟ ૙૚െ૜ ૜ ࢞-coordinate remains unchanged, and the ࢟-coordinate changes signs. ࢞ ૚૙െ૛ ૛ For transformation ࡮, ቂ ቃൌെ૚ቂ ቃቂ ቃൌቂ ቃ. When a point is reflected over the ࢟-axis, the ࢟ ૙െ૚െ૜ െ૜ ࢟-coordinate remains unchanged, and the ࢞-coordinate changes signs. ࢞ ૙െ૚െ૛ െ૜ For transformation ࡯, ቂ ቃൌെ૚ቂ ቃቂ ቃൌቂ ቃ. When a point is reflected over the line ࢟ൌ࢞, the ࢟ െ૚ ૙ െ૜ െ૛ coordinates of the pre-image point are interchanged (࢞ and ࢟ are switched).

8. A system of equations is shown: ૛࢞ ൅ ૞࢟ ൅ ࢠ ൌ ૜ ૝࢞ ൅ ࢟ െ ࢠ ൌ ૞ ૜࢞ ൅ ૛࢟ ൅ ૝ࢠ ൌ ૚ a. Represent this system as a linear transformation in three-dimensional space represented by a matrix equation in the form of ࡸ࢞ ൌ ࢈. ૛૞ ૚ ࢞ ૜ ൥૝૚െ૚൩ቈ࢟቉ ൌ ൥૞൩ ૜૛ ૝ ࢠ ૚

b. What assumption needs to be made to solve the equation in part (a) for ࢞.

To use inverse operations, we need to assume that ࡸ has an inverse.

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c. Use algebraic methods to solve the system.

Adding equations ૚ and ૛ gives ૟࢞ ൅ ૟࢟ ൌ ૡ.

Adding 4 times equation ૛ and equation ૜ gives ૚ૢ࢞ ൅ ૟࢟ ൌ ૛૚.

Subtracting ሺ૟࢞ ൅ ૟࢟ ൌ ૡሻ from ሺ૚ૢ࢞ ൅ ૟࢟ ൌ ૛૚ሻ gives ૚૜࢞ ൌ ૚૜, so ࢞ൌ૚. ૚ ૟࢞ ൅ ૟࢟ ൌ ૡ ૟࢟ ൌ ૛ ࢟ൌ Back substituting into gives , or ૜. ૚ ૛ ࢟ ࢞ ૛ሺ૚ሻ ൅૞ቀ ቁ൅ࢠൌ૜ ࢠൌെ Back substituting for and into the first equation gives ૜ , so ૜. ૚ ې ૚ ۍ ۑ ێ ۑ ૜ ێ࢞ൌ ۑെ૛ێ ے ૜ ۏ

9. Assume ૚ െ૟ ૚ૡ ૟ ࡸି૚ ൌ ൥૚ૢ െ૞ െ૟൩ ૠૡ െ૞ െ૚૚ ૚ૡ Use inverse matrix operations to solve the equation from Problem 8, part (a) for ࢞. Verify that your solution is the same as the one you found in Problem 8, part (c). ૚ ې ૚ ۍ ࢞ െ૟ ૚ૡ ૟ ૜ ۑ ێ ૚ ቈ࢟቉ ൌ ൥૚ૢ െ૞ െ૟൩൥૞൩ ൌ ૜ , ۑ ێ ૠૡ ۑെ૛ێ ࢠ െ૞ െ૚૚ ૚ૡ ૚ ے ૜ ۏ which is the same solution found in part (c).

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A STORY OF FUNCTIONS Lesson 15 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 15: Solving Equations Involving Linear Transformations of the Coordinate Space

Student Outcomes Students represent systems of three or more simultaneous equations as a linear transformation in the form ܾ represents the linear transformation, ݔ represents the coordinates of the pre-image, andܣ ݔ ൌ ܾ, whereܣ represents the coordinates of the image point. Students apply inverse matrix operations to solve systems of linear equations with three equations.

Lesson Notes Students expand upon their previous work from Lesson 14 to represent more complicated systems of equations as linear transformations in coordinate space. They use software to calculate inverse matrices for systems of degree ͵ and apply the inverse of the coefficient matrix to the linear transformation equation to solve systems. Students model real-world situations that can be written as systems of equations and solve the systems in a specific context including finding the intersection points of lines, designing a card game, and determining the number of coaches needed for a school athletic program.

Classwork Opening Exercise (5 minutes) Students should complete the Opening Exercise in pairs. They should find the answer independently and then verify the solution with a partner. One pair could display the algebraic solution process on the board while another pair displays the matrix method of solving the system.

Opening Exercise Scaffolding:

Mariah was studying currents in two mountain streams. She determined that five times the Provide a simpler system: current in stream A was ૡ feet per second stronger than twice the current in stream B. Another ݔ൅ʹݕൌെͳ ͵ day she found that double the current in stream A plus ten times the current in stream B was ૜ ʹݔ െ ݕ ൌ feet per second. She estimated the current in stream A to be ૚Ǥ ૞ feet per second and stream B to be almost still (૙ feet per second). Was her estimate reasonable? Explain your answer after Use scaffolded questions such completing parts (a)–(c). as: a. Write a system of equations to model this problem. Is ሺͳǡ ʹሻ a solution to this ૞࢞ ൌ ૡ ൅ ૛࢟ system of equations? ૛࢞ ൅ ૚૙࢟ ൌ ૜ Explain how you know. Ͷ͵ ͳ Is ቀ ǡെ ቁ a solution to ʹ͹ ͷͶ this system of equations? Explain how you know.

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b. Solve the system using algebra.

Procedures may vary. An example of an appropriate algebraic procedure is shown.

Multiply the first equation by ૞: ૞ሺ૞࢞ െ ૛࢟ ൌ ૡሻ ՜ ૛૞࢞ െ ૚૙࢟ ൌ ૝૙

Add this equation to the second equation:

૛૞࢞ െ ૚૙࢟ ൌ ૝૙ ൅૛࢞ ൅ ૚૙࢟ ൌ ૜ ૛ૠ࢞ ൌ ૝૜ ૝૜ ࢞ൌ ૛ૠ

Back substitute the ࢞-value into the first equation and isolate ࢟. ૝૜ ૞ή െ૛࢟ൌૡ ૛ૠ ૚ െ૛࢟ ൌ ૛ૠ ૚ ࢟ൌെ ૞૝ Solution: ૝૜ ൦ ૛ૠ ൪ ૚ െ ૞૝

c. Solve the system by representing it as a linear transformation of the point ࢞ and then applying the inverse of the transformation matrix ࡸ to the equation. Verify that the solution is the same as that found in part (b). ૞െ૛࢞ ૡ ቂ ቃቂ ቃൌቂ ቃ ૛૚૙࢟ ૜ ࢞ ૞െ૛ି૚ ૡ ቂ ቃൌቂ ቃ ቂ ቃ ࢟ ૛૚૙ ૜ ૝૜ ࢞ ૚ ૚૙ ૛ ૡ ቂ ቃൌ ቂ ቃቂ ቃൌ൦ ૛ૠ ൪ ࢟ ૞૝ െ૛ ૞ ૜ ૚ െ ૞૝

which is the same solution as in part (b). ૝૜ ૚ ࢞ൌ ൎ૚Ǥ૞ૢ܎ܜܛ܍܋Τ ࢟ൌെ ൎ െ૙Ǥ ૙૛ ܎ܜΤ ܛ܍܋ Mariah’s estimate was reasonable because ૛ૠ and ૞૝ . Since the signs are opposite, the currents are moving in opposite directions.

Discussion (3 minutes) Do you prefer the algebraic or matrix method for solving systems of two linear equations? Explain why you prefer one method over the other. Share your thoughts with a partner. à Answers will vary. An example of an appropriate response would be to state a preference for the algebraic method because it does not require calculating the inverse of the transformation matrix.

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Describe when it might be more efficient to use algebra to solve systems of equations. Discuss your ideas with your partner. à Answers will vary. An example of an appropriate response would be that it would be more efficient to use algebra to solve systems of equations when there are a few equations or when it is clear that the system is either inconsistent or dependent. Describe when it might be more efficient to use matrices to solve systems of equations. Discuss your ideas with your partner. à Answers will vary. An example of an appropriate response would be that using matrices might be more efficient to solve systems of equations when the system consists of several equations with several variables.

Example (12 minutes) Scaffolding: Students above grade level can Students should complete parts (a)–(c) in pairs during the first few minutes. One pair complete the entire example could display their procedure and algebraic solution on the board, while another pair with a partner with no leading displays the linear transformation equation. Alternatively, each pair could display first questions from the teacher. their algebraic solution and then the transformation equation on white boards for a Struggling students may need quick check. For part (d), a free software program or graphing calculator app should help setting up equations. be used to demonstrate how to input matrices and calculate their inverses. The Consider giving them questions program should be projected on the board for student viewing. Alternatively, the to lead to the system such as: screen of a graphing calculator could be projected to demonstrate the use of technology to calculate matrix inverses. Part (e) should be completed as a teacher-led Look at the first hand. If ݔ is the exercise. The calculation could first be performed using technology, and then students value of green cards, ݕ the value of ଵ by hand to verify that the software calculation is accurate. yellow cards, and ݖ the value ofିܣ could multiply blue cards, let’s write an equation What features of the system of equations lend it to being solved using for just the first hand. algebra? à Answers will vary but might include that elimination could be used to find the value of the variables quickly because some of the equations in the system have corresponding coefficients that are either identical or opposites, which means that a variable could be eliminated by adding or subtracting two of the equations as they are written. Also, the values of the variables are integers, which facilitate computations when performing back substitution. How did you find matrix ܮ? à The matrix is constructed of the coefficients of the variables of the system. Can you always construct matrix ܮ using the coefficients of the variables in the order in which they appear in the system? Why? à No, the equations must be written in standard form first. When we looked at two-dimensional space, how did we represent a system of equations using matrices? .ܾ ݔ ൌܮ à We wrote the system using the equation What did the equation represent? .ܾ à A linear transformation of the point ݔ to the image point Could all systems be represented as a linear transformation of a point to a desired image point in two- dimensional space? à No

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How can you tell when systems could be solved using the linear transformation matrix equation? à The determinant of the matrix ܮ must be nonzero. ?is nonzero ܮ ݔ ൌ ܾ when the determinant ofܮ How did we solve the equation à We applied ܮିଵto both sides of the equation. What is the result? ଵܾିܮà ݔൌ ?And what does ݔ represent geometrically à The pre-image point that, once ܮ is applied to it, results in image point ܾ. ?ݔ ൌ ܾ if the determinant is Ͳܮ Why can’t we solve the equation à ܮ does not have an inverse when the determinant is Ͳ. What does this imply geometrically? .cannot be reversed to produce point ݔ ܮ à The transformation Now, for three-dimensional space, how can we represent a system of equations as a linear transformation of a point? .ܾ ݔ ൌܮ à The same way that we represent two-dimensional systems, with the equation ?And what condition do you think should be met to find the point ݔ using this equation à The determinant must be nonzero. Right, and how can you determine if a ͵ൈ͵ matrix has a nonzero determinant? à Answers may vary but might include using the diagonal method as shown:

݈ଵǡଵ ݈ଵǡଶ ݈ଵǡଷ If ܮൌ቎݈ଶǡଵ ݈ଶǡଶ ݈ଶǡଷ቏, ݈ଷǡଵ ݈ଷǡଶ ݈ଷǡଷ ݈ଵǡଵ ݈ଵǡଶ ݈ଵǡଷ

ቮ݈ଶǡଵ ݈ଶǡଶ ݈ଶǡଷቮൌ݈ଵǡଵ݈ଶǡଶ݈ଷǡଷ ൅݈ଵǡଶ݈ଶǡଷ݈ଷǡଵ ൅݈ଵǡଷ݈ଶǡଵ݈ଷǡଶ െ݈ଵǡଶ݈ଶǡଵ݈ଷǡଷ െ݈ଵǡଵ݈ଶǡଷ݈ଷǡଶ െ݈ଵǡଷ݈ଶǡଶ݈ଷǡଵ ݈ଷǡଵ ݈ଷǡଶ ݈ଷǡଷ How can we use technology to determine if a matrix ܮ has an inverse? à Answers will vary but are likely to include entering the matrix into the software program and applying the inverse function to it. How do we know matrix ܮ for the system in our example has an inverse? à We can calculate it directly using the software. ?ܾ ݔ ൌܮ How can we use technology to solve the equation .ଵܾ to find ݔିܮ ଵ and then findିܮ à We can use a calculator to find Why does it make sense that our solution to part (e) is the same as the solution we found in part (b)? ݔ ଵܾ, which is equal to ݔ, and ݔ represents the point ቈݕ቉, which is the solutionିܮ à In part (e) we calculated ݖ to the system. In this problem, which method of solving did you prefer and why? Discuss your reasoning with a partner. à Answers will vary but might include that the algebraic method was preferable for this system because the elimination method was easy to use given the coefficients of the variables in the equations.

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What factors would you consider when trying to determine whether to solve a system of equations in three- dimensional space using algebraic methods versus using matrices? Share your ideas with your partner. à Answers may vary but might include the size and relationship between the corresponding coefficients in the equations (e.g., whether elimination could be used easily to find the values of the variables).

Example

Dillon is designing a card game where different colored cards are assigned point values. Kryshna is trying to find the value of each colored card. Dillon gives him the following hints. If I have ૜ green cards, ૚ yellow card, and ૛ blue cards in my hand, my total is ૢ. If I discard ૚ blue card, my total changes to ૠ. If I have ૚ card of each color (green, yellow, and blue), my cards total ૚.

a. Write a system of equations for each hand of cards if ࢞ is the value of green cards, ࢟ is the value of yellow cards, and ࢠ is the value of blue cards.

૜࢞൅࢟൅૛ࢠൌૢ

૜࢞൅࢟൅ࢠൌૠ

࢞൅࢟൅ࢠൌ૚

b. Solve the system using any method you choose.

Answers will vary. An example of an appropriate response is shown.

Subtract the second equation from the first.

૜࢞ ൅ ࢟ ൅ ૛ࢠ ൌ ૢ ൅࢟ ൅ ࢠ ൌ ૠሻ ט െሺ૜࢞ ࢠൌ૛

Subtract the third equation from the second.

૜࢞൅࢟൅ࢠൌૠ െሺ࢞ ൅ ࢟ ൅ ࢠ ൌ ૚ሻ ૛࢞ ൌ ૟ ࢞ൌ૜

Back substitute values of ࢞ and ࢠ into the first equation, and isolate ࢟.

૜ሺ૜ሻ ൅࢟൅૛ሺ૛ሻ ൌૢ ૚૜ െ ࢟ ൌ ૢ ࢟ൌെ૝ ૜ Scaffolding: Solution: ൥െ૝൩ Green cards are worth ૜ points, yellow cards െ૝ points, Provide students with a ૛ flow chart that outlines and blue cards ૛ points. the steps to solving systems of equations using ࢞ ૢ inverse matrix operations. c. Let ࢞ൌቈ࢟቉ and ࢈ൌ൥ૠ൩. Find a matrix ࡸ so that the linear transformation equation ࢠ ૚ Provide written directions ࡸ࢞ ൌ ࢈ would produce image coordinates that are the same as the solution or printed screen shots to to the system of equations. students to aid them in ૜૚૛ ࡸൌ൥૜૚૚൩ entering matrices and ૚૚૚ calculating their inverses using software.

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d. Enter matrix ࡸ into a software program or app, and try to calculate its inverse. Does ࡸ have an inverse? If so, what is it? ૙૙Ǥ૞૙െ૙Ǥ૞૙ ࡸି૚ ൌ ൥െ૚ ૙Ǥ ૞૙ ૚Ǥ ૞૙ ൩ ૚െ૚૙

ૢ e. Calculate ࡸି૚ ൥ૠ൩. Verify that the result is equivalent to the solution to the system you calculated in part (b). ૚ Why should the solutions be equivalent? ૢ ૙૙Ǥ૞૙െ૙Ǥ૞૙ૢ ࡸି૚ ൥ૠ൩ ൌ ൥െ૚ ૙Ǥ ૞૙ ૚Ǥ ૞૙ ൩൥ૠ൩ ૚ ૚െ૚૙ ૚ ૙૙Ǥ૞૙െ૙Ǥ૞૙ૢ ૜ ൥െ૚ ૙Ǥ ૞૙ ૚Ǥ ૞૙ ൩൥ૠ൩ ൌ ൥െ૝൩ , ૚െ૚૙ ૚ ૛

which is equivalent to the solution from part (b). This makes sense because the system of equations can be represented using the linear transformation equation ࡸ࢞ ൌ ࢈. From what we learned in Lesson 14, ࢞ൌࡸି૚࢈, so the solution found using inverse matrix operations should be the same as the solution set when solving the system of equations using algebra.

Exercises 1–3 (15 minutes) Students should be placed into small groups. Each group should solve one of the problems. The group should spend about 5 minutes determining whether to solve the system using algebra or matrices, assisted by technology. They should solve the system using the method they chose and then verify the solution using back substitution. For the next few minutes, they should meet with the other groups assigned the same problem and verify their solutions, as well as discuss their arguments for the solution method they selected. During the last few minutes, each problem should be presented to the entire class. Presenters should display the problem, state the solution method chosen, justify the method selected, and display the solution. Each small group will need access to software that can be used to determine the inverse of matrices.

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Exercises

1. The system of equations is given: ૛࢞ െ ૝࢟ ൅ ૟ࢠ ൌ ૚૝ Scaffolding: ૢ࢞ െ ૜࢟ ൅ ࢠ ൌ ૚૙ Challenge advanced students ૞࢞ ൅ ૢࢠ ൌ ૚ by having them write a system a. Solve the system using algebra or matrix operations. If you use matrix operations, of equations which cannot be include the matrices you entered into the software and the calculations you performed to solve the system. solved using inverse matrix operations, and have them Answers will vary. An appropriate response is included. explain the process they used ૛െ૝૟ in constructing their system. Matrix method: ࡭ൌ൥ૢെ૜૚൩ ૞૙ૢ

૛ૠ ૢ ૠ ې െۍ ۑ૜૝૙ ૡ૞ ૚ૠ૙ ێ ૚ૢ ૜ ૚૜ ۑ െ െ ێ ࡭ି૚ ൌ ۑ ૡ૞ ૡ૞ ૡ૞ ێ ۑ ૜ ૚ ૜ ێ െ ے ૟ૡ ૚ૠ ૜૝ ۏ ૛ૠ ૢ ૠ ૚ ې െۍ ې െۍ ۑ ૡ૞ ێ ۑ૜૝૙ ૡ૞ ૚ૠ૙ ێ ૚ૢ ૜ ૚૜ ૚૝ ૛ૡ૜ ۑ െێ ൥૚૙൩ ൌ ۑ െ െ ێ࢞ൌ࡭ି૚࢈ൌ ۑ ૡ૞ ێ ૚ ۑ ૡ૞ ૡ૞ ૡ૞ ێ ۑ ૛ ێ ۑ ૜ ૚ ૜ ێ െ ے ૚ૠ ۏ ے ૟ૡ ૚ૠ ૜૝ ۏ

b. Verify your solution is correct. ૚ ૛ૡ૜ ૛ ૛൬െ ൰െ૝൬െ ൰൅૟൬ ൰ൌ૚૝ ૡ૞ ૡ૞ ૚ૠ ૚ ૛ૡ૜ ૛ ૢ൬െ ൰െ૜൬െ ൰൅૚൬ ൰ൌ૚૙ ૡ૞ ૡ૞ ૚ૠ ૚ ૛ ૞൬െ ൰൅ૢ൬ ൰ൌ૚ ૡ૞ ૚ૠ

c. Justify your decision to use the method you selected to solve the system.

Answers will vary. An example of an appropriate response is shown: The coefficients of the variables in the system did not contain corresponding variables that were identical or opposites, which might indicate that elimination is a slower method than using the matrix method.

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2. An athletic director at an all-boys high school is trying to find out how many coaches to hire for the football, basketball, and soccer teams. To do this, he needs to know the number of boys that play each sport. He does not have names or numbers but finds a note with the following information listed: The total number of boys on all three teams is ૡ૟. The number of boys that play football is ૠ less than double the total number of boys playing the other two sports. The number of boys that play football is ૞ times the number of boys playing basketball. a. Write a system of equations representing the number of boys playing each sport where ࢞ is the number of boys playing football, ࢟ basketball, and ࢠ soccer.

࢞൅࢟൅ࢠൌૡ૟ ૛ሺ࢟൅ࢠሻ െૠൌ࢞ ࢞ൌ૞࢟

b. Solve the system using algebra or matrix operations. If you use matrix operations, include the matrices you entered into the software and the calculations you performed to solve the system.

Answers will vary. An appropriate response is shown.

Matrix method: ૚૚૚ ૡ૟ ࡭ൌ൥െ૚ ૛ ૛൩ ࢈ ൌ ൥ ૠ ൩ ૚െ૞૙ ૙ ૛ ૚ ې െ ૙ ۍ ۑ ૜ ૜ ێ ૛ ૚ ૚ ۑ െ െ ێ ࡭ି૚ ൌ ۑ૚૞ ૚૞ ૞ێ ۑ ૚ ૛ ૚ ێ ے ૞ ૞ ૞ ۏ ૛ ૚ ې െ ૙ ۍ ۑ ૜ ૜ ێ ૛ ૚ ૚ ૡ૟ ૞૞ ൥ ૠ ൩ ൌ ൥૚૚൩ ۑ െ െ ێ࢞ൌ࡭ି૚࢈ൌ ૙ ૛૙ ۑ૚૞ ૚૞ ૞ێ ۑ ૚ ૛ ૚ ێ ے ૞ ૞ ૞ ۏ

c. Verify that your solution is correct. ૚ሺ૞૞ሻ ൅૚ሺ૚૚ሻ ൅૚ሺ૛૙ሻ ൌૡ૟ ૛ሺ૚૚ ൅ ૛૙ሻ െૠൌ૞૞ ૞૞ ൌ ૞ሺ૚૚ሻ

d. Justify your decision to use the method you selected to solve the system.

Answers will vary. An example of an appropriate response is shown: Using technology to solve the system using matrices generally takes less time than using algebra, especially when the solution set contains fractions.

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3. Kyra had ̈́૛૙ǡ ૙૙૙ to invest. She decided to put the money into three different accounts earning ૜Ψ, ૞Ψ, and ૠΨ simple interest, respectively, and she earned a total of ̈́ૢ૛૙Ǥ ૙૙ in interest. She invested half as much money at ૠΨ as at ૜Ψ. How much did she invest in each account? a. Write a system of equations that models this situation.

࢞ ൅ ࢟ ൅ ࢠ ൌ ૛૙૙૙૙ ૙Ǥ ૙૜࢞ ൅ ૙Ǥ ૙૞࢟ ൅ ૙Ǥ ૙ૠࢠ ൌ ૢ૛૙ ࢞െ૛ࢠൌ૙

b. Find the amount invested in each account.

She invested ̈́ૡǡ ૙૙૙ at ૜Ψ, ̈́ૡǡ ૙૙૙ at ૞Ψ, and ̈́૝ǡ ૙૙૙ at ૠΨ.

Closing (5 minutes) Have students summarize in writing the process of solving a problem using systems of equations in three-dimensional space and matrices. Students could individually write a bulleted list including important steps in the procedure along with suggestions for recognizing special cases and recommendations as to when representing a system as a linear transformation and applying inverse matrix operations is an efficient method of finding the solution. Have students share their results with a partner, and if time permits, have them display a compilation of their bulleted ideas on chart paper or on the board. à Create a system of equations. à Systems of equations in three-dimensional space can be represented as linear transformations using represents the linear transformation of point ݔ, and ܾ represents ܣ ݔ ൌ ܾ, whereܣ the matrix equation the image point after the linear transformation. à The entries of matrix ܣ are the coefficients of the system, and ܾ is the column matrix representing the constants for the system. à By applying inverse matrix operations, the solution to the system can be found by calculating ܣିଵܾ. à Software programs can be used to calculate the inverse matrices for ͵ൈ͵ systems and can be an efficient way to solve systems in three-dimensional space. à As with systems in two-dimensional space, systems in three-dimensional space can only be represented .(is nonzero ܣ has an inverse (e.g., the determinant of ܣ as linear transformations of ݔ when à If the determinant of ܣ is Ͳ, there is no single solution to the system. à When the coefficients of corresponding variables are not identical or opposites, which expedite the process of elimination, it may be quicker to use matrices to solve the system rather than algebraic methods.

Exit Ticket (5 minutes)

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Name Date

Lesson 15: Solving Equations Involving Linear Transformations of the Coordinate Space

Exit Ticket

The lemonade sales at a baseball game are described as follows: The number of small lemonades purchased is the number of mediums sold plus double the number of larges sold. The total number of all sizes sold is ͹Ͳ. One-and-a-half times the number of smalls purchased plus twice the number of mediums sold is ͳͲͲ.

Use a system of equations and its matrix representation to determine the number of small, medium, and large lemonades sold.

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Exit Ticket Sample Solutions

The lemonade sales at a baseball game are described as follows:

The number of small lemonades purchased is the number of mediums sold plus double the number of larges sold.

The total number of all sizes sold is ૠ૙.

One-and-a-half times the number of smalls purchased plus twice the number of mediums sold is ૚૙૙.

Use a system of equations and its matrix representation to determine the number of small, medium, and large lemonades

sold. ૛࢒ ൅ ࢓ ൌ ࢙ ࢙൅࢓൅࢒ൌૠ૙ ૚Ǥ ૞࢙ ൅ ૛࢓ ൌ ૚૙૙ െ૚ ૚ ૛ ࢙ ૙ ൥ ૚૚૚൩ቈ࢓቉ ൌ ൥ ૠ૙ ൩ ૚Ǥ ૞ ૛ ૙ ࢒ ૚૙૙

܌܍ܜሺ࡭ሻ ൌ ૟ െ૝ ૡ െ૛ ې ۍ ۑ ૢ ૢ ૢ ێ ૚ െ૛ ૛ ۑ ێ࡭ି૚ ǣ ۑ ૜ ૜ ૜ ێ ۑ૚ ૠ െ૝ ێ ے ૢ ૢ ૢ ۏ ૝૙ ࡭ି૚࢈ǣ ൥૛૙൩ ૚૙ Based on the system, projected sales are ૝૙ small lemonades, ૛૙ medium lemonades, and ૚૙ large lemonades.

Problem Set Sample Solutions

1. A small town has received funding to design and open a small airport. The airport plans to operate flights from three airlines. The total number of flights scheduled is ૚૙૙. The airline with the greatest number of flights is planned to have double the sum of the flights of the other two airlines. The plan also states that the airline with the greatest number of flights will have ૝૙ more flights than the airline with the least number of flights. a. Represent the situation described with a system of equations. Define all variables. ࢇ ൅ ࢈ ൅ ࢉ ൌ ૚૙૙ ࢇൌ૛ሺ࢈൅ࢉሻ ࢇൌࢉ൅૝૙

ࢇ: number of flights for the airline with the most flights

࢈: number of flights for the airline with the second-greatest number of flights

ࢉ: number of flights for the airline with the least number of flights

Note: Variables used may differ.

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b. Represent the system as a linear transformation using the matrix equation ࡭࢞ ൌ ࢈. Define matrices ࡭, ࢞, and ࢈. Equation: ૚૚ ૚ࢇ ૚૙૙ ൥૚െ૛െ૛൩ቈ࢈቉ ൌ ൥ ૙ ൩ ૚૙െ૚ࢉ ૝૙ ࡭: ૚૚ ૚ ൥૚െ૛െ૛൩ ૚૙െ૚ ࢞: ܉ ቈ܊቉ ܋ ࢈: ૚૙૙ ൥ ૙ ൩ ૝૙

c. Explain how you can determine if the matrix equation has a solution without solving it.

Answers will vary but should indicate that technology can be used to verify that the matrix ࡭ has a nonzero determinant or that it has an inverse.

d. Solve the matrix equation for ࢞. ૛ ې ૟૟ۍ ۑ૜ ێ ૛ ۑ ૟ ێ࢞ൌ࡭ି૚࢈ൌ ۑ ૜ ێ ۑ૛ ێ ૛૟ ے૜ ۏ

e. Discuss the solution in context. ૛ ૛ ૟૟ ૛૟ The solution indicates that the airlines, from greatest to least number of flights, should have ૜, ૜, ૛ ૟ and ૜ flights, respectively. This does not make sense given that the number of flights must be a whole number. Therefore, the number of flights granted should be approximately ૟ૠǡ ૛ૠ, and ૠ, which would satisfy the second and third conditions and would result in only ૚ more than the total number of flights planned for the airport.

2. A new blockbuster movie opens tonight, and several groups are trying to buy tickets. Three types of tickets are sold: adult, senior (over ૟૞), and youth (under ૚૙). A group of ૜ adults, ૛ youths, and ૚ senior pays̈́૞૝Ǥ ૞૙ for their tickets. Another group of ૟ adults and ૚૛ youths pays $૚૞૚Ǥ ૞૙. A final group of ૚ adult, ૝ youths, and ૚ senior pays $૝ૢǤ ૙૙. What is the price for each type of ticket? a. Represent the situation described with a system of equations. Define all variables.

૜ࢇ ൅ ૛࢟ ൅ ૚࢙ ൌ ૞૝Ǥ ૞૙ ૟ࢇ ൅ ૚૛࢟ ൌ ૚૞૚Ǥ ૞૙ ૚ࢇ ൅ ૝࢟ ൅ ૚࢙ ൌ ૝ૢ

ࢇ: price of an adult ticket

࢟: price of a youth ticket

࢙: price of a senior ticket

Note: Variables used may differ.

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b. Represent the system as a linear transformation using the matrix equation ࡭࢞ ൌ ࢈. ૜૛૚ࢇ ૞૝Ǥ ૞૙ ࡭࢞ ൌ ࢈ ൥૟૚૛૙൩ቈ࢟቉ ൌ ൥૚૞૚Ǥ ૞૙൩ ૚૝૚࢙ ૝ૢ

c. Explain how you can determine if the matrix equation has a solution without solving it.

Answers will vary but should indicate that technology can be used to verify that the matrix ࡭ has a nonzero determinant or that it has an inverse.

d. Solve the matrix equation for ࢞. ૚૙Ǥ ૛૞ ࢞ൌ࡭ି૚࢈ൌ൥ ૠǤ ૞૙ ൩ ૡǤ ૠ૞

e. Discuss the solution in context.

An adult ticket costs ̈́૚૙Ǥ ૛૞, a youth ticket costs ̈́ૠǤ ૞૙, and a senior ticket costs ̈́ૡǤ ૠ૞.

f. How much would it cost your family to attend the movie?

Answers will vary.

3. The system of equations is given: ૞࢝ െ ૛࢞ ൅ ࢟ ൅ ૜ࢠ ൌ ૛ ૝࢝ െ ࢞ ൅ ૟࢟ ൅ ૛ࢠ ൌ ૙ ࢝െ࢞െ࢟െࢠൌ૜ ૛࢝ ൅ ૠ࢞ െ ૜࢟ ൅ ૞ࢠ ൌ ૚૛

a. Write the system using a matrix equation in the form ࡭࢞ ൌ ࢈. ૞െ૛૚ ૜ ࢝ ૛ ૝െ૚૟ ૛ ࢞ ૙ ൦ ൪቎ ቏ൌ൦ ൪ ૚െ૚െ૚െ૚࢟ ૜ ૛ૠെ૜૞ࢠ ૚૛

b. Write the matrix equation that could be used to solve for ࢞. Then use technology to solve for ࢞. െૡ ૛૛ ૡ૞ ૚૜ ૜ૢ૞ ې ۍ ې ۍ ۑ ૚૞ૢ ێ ۑ૚૞ૢ ૚૞ૢ ૚૞ૢ ૚૞ૢێ ି૚ ૜૛૟ ۑ ێ ૛ ۑ െ૜ૡ ૛૞ ૝૟ ૛૛ێ ࢝ ૞െ૛૚ ૜ ૛ ۑ ૚૞ૢ ێ ૙ ۑ૚૞ૢ ૚૞ૢ ૚૞ૢ ૚૞ૢێ ࢞ ૝െ૚૟ ૛ ૙ ቎ ቏ൌ ۑ ێ቎࢟቏ൌ൦ ൪ ቎ ቏ൌ ۑ െૠ૜ ێ ૚െ૚െ૚െ૚ ૜ െ૚૚ ૚ૠ െૢ െ૚ ૜ ૚૛ ۑ ێ ࢠ ૚૛ ۑ ૚૙૟ ێ ۑ ૚૙૟ ૚૙૟ ૚૙૟ ૞૜ێ ૛ૠെ૜૞ ۑെ૚ૢૢێ ۑ૜૚ െ૚ૢ െૠ૚ െ૛ ێ ے ૚૙૟ ۏ ے ૚૙૟ ૚૙૟ ૚૙૟ ૞૜ۏ

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c. Verify your solution using back substitution. ૜ૢ૞ ૜૛૟ െૠ૜ െ૚ૢૢ ૞൬ ൰െ૛൬ ൰൅ ൅૜൬ ൰ൌ૛ ૚૞ૢ ૚૞ૢ ૚૙૟ ૚૙૟ ૜ૢ૞ ૜૛૟ െૠ૜ െ૚ૢૢ ૝൬ ൰െ ൅૟൬ ൰൅૛൬ ൰ൌ૙ ૚૞ૢ ૚૞ૢ ૚૙૟ ૚૙૟ ૜ૢ૞ ૜૛૟ െૠ૜ െ૚ૢૢ െ െ െ ൌ૜ ૚૞ૢ ૚૞ૢ ૚૙૟ ૚૙૟ ૜ૢ૞ ૜૛૟ െૠ૜ െ૚ૢૢ ૛൬ ൰൅ૠ൬ ൰െ૜൬ ൰൅૞൬ ൰ൌ૚૛ ૚૞ૢ ૚૞ૢ ૚૙૟ ૚૙૟

d. Based on your experience solving this problem and others like it in this lesson, what conclusions can you draw about the efficiency of using technology to solve systems of equations compared to using algebraic methods?

Answers will vary. An example of an appropriate response would be that, in general, as the number of equations and variables in a system increases, the more efficient it is to use matrices and technology to solve systems when compared to using algebraic methods.

െ૜ 4. In three-dimensional space, a point ࢞ is reflected over the ࢞ࢠ plane resulting in an image point of ൥ ૚ ൩. െ૛ a. Write the transformation as an equation in the form ࡭࢞ ൌ ࢈, where ࡭ represents the transformation of point ࢞ resulting in image point ࢈. ૚૙૙࢞ െ૜ ൥૙െ૚૙൩ቈ࢟቉ ൌ ൥ ૚ ൩ ૙૙૚ࢠ െ૛

b. Use technology to calculate ࡭െ૚. ૚૙૙ ࡭ି૚ ൌ ൥૙െ૚૙൩ ૙૙૚

c. Calculate ࡭ି૚࢈ to solve the equation for ࢞.

െ૜ ࢞ൌ൥െ૚൩ െ૛

d. Verify that this solution makes sense geometrically.

Reflecting a point over the ࢞ࢠ plane effectively reflects it over the plane ࢟ൌ૙, which will change the sign of the ࢟-coordinate and will leave the ࢞- and ࢠ-coordinates unchanged.

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5. Jamie needed money and decided it was time to open her piggy bank. She had only nickels, dimes, and quarters. The total value of the coins was ̈́ૡ૞Ǥ ૞૙. The number of quarters was ૜ૢ less than the number of dimes. The total value of the nickels and dimes was equal to the value of the quarters. How many of each type of coin did Jamie have? Write a system of equations and solve.

Let ࢞ represent the number of quarters, ࢟ represent the number of dimes, and ࢠ represent the number of nickels.

૙Ǥ ૛૞࢞ ൅ ૙Ǥ ૚૙࢟ ൅ ૙Ǥ ૙૞ࢠ ൌ ૡ૞Ǥ ૞૙ ࢞ൌ࢟െ૜ૢ ૙Ǥ ૙૞ࢠ ൅ ૙Ǥ ૚૙࢟ ൌ ૙Ǥ ૛૞࢞ ࢞ ൌ ૚ૠ૚ǡ ࢟ ൌ ૛૚૙ǡ ࢠ ൌ ૝૜૞ Jamie has ૚ૠ૚ quarters, ૛૚૙ dimes, and ૝૜૞ nickels.

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A STORY OF FUNCTIONS Lesson 16 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 16: Solving General Systems of Linear Equations

Student Outcomes Students represent complicated systems of equations, including ͶൈͶ and ͷൈͷ systems of equations, using represents the coefficient matrix, ݔ is the solution to the ܣ ݔ ൌ ܾ, whereܣ matrix equations in the form system, andܾ represents the constant matrix. Students use technology to calculate the inverse of matrices and use inverse matrix operations to solve complex systems of equations.

Lesson Notes In this lesson, students apply their understanding from Lessons 14 and 15 to systems that are higher than 3-by-3 and to systems in two- and three-dimensional space that are more complicated than those presented in previous lessons. They discover that, while it is difficult to geometrically describe linear transformations in four- or higher-dimensional space, the mathematics behind representing systems of equations as linear transformations using matrices is valid for higher- degree space. Students apply this reasoning to represent complicated systems of equations using matrices, and they use technology to solve the systems.

Classwork Example (15 minutes)

In this example, students see that a cubic function can be used to model scientific data comparing side length to volume. Students should complete part (a) with a partner. After a few minutes, a selected pair should display their solution, demonstrating how to use substitution of the data points into the cubic equation to find the equations for the system. After verifying the correct system, students should complete parts (b)–(d) with a partner. Technology that enables students to calculate the inverse of a matrix is needed to complete part (c). Each student should write and solve the matrix equation independently and verify the answer with a partner, but students can work in small groups or pairs, especially if access to technology is limited. Part (e) should be completed as a teacher-led discussion. Students should be encouraged to critically assess the usefulness of the model. If time permits, students could create a two-column chart for display that lists positives and negatives for the model. In the problem, what do the ordered pairs represent? à For each ordered pair, the first number is the greatest linear measurement of the irregularly-shaped object, and the second number represents the volume in cubic centimeters measured by water displacement. So, how could we use the information from the problem to write a system of equations? ,à Substitute each ordered pair into the ݒሺݔሻ equation, and simplify the resulting equations. Specifically .substitute the first coordinate for ݔ and the second coordinate for ݒሺݔሻ How would this look for the first point? à ͵ൌܽሺͳଷሻ ൅ܾሺͳଶሻ ൅ܿሺͳሻ ൅݀

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Once we have performed substitution with all the ordered pairs, what are we left with? à Four equations with four unknowns And how can we represent our system of equations as a matrix equation? represents the coefficient matrix for the system when all the ܣ ݔ ൌ ܾ, whereܣ à Use the equation ܽ ܾ .equations are written in standard form, ݔൌ቎ ቏and ܾ represents the ݒሺݔሻ values for the equations ܿ ݀ How can we use the matrix equation to write a model for the scientist’s data? .ଵ to both sides of the matrix equationିܣ à Isolate ݔ by applying Describe the measurements of the first object measured. à Its greatest linear measurement is ͳ , and its volume is ͵ ଷ. Why did the scientist select a cubic equation to model the data? à He generalized that there is usually a cubic relationship between the linear dimensions of an object and its volume. How else could we determine whether a cubic equation would serve as a good model for the data? à Answers will vary. Examples of appropriate responses would be that the points could be plotted to see if they seem to fit the pattern of a cubic function or that the data could be entered into a software program and a cubic regression performed. How can you find the inverse of matrix ܣ? à Students should mention the software or application that has been used in previous lessons to calculate the inverse of 3-by-3 matrices. Does ܣ have an inverse? If so, what is it? ͳ ͳ ͳ ͳ െ െ ې ͳͷ ͺ ͳʹ ͶͲ ۍ ۑ Ͷ ͳͳ ͵ ͹ ێ ۑ െ െ ێ ͷ ͺ Ͷ ͶͲ ۑ ێ à Yes, it is ͶͶ ͳ͹ ͷ ͹ ۑ െ െێ ۑ ͳͷ Ͷ ͵ ʹͲ ێ ۑ ͳ͸ ͳ ێ െ͵ ͳ െ ے ͷ ͷ ۏ And how do we use the inverse matrix to solve the system? ͲǤͳ͹ͷ െͳǤʹʹͷ ଵܾൌ൦ ൪ିܣà ݔൌ ͶǤͶͷ െͲǤͶ How do we use the solution to the system to write a cubic equation to model the data? à Substitute the solution for ܽ, ܾ, ܿ, and ݀ in the equation. The cubic equation that models the data is .ݒሺݔሻ ൌͲǤͳ͹ͷݔଷ െ ͳǤʹʹͷݔଶ ൅ͶǤͶͷݔെͲǤͶ How well does this model represent the scientist’s data? à It represents the data well because all the points fit the model. Does this mean that a cubic model is the best model for this data? à Not necessarily. It is possible that other models might fit the data also.

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What are some limitations of the model? à Answers will vary but might include that there are only four data points and no indication that they are based on repeated measurements. What could we recommend to the scientist if he wanted to strengthen his argument that a cubic model should represent the relationship between the greatest linear measure and the volume of the irregular objects? à Answers will vary but might include having him take additional measurements and assess the fit of several types of models to the data set to see which model is the best fit.

Example

A scientist measured the greatest linear dimension of several irregular metal objects. He then used water displacement to calculate the volume of each of the objects. The data he collected are ሺ૚ǡ ૜ሻ,ሺ૛ǡ ૞ሻ,ሺ૝ǡ ૢሻ, and ሺ૟ǡ ૛૙ሻ, where the first coordinate represents the linear measurement of the object in centimeters, and the second coordinate represents the volume in cubic centimeters. Knowing that volume measures generally vary directly with the cubed value of linear measurements, he wants to try to fit this data to a curve in the form of ࢜ሺ࢞ሻ ൌࢇ࢞૜ ൅࢈࢞૛ ൅ࢉ࢞൅ࢊ.

a. Represent the data using a system of equations. Scaffolding: ૜ൌࢇ൅࢈൅ࢉ൅ࢊ Advanced students can ૞ ൌ ૡࢇ ൅ ૝࢈ ൅ ૛ࢉ ൅ ࢊ select an additional ૢ ൌ ૟૝ࢇ ൅ ૚૟࢈ ൅ ૝ࢉ ൅ ࢊ function type and use ૛૙ ൌ ૛૚૟ࢇ ൅ ૜૟࢈ ൅ ૟ࢉ ൅ ࢊ matrices to try to fit the data to their selected b. Represent the system using a matrix equation in the form ࡭࢞ ൌ ࢈. model. ૚૚૚૚ࢇ ૜ Provide students with a ૡ૝૛૚࢈ ૞ ൦ ൪ ቎ ቏ൌ൦ ൪ written copy of a worked ૟૝ ૚૟ ૝ ૚ ࢉ ૢ ૛૚૟ ૜૟ ૟ ૚ ࢊ ૛૙ example using a 3-by-3 system, and suggest they c. Use technology to solve the system. follow the same procedure to write and solve the ି૚ ࢇ ૚૚૚૚ ૜ ૙Ǥ ૚ૠ૞ matrix equation for this ࢈ ૡ૝૛૚ ૞ െ૚Ǥ ૛૛૞ ቎ ቏ൌ൦ ൪ ൦ ൪ൌ൦ ൪ ࢉ ૟૝ ૚૟ ૝ ૚ ૢ ૝Ǥ ૝૞ problem. ࢊ ૛૚૟ ૜૟ ૟ ૚ ૛૙ െ૙Ǥ ૝ Have students compute the cubic regression d. Based on your solution to the system, what cubic equation models the data? equation using software to ࢜ሺ࢞ሻ ൌ ૙Ǥ ૚ૠ૞࢞૜ െ ૚Ǥ ૛૛૞࢞૛ ൅ ૝Ǥ ૝૞࢞ െ ૙Ǥ ૝ verify that it matches the equation found using matrices. e. What are some of the limitations of the model?

It is based off of only four points, so the equation goes exactly through all the points.

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Exercises (20 minutes)

Allow students to complete the work in pairs. They should each complete the work and then compare answers. Some pairs may need more help or additional instruction. After a few minutes, discuss Exercise 1 part (a) to ensure that students are able to represent the situation using a matrix. After about 5 minutes, Exercise 1 could be reviewed and discussed to help students who are struggling with understanding how to apply the matrix to find the solutions in parts (b) and (c). After the discussion, students could complete Exercise 2 in pairs or small groups. It might be necessary to model for students how to compute matrices to powers (e.g., how to use software to find ܣଷ଴).

Exercises

1. An attendance officer in a small school district noticed a trend among the four elementary schools in the district. This district used an open enrollment policy, which means any student within the district could enroll at any school in the district. Each year, ૚૙Ψ of the students from Adams Elementary enrolled at Davis Elementary, and ૚૙Ψ of the students from Davis enrolled at Adams. In addition, ૚૙Ψ of the students from Brown Elementary enrolled at Carson Elementary, and ૛૙Ψ of the students from Brown enrolled at Davis. At Carson Elementary, about ૚૙Ψ of students enrolled at Brown, and ૚૙Ψ enrolled at Davis, while at Davis, ૚૙Ψ enrolled at Brown, and ૛૙Ψenrolled at Carson. The officer noted that this year, the enrollment was 490, 250, 300, and 370 at Adams, Brown, Carson, and Davis, respectively. a. Represent the relationship that reflects the annual movement of students among the elementary schools using a matrix. ૙Ǥૢ૙૙૙Ǥ૚ Scaffolding: ૙૙Ǥૠ૙Ǥ૚૙Ǥ૛ ࡭ൌ൦ ൪ ૙૙Ǥ૚૙Ǥૡ૙Ǥ૚ Define ܣ as applying the ૙Ǥ ૚ ૙Ǥ ૚ ૙Ǥ ૛ ૙Ǥ ૟ enrollment trend from one year to the next year. b. Write an expression that could be used to calculate the attendance one year prior to Then have students the year cited by the attendance officer. Find the enrollment for that year. determine an expression Expression ൌ࡭ି૚࢈, where to represent applying the ૝ૢ૙ enrollment trend for 2 ૛૞૙ ࢈ൌ൦ ൪ years, 3 years, and ݊ years. ૜૙૙ ૜ૠ૙ Work with students to ૞૙૙ define ܣିଵ as representing ૛૙૙ ࡭ି૚࢈ൌ൦ ൪ the enrollment trend ૜૙૙ ૝૙૙ backward one year.

Enrollment one year prior to cited data: ૞૙૙at Adams, ૛૞૙ at Brown, ૜૙૙ at Work step-by-step through Carson, and ૝૙૙ at Davis part (a) with struggling students. Then encourage

them to use similar c. Assuming that the trend in attendance continues, write an expression that could be used to calculate the enrollment two years after the year cited by the attendance reasoning to what we officer. Find the attendance for that year. applied in Exercise 1 parts Expression ൌ࡭૛࢈ (b) and (c) to find the ૝૟૞Ǥ ૡ values in Exercise 2 parts ૛ૢ૟Ǥ ૠ ࡭૛࢈ൌ൦ ൪ (b) and (c). ૜૙૞Ǥ ૚ ૜૝ૢǤ ૠ

d. Interpret the results to part (c) in context.

The approximate enrollment at the schools would be 466, 297, 305, and 350 for Adams, Brown, Carson, and Davis, respectively.

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2. Mrs. Kenrick is teaching her class about different types of polynomials. They have just studied quartics, and she has offered 5 bonus points to anyone in the class who can determine the quartic that she has displayed on the board. ૠ ሺെ૟ǡ ૛૞ሻ ሺെ૜ǡ ૚ሻ ቀെ૛ǡ ቁ ሺ૙ǡ െ૞ሻ ሺ૜ǡ ૚૟ૢሻ The quartic has 5 points identified: , , ૜ , , and . Logan really needs those bonus points and remembers that the general form for a quartic is ࢟ൌࢇ࢞૝ ൅࢈࢞૜ ൅ࢉ࢞૛ ൅ࢊ࢞൅ࢋ. Can you help Logan determine the equation of the quartic? a. Write the system of equations that would represent this quartic. ૛૞ ൌ ૚૛ૢ૟ࢇ െ ૛૚૟࢈ ൅ ૜૟ࢉ െ ૟ࢊ ൅ ࢋ ૚ ൌ ૡ૚ࢇ െ ૛ૠ࢈ ൅ ૢࢉ െ ૜ࢊ ൅ ࢋ ૠ ൌ ૚૟ࢇ െ ૡ࢈ ൅ ૝ࢉ െ ૛ࢊ ൅ ࢋ ૜ െ૞ ൌ ࢋ ૚૟ૢ ൌ ૡ૚ࢇ ൅ ૛ૠ࢈ ൅ ૢࢉ ൅ ૜ࢊ ൅ ࢋ

b. Write a matrix that would represent the coefficients of this quartic.

૚૛ૢ૟ െ૛૚૟ ૜૟ െ૟ ૚ ېૡ૚ െ૛ૠ ૢ െ૜ ૚ ۍ ۑ ێ ۑ૚૟ െૡ ૝ െ૛ ૚ ێ࡭ൌ ۑ૙૙૙૙૚ ێ ےૡ૚ ૛ૠ ૢ ૜ ૚ ۏ

c. Write an expression that could be used to calculate coefficients of the equation.

Expression ൌ࡭ି૚࢈, where ૛૞ ې ૚ ۍ ۑ ૠ ێ ۑ ێ࢈ൌ ۑ ૜ ێ ۑ െ૞ ێ ے૚૟ૢۏ ૚ ې ۍ ۑ ૜ ێ ૜ ۑ ێ࡭ି૚࢈ൌ ۑ ૠ ێ ۑ ૚ ێ ےെ૞ۏ

d. Explain the answer in the context of this problem. ૚ ࢇൌ ,࢈ ൌ ૜,ࢉ ൌ ૠ,ࢊ ൌ ૚,ࢉ ൌ െ૞ ૜ These are the coefficients of the quartic. The equation of the quartic is ૚ ࢟ൌ ࢞૝ ൅૜࢞૜ ൅ૠ࢞૛ ൅࢞െ૞ ૜ .

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3. The Fibonacci numbers are the numbers ૚, ૚, ૛, ૜, ૞, ૡ, ૚૜, ૛૚, ૜૝, … Each number beyond the second is the sum of the previous two. ૚ ૚ ૛ ૜ ૞ Let ࢛ ൌቂ ቃ,࢛ ൌቂ ቃ,࢛ ൌቂ ቃ,࢛ ൌቂ ቃ,࢛ ൌቂ ቃ, and so on. ૚ ૚ ૛ ૛ ૜ ૜ ૝ ૞ ૞ ૡ ૙૚ a. Show that ࢛ ൌቂ ቃ࢛ . ࢔ା૚ ૚૚ ࢔

ࢌ૚ ࢌ૛ If we define the terms in the Fibonacci sequence as ࢌ࢔, where ࢔ൌ૚,૛,૜,૝,૞, …, then ࢛૚ ൌ൤ ൨, ࢛૛ ൌ൤ ൨, ࢌ૛ ࢌ૜ ࢌ૜ ࢌ࢔ ࢌ࢔ା૚ ࢌ࢔ା૚ ࢛૜ ൌ൤ ൨, ࢛࢔ ൌ൤ ൨, and ࢛࢔ା૚ ൌ൤ ൨. By definition, ࢌ࢔ା૛ ൌࢌ࢔ା૚ ൅ࢌ࢔, so ࢛࢔ା૚ ൌ൤ ൨. ࢌ૝ ࢌ࢔ା૚ ࢌ࢔ା૛ ࢌ࢔ା૚ ൅ࢌ࢔ ૙૚ ૙૚ ࢌ࢔ ࢌ࢔ା૚ Now, ቂ ቃ࢛࢔ ൌቂ ቃ൤ ൨ൌ൤ ൨, which is equivalent to ࢛࢔ା૚. ૚૚ ૚૚ࢌ࢔ା૚ ࢌ࢔ା૚ ൅ࢌ࢔

b. How could you use matrices to find ࢛૜૙? Use technology to find ࢛૜૙. ૙૚૛ૢ ૚ ૡ૜૛૙૝૙ ࢛ ൌቂ ቃ ቂ ቃൌቂ ቃ ૜૙ ૚૚ ૚ ૚૜૝૟૛૟ૢ

૚૟૞૞ૡ૙૚૝૚ c. If ࢛ ൌቂ ቃ, find ࢛ . Show your work. ࢔ ૛૟ૠૢ૚૝૛ૢ૟ ࢔ି૚ ૙૚ି૚ ૚૙૛૜૜૝૚૞૞ ࢛ ൌቂ ቃ ࢛ ൌቂ ቃ ࢔ି૚ ૚૚ ࢔ ૚૟૞૞ૡ૙૚૝૚

Closing (5 minutes) Review Exercise 3 as a teacher-led discussion. Students should be encouraged to respond in writing to the questions provided: In what ways can matrix operations be useful in modeling real-world situations? à Answers will vary but might include that they can be used to create models for data or to project trends forward and backward in time. In what ways can matrix operations be useful in representing mathematical relationships like the numbers in the Fibonacci sequence? à Answers will vary but might include that matrix operations can be used to determine values that would be very cumbersome to calculate by hand.

Exit Ticket (5 minutes)

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Name Date

Lesson 16: Solving General Systems of Linear Equations

Exit Ticket

Anabelle, Bryan, and Carl are playing a game using sticks of gum. For each round of the game, Anabelle gives half of her sticks of gum to Bryan and one-fourth to Carl. Bryan gives one-third of his sticks to Anabelle and keeps the rest. Carl gives ͶͲ percent of his sticks of gum to Anabelle and ͳͲ percent to Bryan. Sticks of gum can be cut into fractions when necessary. a. After one round of the game, the players count their sticks of gum. Anabelle has ͷʹͷ sticks, Bryan has ͸ͲͲ, and Carl has ͶͷͲ. How many sticks of gum would each player have afterʹmore rounds of the game? Use a matrix equation to represent the situation, and explain your answer in context.

b. How many sticks of gum did each player have at the start of the game? Use a matrix equation to represent the situation, and explain your answer in context.

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Exit Ticket Sample Solutions

a. After one round of the game, the players count their sticks of gum. Anabelle has ૞૛૞ sticks, Bryan has ૟૙૙, and Carl has ૝૞૙. How many sticks of gum would each player have after ૛ more rounds of the game? Use a matrix equation to represent the situation, and explain your answer in context.

૚ ૚ ૚ ૛ ې ۍ ۑ૝ ૛ ૝ێ ૞૛૞ ۑ ૚ ૛ێ ૙ ൥૟૙૙൩ ൌ࢞ ۑ ૜ ૜ێ ૝૞૙ ۑ ێ ۑ૛ ૚ ૚ێ ے૞ ૚૙ ૛ۏ ૜ ې ૞૝ૠۍ ۑ૚૟ ێ ۑ ૠ ێ ࢞ൌ ૞૟૝ ۑ૚૛ ێ ۑ ێ ۑ ૚ ێ ૞૛૛ ے ૛ ۏ

૜ ૠ ૞૝ૠ ૞૟૝ Anabelle would have ૚૟ sticks of gum, Bryan would have ૚૛ sticks, and Carl would have ૚ ૞૛૛ ૛ sticks.

b. How many sticks of gum did each player have at the start of the game? Use a matrix equation to represent the situation, and explain your answer in context.

૚ ૚ ૚ ି૚ ې ۍ ۑ૝ ૛ ૝ێ ૞૛૞ ۑ ૚ ૛ێ ૙ ൥૟૙૙൩ ൌ࢞ ۑ ૜ ૜ێ ૝૞૙ ۑ ێ ۑ૛ ૚ ૚ێ ے૞ ૚૙ ૛ۏ ૟૙૙ ࢞ൌ൥૟૙૙൩ ૜૙૙ At the start of the game, Anabelle and Bryan each had ૟૙૙ sticks of gum, and Carl had ૜૙૙ sticks of gum.

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Problem Set Sample Solutions

1. The system of equations is given: ૚Ǥ ૛࢞ ൅ ૜࢟ െ ૞ࢠ ൅ ૝Ǥ ૛࢝ ൅ ࢜ ൌ ૙ ૟࢞ ൌ ૞࢟ ൅ ૛࢝ ૜࢟ ൅ ૝Ǥ ૞ࢠ െ ૟࢝ ൅ ૛࢜ ൌ ૚૙ ૢ࢞െ࢟൅ࢠ൅૛࢜ൌെ૜ െ૝࢞ ൅ ૛࢟ െ ࢝ ൅ ૜࢜ ൌ ૚ a. Represent this system using a matrix equation. ૚Ǥ ૛ ૜ െ૞ ૝Ǥ ૛ ૚ ࢞ ૙ ې ૙ ۍ ې࢟ۍ ې૟െ૞૙െ૛૙ ۍ ۑ ێ ۑ ێ ۑ ێ ۑ૚૙ێ ൌ ۑ ࢠ ێ ൌ ۑ૙૜૝Ǥ૞െ૟૛ ێ ۑെ૜ێ ۑ࢝ێ ۑെ૚૚ ૙૛ૢ ێ ے ૚ ۏ ے࢜ۏ ےെ૝ ૛ ૙ െ૚ ૜ۏ

b. Use technology to solve the system. Show your solution process, and round your entries to the tenths place.

࢞ ૚Ǥ ૛ ૜ െ૞ ૝Ǥ ૛ ૚ ି૚ ૙ ૙Ǥ ૛ ې ૚Ǥ ૜ ۍ ې ૙ ۍ ې૟െ૞૙െ૛૙ ۍ ې࢟ۍ ۑ ێ ۑ ێ ۑ ێ ۑ ێ ۑെ૚Ǥ ૞ێ ൌ ۑ૚૙ێ ۑ૙ ૜ ૝Ǥ૞െ૟૛ ێ ൌ ۑ ࢠ ێ ۑെ૛Ǥ ૞ێ ۑെ૜ێ ۑെ૚૚ ૙૛ૢ ێ ۑ࢝ێ ےെ૚Ǥ ૚ۏ ے ૚ ۏ ےെ૝ ૛ ૙ െ૚ ૜ۏ ے࢜ۏ

4. A caterer is preparing a fruit salad for a party. She decides to use strawberries, blackberries, grapes, bananas, and kiwi. The total weight of the fruit is ૚૙ pounds. Based on guidelines from a recipe, the weight of the grapes is equal to the sum of the weight of the strawberries and blackberries; the total weight of the blackberries and kiwi is ૛ pounds; half the total weight of fruit consists of kiwi, strawberries, and bananas; and the weight of the grapes is twice the weight of the blackberries. a. Write a system of equations to represent the constraints placed on the caterer when she makes the fruit salad. Be sure to define your variables.

ࡿ = pounds of strawberries

࡮ = pounds of blackberries

ࡳ = pounds of grapes

ࡷ = pounds of kiwi

࡮ࢇ = pounds of bananas

ࡿ൅࡮൅ࡳ൅ࡷ൅࡮ࢇൌ૚૙ ࡳൌࡿ൅࡮ ࡮൅ࡷൌ૛ ࡷ൅ࡿ൅࡮ࢇൌ૞ ࡳൌ૛࡮

b. Represent the system using a matrix equation. ૚ ૚ ૚૚૚ ࡿ ૚૙ ې ૙ ۍ ې ࡮ ۍ ېെ૚ െ૚ ૚ ૙ ૙ۍ ۑ ێ ۑ ێ ۑ ێ ۑ ૛ ێ ൌ ۑ ࡳ ێ ۑ૙ ૚ ૙૚૙ ێ ۑ ૞ ێ ۑ ࡷ ێ ۑ૚ ૙ ૙૚૚ ێ ے ૙ ۏ ے࡮ࢇۏ ے૙ െ૛૚૙૙ ۏ

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c. Solve the system using the matrix equation. Explain your solution in context. ૞ ې ۍ ۑ ૜ ێ ۑ ૞ ێ ି૚ ۑ ૜ ێ ࡿ ૚૚૚૚૚૚૙ ۑ ێ ې ૙ ۍ ېെ૚ െ૚ ૚ ૙ ૙ۍ ې ࡮ ۍ ۑ૚૙ێ ۑ ێ ۑ ێ ۑ ێ ൌ ۑ ૛ ێ ۑ૙૚૙૚૙ ێ ൌ ۑ ࡳ ێ ۑ ૜ ێ ۑ ૞ ێ ۑ૚૙૙૚૚ ێ ۑ ࡷ ێ ۑ ێ ے ૙ ۏ ے૙ െ૛૚૙૙ ۏ ے࡮ࢇۏ ۑ ૚ ێ ۑ ૜ ێ ۑ ێ ے ૜ ۏ ૞ ૞ ૚૙ The fruit salad consists of pounds of strawberries, pounds of blackberries, pounds of grapes, ૜ ૜ ૜ ૚ pound of kiwi, and ૜ pounds of bananas. ૜

d. How helpful would the solution to this problem likely be to the caterer as she prepares to buy the fruit?

It is useful as a general guideline, but the caterer is unlikely to buy the fruit in exactly the amount indicated by ૚ the problem. For instance, it is unlikely that she could purchase exactly pound of kiwi because it generally ૜ has to be purchased per fruit, not per ounce.

5. Consider the sequence ૚, ૚, ૚, ૜, ૞, ૢ, ૚ૠ, ૜૚, ૞ૠ, … where each number beyond the third is the sum of the th th th previous three. Let ࢝࢔ be the points with the ࢔ , ሺ࢔ ൅ ૚ሻ , and ሺ࢔൅૛ሻ terms of the sequence.

a. Find a ૜ൈ૜ matrix ࡭ so that ࡭࢝࢔ ൌ࢝࢔ା૚ for each ࢔. ૙૚૙ ࡭ൌ൥૙૙૚൩ ૚૚૚

b. What is the ૜૙th term of the sequence?

૛ૢ ࢝૜૙ ൌ࡭ ࢝૚ ૙૚૙૛ૢ ૚ ૛૙૟૙૜૜૟૚ ൌ ൥૙૙૚൩ ൥૚൩ ൌ ൥૜ૠૡૢ૞૝ૡૢ൩ ૚૚૚ ૚ ૟ૢૠ૙૙૟ૠ૚

ି૚ ି૚ c. What is ࡭ ? Explain what ࡭ represents in terms of the sequence. In other words, how can you find ࢝࢔ି૚ if you know ࢝࢔? െ૚ െ૚ ૚ ࡭ି૚ ൌ ൥ ૚૙૙൩ ૙૚૙

The first entry of࢝࢔ି૚ is the third entry from ࢝࢔ minus the sum of the first two entries of ࢝࢔, the second entry of ࢝࢔ି૚ is the first entry of ࢝࢔, and the third entry of ࢝࢔ି૚ is the second entry of ࢝࢔.

d. Could you find the െ૞th term in the sequence? If so, how? What is its value?

Yes െ૚ െ૚ ૚ ି૟ ૚ ૚ ࢝ି૞ ൌ ൥ ૚૙૙൩ ൥૚൩ ൌ ൥ ૜ ൩ ૙૚૙ ૚ െ૜

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6. Mr. Johnson completes a survey on the number of hours he spends weekly watching different types of television programs. He determines that he spends ૜૙ hours a week watching programs of the following types: comedy, drama, movies, competition, and sports. He spends half as much time watching competition shows as he does watching dramas. His time watching sports is double his time watching dramas. He spends an equal amount of time watching comedies and movies. The total amount of time he spends watching comedies and movies is the same as the total amount of time he spends watching dramas and competition shows. Write and solve a system of equations to determine how many hours Mr. Johnson watches each type of programming each week.

Let ࢌrepresent the hours watching comedy, ࢊrepresent the hours watching drama, ࢓represent the hours watching movies, ࢉrepresent the hours watching competition shows, and ࢙ represent the hours watching sports.

ࢌ൅ࢊ൅࢓൅ࢉ൅࢙ൌ૜૙ ૙Ǥ ૞ࢊ ൌ ࢉ ࢙ൌ૛ࢊ ࢌൌ࢓ ࢌ൅࢓ൌࢊ൅ࢉ ࢌ ൌ ૝Ǥ ૞ǡ ࢊ ൌ ૟ǡ ࢓ ൌ ૝Ǥ ૞ǡ ࢉ ൌ ૜ǡ ࢙ ൌ ૚૛ ૚૚ ૚ ૚૚ࢌ ૜૙ ૙૙Ǥ૞૙ െ૚૙ࢊ ૙ ተ ተ ተ ተ ተ ተ ተ૙૙ ૙ ૙૙ተ ተ࢓ተ ൌ ተ ૙ ተ ૚૙െ૚૙૙ࢉ ૙ ૚െ૚૚ െ૚૙࢙ ૙

Mr. Johnson spends ૝Ǥ ૞ hours watching comedies, ૟ hours watching dramas, ૝Ǥ ૞ hours watching movies, ૜ hours watching competition shows, and ૚૛ hours watching sports each week.

5. A copper alloy is a mixture of metals having copper as their main component. Copper alloys do not corrode easily and conduct heat. They are used in all types of applications including cookware and pipes. A scientist is studying different types of copper alloys and has found one containing copper, zinc, tin, aluminum, nickel, and silicon. The alloy weighs ૜Ǥ ૛ kilograms. The percentage of aluminum is triple the percentage of zinc. The percentage of silicon is half that of zinc. The percentage of zinc is triple that of nickel. The percentage of copper is fifteen times the sum of the percentages of aluminum and zinc combined. The percentage of copper is nine times the combined percentages of all the other metals. a. Write and solve a system of equations to determine the percentage of each metal in the alloy.

Let ࢉrepresent the percentage of copper, ࢠrepresent the percentage of zinc, ࢚represent the percentage of tin, ࢇrepresent the percentage of aluminum, ࢔represent the percentage of nickel, ࢙represent the percentage of silicon.

ࢉ ൅ ࢠ ൅ ࢚ ൅ ࢇ ൅ ࢔ ൅ ࢙ ൌ ૚૙૙ ࢇൌ૜ࢠ ࢙ൌ૙Ǥ૞ࢠ ࢠൌ૜࢔ ࢉൌ૚૞ሺࢇ൅ࢠሻ ࢉൌૢሺࢠ൅࢚൅ࢇ൅࢔൅࢙ሻ ࢉൌૢ૙Ψ,ࢠ ൌ ૚Ǥ ૞Ψ,࢚ ൌ ૛Ǥ ૠ૞Ψ, ࢇ ൌ ૝Ǥ ૞Ψ,ࡺ ൌ ૙Ǥ ૞Ψ,࢙ ൌ ૙Ǥ ૠ૞Ψ

The alloy has ૢ૙Ψ copper, ૚Ǥ ૞Ψ zinc, ૛Ǥ ૠ૞Ψ tin, ૝Ǥ ૞Ψ aluminum, ૙Ǥ ૞Ψ nickel, and ૙Ǥ ૠ૞Ψ silicon. ૚૚ ૚ ૚ ૚૚ࢉ ૚૙૙ ૙െ૜૙૚૙૙ࢠ ૙ ተ૙െ૙Ǥ૞૙૙૙૚ተ ተ࢚ተ ተ ૙ ተ ൌ ተ૙૚ ૙ ૙െ૜૙ተ ተࢇተ ተ ૙ ተ ૚ െ૚૞ ૙ െ૚૞ ૙ ૙ ࢔ ૙ ૚ െૢെૢെૢെૢെૢ࢙ ૙

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b. How many kilograms of each alloy are present in the sample?

Copper: ૙Ǥ ૢ૙ሺ૜Ǥ ૛ܓ܏ሻ ൌ૛Ǥૡૡܓ܏

Zinc: ૙Ǥ ૙૚૞ሺ૜Ǥ ૛ܓ܏ሻ ൌ ૙Ǥ ૙૝ૡܓ܏

Tin: ૙Ǥ ૙૛ૠ૞ሺ૜Ǥ ૛ܓ܏ሻ ൌ ૙Ǥ ૙ૡૡܓ܏

Aluminum: ૙Ǥ ૙૝૞ሺ૜Ǥ ૛ܓ܏ሻ ൌ ૙Ǥ ૚૝૝ܓ܏

Nickel: ૙Ǥ ૙૙૞ሺ૜Ǥ ૛ܓ܏ሻ ൌ ૙Ǥ ૙૚૟ܓ܏

Silicon: ૙Ǥ ૙૙ૠ૞ሺ૜Ǥ ૛ܓ܏ሻ ൌ ૙Ǥ ૙૛૝ܓ܏

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^dKZzK&&hEd/KE^

DĂƚŚĞŵĂƚŝĐƐƵƌƌŝĐƵůƵŵ

PRECALCULUS AND ADVANCED TOPICS ͻDKh>2

Topic D sĞĐƚŽƌƐŝŶWůĂŶĞĂŶĚ^ƉĂĐĞ

&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors .(ȁȁ, ݒܞȁ, ȁȁܞȁ ,ܞ ,.and their magnitudes (e.g

(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

(+) Add and subtract vectors. à Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that magnitude of a sum of two vectors is typically not the sum of the magnitudes. à Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. à Understand vector subtraction ܞെܟ as ܞ൅ሺെܟሻ, where െܟ is the additive inverse of ܟ, with the same magnitude as ܟ and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

(+) Multiply a vector by a scalar. à Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as

.ሺݒ௫ǡݒ௬ሻൌሺܿݒ௫ǡܿݒ௬ሻܿ ȁȁ ൌ ȁܿȁݒ. Compute theܞusing ȁȁܿ ܞܿ à Compute the magnitude of a scalar multiple ܞ is either along ܞܿ knowing that when ȁܿȁݒ ് Ͳ, the direction of ܞܿ direction of for ሺܿ ൐ Ͳሻ or against ܞ (for ܿ൏Ͳ).

282 dŽƉŝĐ: Vectors in Plane and Space

>ĞƐƐŽŶϭ7: Vectors in the Coordinate Plane (P)1 >ĞƐƐŽŶ18: Vectors and Translation Maps (P) >ĞƐƐŽŶϭϵ: Directed Line Segments and Vectors (P) >ĞƐƐŽŶϮϬ: Vectors and Stone Bridges (E) >ĞƐƐŽŶϮϭ: Vectors and the Equation of a Line (S) >ĞƐƐŽŶϮϮ͗ Linear Transformations of Lines (S) >ĞƐƐŽŶƐϮϯ–24: Why Are Vectors Useful? (P, S)

Topic D opens with a formal definition of a vector (the motivation and context for it is well in place at this point), and the arithmetical work for vector addition, subtraction, scalar multiplication, and vector magnitude is explored along with the geometrical frameworks for these operations.

Lesson 17 introduces vectors in terms of translations. Students use their knowledge of transformations to represent vectors as arrows with an initial point and a terminal point. They calculate the magnitude of a vector, add and subtract vectors, and multiply a vector by a scalar. Students interpret these operations geometrically and compute them component-wise. Lesson 18 builds on vectors as shifts by relating them to translation maps studied in prior lessons. The connection between matrices and vectors becomes apparent in this lesson as a notation for vectors is introduced that recalls matrix notation. This lesson’s focus extends vector addition, subtraction, and scalar multiplication to Թଷ.

Lesson 19 introduces students to directed line segments and how to subtract initial point coordinates from terminal point coordinates to find the components of a vector. Vector arithmetic operations are reviewed, and the parallelogram rule is introduced. Students study the magnitude and direction of vectors. In Lesson 20, students apply vectors to real-world applications as they look at vector effects on ancient stone arches and try to create their own. In Lesson 21, students describe a line in the plane using vectors and parameters, and then they apply this description to lines in Թଷ. The shift to describing a line using vectors to indicate the direction of the line requires students to think geometrically about lines in the plane instead of algebraically. Lesson 21 poses the question, “Is the image of a line under a linear transformation a line?” Students must extend the process of finding parametric equations for a line in Թଶ and Թଷ. Lessons 21 and 22 help students see the coherence between the work that they have done with functions and how that relates to vectors written using parametric equations. This sets the mathematical foundation that students will need to understand the definition of vectors. Vectors are generally described as a quantity that has both a magnitude and a direction. In Lessons 21 and 22, students perform linear transformations in two and three dimensions by writing parametric equations. The most basic definition of a vector is that it is a description of a shift or translation. Students will see that any physical operation that induces a shift of some kind is often thought of as a vector. Hence, vectors are prevalent in mathematics, science, and engineering. For example, force is often interpreted as a vector in physics because a force exerted on an object is a push of some magnitude that causes the object to shift in some direction. In Lessons 23 and 24, students solve problems involving velocity as well as other quantities (e.g., force) that can be represented by vectors. Students work with non-right triangles and continue to work on adding and

1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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©201ϴ Great Minds ®. eureka-math.org A STORY OF FUNCTIONS dŽƉŝĐ DϮ PRECALCULUS AND ADVANCED TOPICS subtracting vectors, but they interpret the resulting magnitude and direction within a context (e.g., with particle motion). Students use pictures to understand that transforming systems of linear equations while preserving the solution can be rephrased in terms of a series of linear transformations and translations.

In Topic D, students make sense of vectors by using them to represent real-world situations such as earthquakes, velocity, and force. Students focus on vector arithmetic, determining the magnitude and direction of vectors and vector sums and products, and constructing arguments about why the operations work the way that they do.

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Lesson 17: Vectors in the Coordinate Plane

Student Outcomes Students add and subtract vectors and understand those operations geometrically and component-wise. Students understand scalar multiplication graphically and perform it component-wise.

Lesson Notes This lesson introduces translation by a vector in the coordinate plane. In this lesson, we represent a vector as an arrow with an initial point and a terminal point. Students learn vector notation and the idea that a vector can represent a shift (i.e., a translation). They calculate the magnitude of a vector, add and subtract vectors, and multiply a vector by a scalar. Students interpret these operations geometrically and compute them component-wise. However, students are not yet working with directed line segments or vectors in Թଷ. Later lessons also represent vectors in magnitude and direction form. Students are making sense of vectors and relating them to a real-world situation. Later, the lesson focuses on vector arithmetic and making sense of why the operations work the way that they do. The study of vectors is a vital part of this course; notation for vectors varies across different contexts and curricula. which, in column format is ۄͶǡ ͷۃ lowercase, bold, non-italicized); or as) ܞ These materials refer to a vector as Ͷ Ͷ ቀ ቁ or ቂ ቃ. ͷ ͷ .ۄͶǡ ͷۃ to establish a name for the vector ”ۄͶǡͷۃൌܞ We use “let without the word let preceding the equation, unless it is ۄͶǡͷۃൌܞ When naming a vector, this curriculum avoids stating absolutely clear from the context that we are naming a vector. However, as we have done in other grades, we continue using ൌ to describe vector equations, such as ܞ൅ܟൌܟ൅ܞ.

We refer to the vector from ܣ to ܤ as vector ሬሬሬሬሬԦܣܤ; notice that this is a ray with a full arrow. This notation is consistent with how vectors were introduced in Grade 8 and is also widely used in post-secondary textbooks to describe both rays and vectors, depending on the context. To avoid confusion, this curriculum provides (or strongly implies) the context, to make it clear whether the full arrow indicates a vector or a ray. For example, when referring to a ray from ܣ passing through ܤ, we say “ray ሬሬሬሬሬԦܣܤ,” and when referring to a vector from ܣ to ܤ, we say “vector ሬሬሬሬሬԦܣܤ.” Encourage students to think about the context of the problem and not just rely on a hasty inference based on the symbol. The magnitude of a vector is signified as ԡܞԡ (lowercase, bold, non-italicized). Since Grade 6, we have been using the term vector in two slightly different ways. Until now, the difference was subtle and it did not matter where the term was used in the discussions. However, in Precalculus and Advanced Topics, we need to distinguish between a bound vector and a free vector:

BOUND VECTOR: A bound vector is a directed line segment (i.e., an arrow). For example, the directed line segment ሬሬሬሬሬԦܣܤ is a bound vector whose initial point (i.e., tail) is ܣ and terminal point (i.e., tip) is ܤ.

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Bound vectors are bound to a particular location in space. A bound vector ሬሬሬሬሬԦܣܤ has a magnitude given by the length of segment ܣܤ and a direction given by the ray ሬሬሬሬሬԦܣܤ. Often only the magnitude and direction of a bound vector matter, not its position in space. In such cases, we consider any translation of that bound vector to represent the same free vector.

FREE VECTOR: A free vector is the equivalence class of all directed line segments (i.e., arrows) that are equivalent to each other by translation. For example, scientists often use free vectors to describe physical quantities that have magnitude and direction only, freely placing an arrow with the given magnitude and direction anywhere it is needed in a diagram. For any directed line segment in the equivalence class defining a free vector, the directed line segment is said to be a representation of the free vector (i.e., it represents the free vector). Any other .ܞ Free vectors are usually notated by a lowercase letter with an arrow ݒԦ or by a boldface lowercase letter representation of a free vector is also labeled by the same lowercase letter (i.e., any two arrows in a diagram/Euclidean plane/Cartesian plane with the same magnitude and direction are labeled ܞ). We can also use the notation ሬሬሬሬሬԦܣܤ to represent a free vector ܞ, but it still is the specific directed line segment from ܣ to ܤ. (One would not label another representation of the free vector ܞ at a different location by ሬሬሬሬሬԦܣܤ, for example.) Unless specifically stated, the term vector refers to free vector throughout this module. Cartesian coordinates are useful for representing free vectors as points in coordinate space. Such representations provide simple formulas for adding/subtracting vectors and for finding a scalar multiple of a vector, as well as making it easy to write formulas for translations.

Classwork Opening (2 minutes) Have students read the opening paragraph, and then ask them what they know about earthquakes. Hold a brief discussion to activate prior knowledge. Emphasize the shifting ground that can occur during an earthquake, and explain that this lesson considers a way to represent such a shift mathematically. Consider referencing this article (http://www.cnn.com/2011/WORLD/asiapcf/03/12/japan.earthquake.tsunami.earth/) that discusses the 2011 Japan earthquake that shifted the coastline by eight feet.

Opening Exercise (2 minutes) Students should complete this exercise working with a partner. Have one or two students share their responses with the whole class. The idea is to help students see that the shift is independent of a starting point and that all points in the plane shift by the same amount. If students mention the force of the earthquake, make a connection back to that language when discussing magnitude.

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Opening Exercise Scaffolding: When an earthquake hits, the ground shifts abruptly due to forces created when the tectonic Use a graphic organizer plates along fault lines rub together. As the tectonic plates shift and move, the intense shaking can even cause the physical movement of objects as large as buildings. (like a Frayer model) to help students make sense Suppose an earthquake causes all points in a town to shift ૚૙ feet to the north and ૞ feet to the east. of this new concept (i.e., vector). See Precalculus and Advanced Topics, Module 1, Lesson 5 for an example of a Frayer diagram. Have advanced students draw vectors representing different shifts that are not whole numbers, such as S (ʹǤʹͷ feet) and W (͹ feet) N (ͻ feet) and E (ͳǤͷ feet)

S (ͳǤ͹ͷ feet) and E a. Explain how the diagram shown above could be said to represent the shifting caused (ͷǤͷfeet) by the earthquake.

The arrows show the amount and direction of the shift. Each point, not just the ones represented in this diagram, would be shifted ૞ feet east and ૚૙ feet north.

b. Draw another arrow that shows the same shift. Explain how you drew your arrow.

The new arrow would be shifted ૞ feet east and ૚૙ feet north from its initial point.

Discussion (5 minutes) Lead a brief discussion to introduce the notion and notation of a vector. Clarify the vector notation used in this module. Vectors are denoted by a bold, lowercase letter when typed, with the components enclosed in angled brackets. Some texts simply use parentheses. When writing a vector by hand, direct students to draw an arrow over the symbol that represents the vector. In mathematics, a shift like the one described in the Opening can be represented by a vector. This vector has a horizontal component of ͷ and a vertical component of ͳͲ. We use the following notation for a vector: ۄͷǡͳͲۃൌܞ When writing a vector that starts at point ܣ and goes through point ܤ, use this notation:

ۄͷǡͳͲۃ ൌ ܤܣሬሬሬሬሬԦ

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Are the points in the diagram above the only points that were shifted during the earthquake? Explain your thinking. à No, every point in the area affected by the earthquake would have been shifted. In reality, how far a point shifts depends on how far it is from a quake’s epicenter. When we consider vectors, the precise location of the arrow that represents the vector does not matter because the vector represents the translation of an object (e.g., points in the plane) by the given horizontal and vertical components. What other objects could be translated by a vector? à You could shift any figure by a vector, such as a line, a circle, or another geometric figure.

Exercises 1–3 (10 minutes) The next exercises let students compare different vectors in the coordinate plane, showing them that vectors are defined as having both a magnitude and a direction, and that two arrows with the same magnitude and direction represent the same vector. Students explore the notion of magnitude in Exercise 3, and the example that follows formally defines the magnitude of a vector. Have students work in small groups or with a partner, but give them time to work individually first so that each student has a chance to think about and make sense of these ideas.

Exercises 1–3

Several vectors are represented in the coordinate plane below using arrows.

1. Which arrows represent the same vector? Explain how you know.

Arrows ܟ, ܝ, and ܉ represent the same vector because they indicate a translation of ૚ unit right and ૜ units up. Arrows ܞ and ܊ represent the same vector because these arrows indicate a translation of ૜ units right and ૚ unit up. Arrows ܋ and ܌ represent the same vector because they represent a translation of ૚ unit left and ૜ units down.

2. Why do arrows ܋ and ܝ not represent the same vector?

These arrows have the same magnitude but opposite directions.

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Discuss the following questions after Exercise 2 with students. How many different vectors are in the diagram? What are the components of each one? ۄ͵െͳǡ െۃൌ܌ and ,ۄǡͳ͵ۃൌܞ ,ۄ͵ͳǡۃൌܝ :à There are really just three vectors Why might we only draw one arrow to represent a vector even though all points are shifting by the components of the vector? à Since a vector represents a translation, no matter where in the coordinate plane we represent it, it will still describe the same shifting. Why does the location of the tip of the arrow matter when representing a vector in the coordinate plane? would appear ۄ͵െͳǡ െۃൌ܌ and ۄ͵ͳǡۃൌܝ à The tip is used to indicate direction. Otherwise, vectors like to be the same. If needed, before starting Exercise 3, redirect students’ attention to the Opening. Give them time to collaborate on their approach to part (b) in groups or with a partner. Students may benefit from guided questioning like that listed below if they appear stuck. What quantities would you need to measure to determine which earthquake shifted the points further? à For each earthquake, you would need to find the distance between the starting location and the stopping location for one point, since all points shifted the same amount. How could you use a familiar formula to determine these quantities? à Since we know the horizontal and vertical displacement, we can use the Pythagorean theorem (or distance formula) to calculate the distance, which is represented by the length of the arrow.

3. After the first earthquake shifted points ૞ feet east and ૚૙ feet north, suppose a second earthquake hits the town and all points shift ૟ feet east and ૢ feet south. a. Write and draw a vector ܜ that represents this shift caused by the second earthquake.

b. Which earthquake, the first one or the second one, shifted all the points in the town further? Explain your reasoning.

The length of the arrow that represents the vector ܞ is ξ૞૛ ൅૚૙૛, or ξ૚૛૞ feet. The length of the arrow that represents the vector ܜ is ඥ૟૛ ൅ ሺെૢሻ૛, or ξ૚૚ૠ feet. The first quake shifted the points further. You can also see from the diagram that if we rotated ܜ from the tip of ܞ to align with ܞ, then ܜ would be slightly shorter.

Explain what you have just learned to your neighbor. Use this as an informal way to check student understanding.

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Example (3 minutes): The Magnitude of a Vector

This example introduces the formula to calculate the magnitude of a vector. Students already likely used the Pythagorean theorem in their work in Exercise 3. This example provides a way to formalize what we mean when we refer to the magnitude of a vector. For now, the direction of a vector is simply indicated by the arrow tip showing the direction as well as the sign and size of the components. Later in this module, students use trigonometry to calculate an angle that represents the direction of a vector.

Example: The Magnitude of a Vector

is the length of the line segment from the origin ۄࢇǡ ࢈ۃൌܞ The magnitude of a vector to the point ሺࢇǡ ࢈ሻ in the coordinate plane, which we denote by ԡܞԡ. Using the language of translation, the magnitude of ܞ is the distance between any point and its image under the translation ࢇ units horizontally and ࢈ units vertically. It is denoted ԡܞԡ.

Explain your .ۄ૟ǡ െૢۃൌܜ and ۄ૞ǡ ૚૙ۃൌܞ a. Find the magnitude of reasoning.

ԡܞԡ ൌ ඥ૞૛ ൅૚૙૛ ൌ ξ૚૛૞

ԡܜԡ ൌ ඥሺ૟ሻ૛ ൅ ሺെૢሻ૛ ൌ ξ૚૚ૠ

We use the Pythagorean theorem (or distance formula) to find the length of the hypotenuse of a triangle with side lengths ࢇ and ࢈.

b. Write the general formula for the magnitude of a vector.

ԡܞԡ ൌ ඥࢇ૛ ൅࢈૛

How does this example confirm or refute your work in Exercise 3, part (b)? à We got the same results and we can now see that the magnitude of a vector is a measure of how much shifting occurred.

Discussion (5 minutes): Vector Addition Explain why this drawing shows the overall shifting caused by both earthquakes. à The points shift from the starting point or pre-image to the image translated ͷ right and ͳͲ up and then those points are translated again, ͸ right and ͻ down to their final location.

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What is the resulting vector that represents the new location of all the points? Explain how you got your answer. You can see that the total horizontal shift is ͳͳ units .ۄͳͳǡͳۃ are ܍ à The components of the new vector right and the total vertical shift is ͳ unit up.

Did you need to grid lines to calculate the components of the overall shift caused by the two earthquakes? Explain why or why not. à No. We can simply add the horizontal components together followed by the vertical components to find the resulting components of the final vector. Next, introduce the idea of vector addition. Students can add this information to their notes.

Vector addition is defined by the rule: .ۄ൅ܿǡܾ൅݀ܽۃൌܟ൅ܞ then ,ۄ݀ ǡܿۃൌܟ and ۄܾ ǡܽۃൌܞ If

The rule stated above shows that two vectors are added simply by adding their horizontal and vertical components. Explain the definition of vector addition geometrically using transformations and by modeling, in general, that the addition of two vectors has the same effect as two horizontal and two vertical translations of a point or other object in the coordinate plane. The diagram shown below illustrates this idea. This method of adding vectors can be referred to as end-to-end. The parallelogram rule for adding vectors is presented in Lesson 19.

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Exercises 4–7 (5 minutes) Students should work these exercises with their small group. As students are working, circulate around the room to check their progress. Lead a short debriefing by having one or two students share their answers with the group. Students should realize that, as with matrices and complex numbers, properties of arithmetic such as the associative property can be extended to vector operations. The next lesson makes these connections even more explicit. College- level mathematics courses use matrices frequently to represent higher-dimension vectors because they make calculations simpler and more efficient.

Exercises 4–10

ۄെ૞ǡ ૛ۃൌܜ and ۄ૜ǡ ૠۃൌܞ Given that .4 a. What is ܞ൅ܜ?

ۄૢ െ૛ǡۃ ൌ ۄ૜൅ሺെ૞ሻǡૠ൅૛ۃൌܜ൅ܞ

b. Draw a diagram that represents this addition and shows the resulting sum of the two vectors.

c. What is ܜ൅ܞ?

ۄૢ െ૛ǡۃൌܞ൅ܜ

d. Draw a diagram that represents this addition and shows the resulting sum of the two vectors.

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5. Explain why vector addition is commutative.

Since we are combining two horizontal and two vertical translations when we add vectors, the end result will be the same regardless of the order in which we apply the translations. Thus, when we add two vectors, it doesn’t matter which comes first and which comes second. Using the rule we were given, we can see that the components represent real numbers, and thus, the commutative property should apply to each component of the resulting sum vector.

.ۄെ૞ǡ ૛ۃൌܜ and ۄ૜ǡ ૠۃൌܞ Given .6 a. Show numerically that ԡܞԡ ൅ ԡܜԡ ് ԡܞ൅ܜԡ.

ԡܞԡ ൌ ξ૜૛ ൅ૠ૛ ൌ ξ૞ૡ and ԡܜԡ ൌ ඥሺെ૞ሻ૛ ൅૛૛ ൌ ξ૛ૢ

ԡ ൌ ඥሺെ૛ሻ૛ ൅ሺૢሻ૛ ൌ ξૡ૞ܜ൅ܞand ԡ ۄૢ െ૛ǡۃൌܜ൅ܞ

ξ૞ૡ ൅ ξ૛ૢ ് ξૡ૞. This can be confirmed quickly using approximations for each square root.

b. Provide a geometric argument to explain, in general, why the sum of the magnitudes of two vectors is not equal to the magnitude of the sum of the vectors.

When added end-to-end, two vectors and the resulting sum vector lie on the sides of a triangle. Since the sum of any two sides of a triangle must be longer than the third side, and the magnitude of the vectors would correspond to the lengths of the sides of the triangle, this statement cannot be true.

c. Can you think of an example of when the statement would be true? Justify your reasoning.

This statement would be true if one of the vectors had a magnitude of ૙. The sum of the vectors would be equal to the original nonzero vector, so they would have the same magnitude.

.called the zero vector? Describe its geometric effect when added to another vector ۄ૙ǡ ૙ۃൌܗ Why is the vector .7

The magnitude of this vector is ܗ, and its components are ૙. It maps the pre-image vector onto itself and essentially has no translational effect on the original vector. It has the same effect as adding the real-number ૙ to any other real number.

Exercises 8–10 (8 minutes) These problems introduce the scalar multiplication of a vector. Students have already examined the effect of scalar multiplication with complex numbers and matrices. They should be familiar with the idea of dilation from their work in Grade 8 and Geometry. We want them to see that if we multiply a vector ܞ by a scalar ܿ, it produces a new vector ܿܞ, which is a dilation of the vector ܞ by a scale factor ܿ. This effect maintains the direction of the vector and multiplies its magnitude by a factor of ܿ. Students may need the terminology initial point clarified before beginning. Explain that correspond to the point in the Cartesian plane ۄ ǡۃ when the initial point is ሺͲǡͲሻ then the components of the vector ሺܽǡ ܾሻ. For this reason, we often draw arrows that represent vectors from the origin.

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8. Given the vectors shown below

ۄ૜ǡ ૟ۃൌܞ ۄǡ ૚ૡૢۃൌܝ ۄെ૜ǡ െ૟ۃൌܟ ۄ૚ǡ ૛ۃൌܛ ۄെ૚Ǥ ૞ǡ െ૜ۃൌܜ ۄ૟ǡ ૚૛ۃൌܚ

a. Draw each vector with its initial point located at ሺ૙ǡ ૙ሻ. The vector ܞ is already shown. How are all of these vectors related?

All of these vectors lie on the same line that passes through the origin. If you dilate with center ሺ૙ǡ ૙ሻ, then each vector is a dilation of every other vector in the list.

b. Which vector is ૛ܞ? Explain how you know.

The vector ૛ܞ is twice as long as ࢜ and points in the same direction. The components would be doubled so ࢘ൌ૛ܞ.

Introduce scalar multiplication at this point. Have students record the rule in their notes.

Vector multiplication by a scalar is defined by the rule: .ۄܾܿ ǡܽܿۃ ൌ ܞܿ and ܿ is a real number, then ۄܾ ǡܽۃൌܞ If

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If time permits, ask early finishers to show that ԡܿܞԡ ൌܿξܽଶ ൅ܾଶ by showing the following: then by the definition of magnitude of a vector, the properties of real numbers and the ,ۄܾܿ ǡܽܿۃ ൌ ܞܿ If properties of radicals,

ԡܿܞԡ ൌ ඥሺܿܽሻଶ ൅ ሺܾܿሻଶ ൌ ඥܿଶሺܽଶ ൅ܾଶሻ ൌܿඥܽଶ ൅ܾଶ

.and explain your reasoning ۄ૜ǡ ૟ۃൌܞ c. Describe the remaining vectors as a scalar multiple of

ܝൌ૜ܞ ܟൌെܞ ૚ ܛൌ ܞ ૜ ૚ ܜൌെ ܞ ૛

.Explain ?ܞ a scalar multiple of ۄ૜ξ૛ǡ૟ξ૛ۃൌܘ d. Is the vector

Yes; ࢖ൌξ૛ܞ. You can see that if the initial point were located at ሺ૙ǡ ૙ሻ, then the vector ࢖ would also lie on the line through the origin that contains the other vectors in this exercise.

?ۄ૜ǡ ૟ۃൌܞ Which vector from Exercise 8 would it make sense to call the opposite of .9

You could call ܟ the opposite of ܞ because ܟ has the same length as ܞ, and it has the opposite direction.

.to support your reasoning ۄ૟ǡ ૜ۃൌܝ and ۄ૞ǡ ૠۃൌܞ Describe a rule that defines vector subtraction. Use the vectors .10

Since adding two real numbers is adding the opposite of the second number to the first number, it makes sense that vector subtraction would work in a similar way. To subtract two vectors, you add the opposite of the second vector or, more simply, just subtract the components.

We can create the opposite of ܝ by multiplying by the scalar െ૚.

ۄെ૚ǡ ૝ۃ ൌ ۄ૞൅ሺെ૟ሻǡૠ൅ሺെ૜ሻۃሻൌܝ൅ሺെܞൌܝെܞ Or, simply subtracting the components gives

ۄെ૚ǡ ૝ۃ ൌ ۄ૞െ૟ǡૠെ૜ۃൌܝെܞ

If time permits, you may discuss the geometric effect of subtraction as adding the opposite of the second vector when placing the vectors end-to-end.

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Closing (2 minutes) Give students one minute to brainstorm the top three things they learned about vectors in this lesson. Have them share briefly with a partner, and then ask for a few volunteers to share with the entire class. Use the Lesson Summary below to clarify any misunderstandings that may arise when students report out.

Lesson Summary

A vector can be used to describe a translation of an object. It has a magnitude and a direction based on its can represent a translation of ࢇ units horizontally and ࢈ ۄࢇǡ ࢈ۃൌܞ horizontal and vertical components. A vector units vertically with magnitude given by ԡܞԡ ൌ ξࢇ૛ ൅࢈૛.

To add two vectors, add their respective horizontal and vertical components. To subtract two vectors, subtract their respective horizontal and vertical components. Multiplication of a vector by a scalar multiplies the horizontal and vertical components of the vector by the value of the scalar.

Exit Ticket (3 minutes)

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Name Date

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Exit Ticket

is represented by the arrow shown below. How are the vectors the same? How ܝ and the vector ,ۄǡͶ͵ۃൌܞ Vector .1 are they different?

.Write each vector in component form and draw an arrow to represent the vector .ۄʹǡ െ͵ۃൌܞ and ۄͳǡͷۃൌܝ Let .2 a. ܝ൅ܞ

b. ܝെܞ

c. ʹܝ ൅ ͵ܞ

?ܞ൅ܝ as in Problem 2, part (a), what is the magnitude of ۄʹǡ െ͵ۃൌܞ and ۄͳǡͷۃൌܝ For .3

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Exit Ticket Sample Solutions

is represented by the arrow shown below. How are the vectors the same? How ܝ and the vector ۄ૜ǡ ૝ۃൌܞ Vector .1 are they different?

Both ܝ and ܞ have the same length and direction, so they are different representations of the same vector. They both represent a translation of ૜ units right and ૝ units up.

.Write each vector in component form and draw an arrow to represent the vector .ۄ૜ǡ െ૛ۃൌܞ and ۄ૚ǡ ૞ۃൌܝ Let .2 a. ܝ൅ܞ ۄ૝ǡ ૜ۃൌܞ൅ܝ

b. ܝെܞ ۄെ૛ǡ ૠۃ ሻ ൌܞ൅ሺെܝൌܞെܝ

c. ૛ܝ ൅ ૜ܞ ۄ૚૚ǡ ૝ۃ ൌ ܞ൅ ૜ ܝ૛

? ܞas in Problem 2, part (a), what is the magnitude of ൅ ۄ૜ǡ െ૛ۃൌܞ and ۄ૚ǡ ૞ۃൌܝ For .3 ԡ ൌ૞ۄ૝ǡ ૜ۃԡ ൌ ԡܞ൅ܝԡ

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Problem Set Sample Solutions

1. Sasha says that a vector has a direction component in it; therefore, we cannot add two vectors or subtract one from the other. His argument is that we cannot add “east” to “north” nor subtract “east” from “north,” for instance. Therefore, he claims, we cannot add or subtract vectors. a. Is he correct? Explain your reasons.

No, Sasha is not correct. Although a vector has a magnitude and direction, and it is numerically suited to do translation of an object, it has horizontal and vertical components indicating how many units for translation. Therefore, we can add and subtract vectors.

b. What would you do if you need to add two vectors, ܝ and ܞ, or subtract vector ܞ from vector ܝ arithmetically?

ۄ૛ܞ૚ǡܞۃൌ ܞǡۄ૛ܝ૚ǡܝۃൌܝ ,For addition, we add the same corresponding vector components. For example

.ۄ૛ܞ૛ ൅ܝ૚ǡܞ૚ ൅ܝۃൌܞ൅ܝ

.ۄ૛ܞ૛ െܝ૚ǡܞ૚ െܝۃ ൌ ۄ૛ሻܞ૛ ൅ ሺെܝ૚ሻǡܞ૚ ൅ ሺെܝۃ ሻ ൌܞ൅ሺെܝൌܞെܝ ,For subtraction

.write each vector in component form, graph it, and explain the geometric effect ,ۄെ૝ǡ ૛ۃൌܞ and ۄ૜ǡ ૚ۃൌܝ Given .2 a. ૜ܝ ۄǡ ૜ૢۃ ൌ ܝ૜ The vector is dilated by a factor of ૜ and the direction stays the same.

૚ b. ܞ ૛ ૚ ۄെ૛ǡ ૚ۃൌܞ ૛ ૚ The vector is dilated by a factor of , and the direction stays the same. ૛

c. Ȃ૛ܝ ۄെ૟ǡ െ૛ۃ ൌ ܝ૛ The vector is dilated by a factor of ૛, and the direction is reversed

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d. െܞ .ۄ૝ǡ െ૛ۃ ൌ ܞെ The length of the vector is unchanged, and the direction is reversed.

e. ܝ൅ܞ ۄെ૚ǡ ૜ۃൌܞ൅ܝ we move ૜ units to the right ,ܞ from the tip of vector ,ۄെ૝ǡ ૛ۃൌܞ onto vector ۄ૜ǡ ૚ۃൌܝ When adding vector .ۄെ૚ǡ ૜ۃ and ૚ unit upward, and the resultant vector is

f. ૛ܝ ൅ ૜ܞ ۄെ૟ǡ ૡۃ ൌ ܞ൅ ૜ ܝ૛ we move ૟ units to the ,ܞfrom the tip of vector ૜ ,ۄെ૚૛ǡ ૟ۃ ൌ ܞonto vector ૜ ۄ૟ǡ ૛ۃ ൌ ܝWhen adding vector ૛ .ۄെ૟ǡ ૡۃ right and ૛ units upward, and the resultant vector is

g. ૝ܝ െ ૜ܞ ۄ૛૝ǡ െ૛ۃ ሻ ൌܞ൅ ሺെ૜ ܝൌ ૝ ܞെ ૜ ܝ૝ we move ૚૛ units ,ܞfrom the tip of vector െ૜ ,ۄ૚૛ǡ െ૟ۃ ൌ ܞonto vector െ૜ ۄ૚૛ǡ ૝ۃ ൌ ܝWhen adding vector ૝ .ۄ૛૝ǡ െ૛ۃ to the right and ૝ units upward. The resultant vector is

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૚ ૚ h. ܝെ ܞ ૛ ૜ ૚ ૚ ૚ ૚ ૚ૠ ૚ ۄ ǡെ ۃ ൰ ൌܞ ൅൬െܝ ൌܞ െܝ ૛ ૜ ૛ ૜ ૟ ૟ ૚ ૜ ૚ ૚ ૝ ૚ ૜ we move units ,ܞ from the tip of vector െ ,ۄെ ǡ૚ۃൌܞ onto vector െ ۄ ǡ ۃൌܝ When adding vector ૛ ૛ ૛ ૜ ૜ ૜ ૛ ૚ ૚ૠ ૚ .ۄ ǡെ ۃ to the right and unit upward, and the resultant vector is ૛ ૟ ૟

.find the following ,ۄെ૝ǡ ૛ۃൌܞ and ۄ૜ǡ ૚ۃൌܝ Given .3 a. ԡܝԡ.

ξ૚૙

b. ԡܞԡ.

૛ξ૞

c. ԡ૛ܝԡ and ૛ԡܝԡ.

ԡ૛ԡ ൌ ξ૝૙ ൌ૛ξ૚૙.૛ԡԡ ൌ૛ξ૚૙.

૚ ૚ d. ቛ ܞቛ and ԡܞԡ ૛ ૛ ૚ ૚ ૚ ૚ ԡ ൌ ξ૞ܞฯ ൌ ξ૞. ԡܞ ฯ ,ۄെ૛ǡ ૚ۃൌܞ ૛ ૛ ૛ ૛

e. Is ԡܝ൅ܝԡ equal to ԡܝԡ ൅ ԡܝԡ? Explain how you know.

Yes. We have

ԡܝ൅ܝԡ ൌ ԡ૛ܝԡ ൌ૛ξ૚૙ and

ԡܝԡ ൅ ԡܝԡ ൌ૛ԡܝԡ ൌ૛ξ૚૙.

f. Is ԡܝ൅ܞԡ equal to ԡܝԡ ൅ ԡܞԡ? Explain how you know.

No. We have

.ԡ࢛൅࢜ԡ ൌ ξ૚૙ and ԡ࢛ԡ ൅ ԡ࢜ԡ ൌ ξ૚૙ ൅૛ξ૞ۄെ૚ǡ ૜ۃ ൅ൌ

g. Is ԡܝെܞԡ equal to ԡܝԡ െ ԡܞԡ? Explain how you know.

.We have ԡ࢛െ࢜ԡ ൌ ξ૞૙ ൌ૞ξ૛, but ԡ࢛ԡ െ ԡ࢜ԡ ൌ ξ૚૙ െ૛ξ૞ .ۄૠǡ െ૚ۃ No. െൌ

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.ሻܟ൅ܞ൅ሺܝൌܟሻ ൅ܞ൅ܝshow that ሺ ,ۄെ૝ǡ ૟ۃൌܟ and ,ۄ૜ǡ െ૝ۃൌܞ ,ۄ૚ǡ ૛ۃൌܝ Given .4

ۄ૙ǡ ૝ۃ ൌ ۄെ૝ǡ ૟ۃ ൅ ۄ૝ǡ െ૛ۃൌܟሻ ൅ܞ൅ܝሺ ۄ૙ǡ ૝ۃ ൌ ۄെ૚ǡ ૛ۃ ൅ ۄ૚ǡ ૛ۃ ሻ ൌܟ൅ܞ൅ሺܝ

5. Tyiesha says that if the magnitude of a vector ܝ is zero, then ܝ has to be a zero vector. Is she correct? Explain how you know.

૛ ૛ ૛ ૛ ૛ ૛ ૛ሻ areܝ૚ሻ and ሺܝ૛ሻ ൌ૙. Then ሺܝ૚ሻ ൅ ሺܝ૛ሻ ൌ૙, so ሺܝ૚ሻ ൅ ሺܝԡ ൌ ඥሺܝand ԡ ,ۄ૛ܝ૚ǡܝۃൌܝ Yes. Suppose that non-negative numbers, and if two non-negative numbers sum to zero, then both numbers must be zero. Then

ܝ૚ ൌ૙ and ܝ૛ ൌ૙, which proves ࢛ is a zero vector.

6. Sergei experienced one of the biggest earthquakes when visiting Taiwan in 1999. He noticed that his refrigerator moved on the wooden floor and made marks on it. By measuring the marks he was able to trace how the refrigerator moved. The first move was northeast with a distance of ૛૙܋ܕ. The second move was northwest with a distance of ૚૙܋ܕ. The final move was northeast with a distance of ૞܋ܕ. Find the vectors that would re-create the refrigerator’s movement on the floor and find the distance that the refrigerator moved from its original spot to its resting place. Draw a diagram of these vectors.

.ۄ૚૙ξ૛ǡ૚૙ξ૛ۃ ૚ ൌܞ The first vector

.ۄെ૞ξ૛ǡ૞ξ૛ۃ ૛ ൌܞ The second vector

૞ඥ૛ ૞ඥ૛ ۄ ǡ ۃ ൌ ܞ The third vector ૜ ૛ ૛ .

The resultant vector is ૚૞ξ૛ ૜૞ξ૛ . ۄ ǡ ۃ ൌ ܞ൅ ܞ൅ ܞൌܞ ૚ ૛ ૜ ૛ ૛

૛ ૛ ૚૞ξ૛ ૜૞ξ૛ ԡܞԡ ൌ ඨቆ ቇ ൅ ቆ ቇ ൌ૞ξ૛ૢ ૛ ૛

The magnitude is ૞ξ૛ૢ܋ܕ.

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A STORY OF FUNCTIONS Lesson 18 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 18: Vectors and Translation Maps

Student Outcomes Students use vectors to define a translation map and translate geometric figures in Թଶ and Թଷ. Students represent and perform calculations with three-dimensional vectors and understand operations in three dimensions that are analogous to operations in two dimensions.

Lesson Notes This lesson builds on the use of vectors to represent shifts by using vectors to define translation maps studied in prior lessons. The connection between matrices and vectors should become more apparent in this lesson. A notation for vectors that recalls matrix notation is used. This lesson focuses on extending vector addition, subtraction, and scalar multiplication to Թଷ. This lesson lends itself to working in a program capable of creating three-dimensional graphs, such as GeoGebra (v5.0 or later). Consider modeling these lessons on one computer or, if access to a lab is available, have students work on their own computers in the latter portion of this lesson.

Classwork Opening Exercise (5 minutes) Activate student knowledge about the previous day’s lesson with this short Opening Exercise. Students should work independently at first and then check their solutions with a partner. Students could draw their vectors using GeoGebra. If using GeoGebra, type vector ሾሺ܉,܊)ሿ in the input box where ܉ and ܊ are the horizontal and vertical components of the ͳ .ൌቀ ቁܞ appears as ۄʹͳǡۃൌܞ vector. In GeoGebra, the vector ʹ

Opening Exercise

Write each vector described below in component form and find its magnitude. Draw an arrow originating from ሺ૙ǡ ૙ሻ to represent each vector’s magnitude and direction.

a. Translate ૜ units right and ૝ units down.

ԡ ൌ૞ܝԡ ,ۄ૜ǡ െ૝ۃൌܝ

b. Translate ૟ units left.

ԡ ൌ૟ܞԡ ,ۄെ૟ǡ ૙ۃൌܞ

c. Translate ૛ units left and ૛ units up.

ԡ ൌ૛ξ૛ܟǡ ԡۄെ૛ǡ ૛ۃൌܟ

d. Translate ૞ units right and ૠ units up.

ԡ ൌ ξ૛૞ ൅ ૝ૢ ൌ ξૠ૝ܜǡ ԡۄ૞ǡ ૠۃൌܜ

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Discussion (5 minutes) Lead a short discussion to model how to write a vector as a translation map. Introduce Scaffolding: the column notation for vectors, and ask students to recall when they have seen something similar in previous lessons. They should recall it from Module 1 and from the GeoGebra or other beginning of this module. graphing software can be a powerful tool for We can use a translation map to represent a vector. In fact, we can make this scaffolding this lesson, the definition of a vector: A vector is a translation map. particularly for students

that who are struggling to ܞܶ ,is identified with the translation map ۄǡͳʹۃൌܞ For example, the vector ݔ ʹ ݔ ʹ ݔ൅ʹ quickly sketch graphs of maps a point ࢞ൌቂ ቃ to the point ࢞൅ቂ ቃൌቂ ቃ൅ቂ ቃൌ൤ ൨; thus .ݕ ͳ ݕ ͳ ݕ൅ͳ circles and lines by hand ݔ ݔ൅ʹ Post the formulas for a line ܶ ቀቂ ቃቁ ൌ ൤ ൨ . ݕ ݕ൅ͳ and a circle on the board ܞ Why would it make sense to think of a vector as a translation map? prior to this lesson, and have students pair share à Because vectors accomplish the same thing as a translation map. An with a partner what they object is translated by the components of the vector in the horizontal and recall about these vertical direction. The object’s size and shape would remain unchanged equations and the related under this type of transformation. graphs. Explain how writing vectors in a column makes it easier to think of them as ܽ A line with slope െ and translation maps. ܾ ܿ ݕ-intercept ቀͲǡ ቁ is given à We are using the same kind of notation we used when working with ܾ .ܿ translation maps and linear transformations in the previous module and by ܽݔ ൅ ܾݕ ൌ earlier in this module. A circle with radius ݎ and Give students time to summarize this information in their notes or on the student center ሺ݄ǡ ݇ሻ is given by .ଶݎmaterials. ሺݔെ݄ሻଶ ൅ ሺݕെ݇ሻଶ ൌ For the following equations, have students Exercises 1–3 (5 minutes) sketch the graph and state Provide students with the opportunity to work in small groups on the next exercises. the key features. ʹCheck for precision and accuracy in their work with the translation map notation applied ʹݔ െ ͸ݕ ൌ ͳ to vectors. Make sure they are thinking carefully about the transformations of the circle ݔ൅ʹݕൌെͷ under the mapping in Exercises 2 and 3, and check to make sure they are creating the ሺݔെͳሻଶ ൅ ሺݕെ͵ሻଶ ൌͻ equations of the image figures correctly.

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Exercises 1–3

1. Write a translation map defined by each vector from the opening.

:and its associated translation map ۄെ૛ǡ ૞ۃൌܞ Consider the vector ࢞ ࢞െ૛ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ܞ ࢟ ࢟൅૞

Vector in Component Form Translation Map, ࢀܞ ࢞ ࢞൅૜ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ۄ૜ǡ െ૝ۃൌܝ ܝ ࢟ ࢟െ૝ ࢞ ࢞െ૟ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ۄെ૟ǡ ૙ۃൌܞ ܞ ࢟ ࢟ ࢞ ࢞െ૛ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ۄെ૛ǡ ૛ۃൌܟ ܟ ࢟ ࢟൅૛ ࢞ ࢞൅૞ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ۄ૞ǡ ૠۃൌܜ ܜ ࢟ ࢟൅ૠ

૛ ૛ 2. Suppose we apply the translation map ࢀܞ to each point on the circle ሺ࢞൅૝ሻ ൅ ሺ࢟െ૜ሻ ൌ૛૞. a. What is the radius and center of the original circle?

The radius is ૞ and the center is ሺെ૝ǡ ૜ሻ.

b. Show that the image points satisfy the equation of another circle. ࢞ ࢞ᇱ Suppose that ࢀ ቀቂ ቃቁ ൌ ൤ ൨. Then ࢞ᇱ ൌ࢞െ૛ and ࢟ᇱ ൌ࢟൅૞, so ࢞ൌ࢞ᇱ ൅૛ and ࢟ൌ࢟ᇱ െ૞. ܞ ࢟ ࢟Ԣ Since ሺ࢞൅૝ሻ૛ ൅ ሺ࢟െ૜ሻ૛ ൌ૛૞, we have

ሺ࢞൅૝ሻ૛ ൅ ሺ࢟െ૜ሻ૛ ൌ૛૞ ૛ ૛ ൫ሺ࢞ᇱ ൅૛ሻ ൅૝൯ ൅ ൫ሺ࢟ᇱ െ૞ሻ െ૜൯ ൌ૛૞ ሺ࢞ᇱ ൅૟ሻ૛ ൅ ሺ࢟ᇱ െૡሻ૛ ൌ૛૞ ࢞ᇱ So, the image points ൤ ൨ lie on a circle. ࢟Ԣ

c. What are the center and radius of this image circle?

The radius of ૞ remains unchanged. The new center is ሺെ૟ǡ ૡሻ.

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3. Suppose we apply the translation map ࢀܞ to each point on the line ૛࢞ െ ૜࢟ ൌ ૚૙. a. What are the slope and ࢟-intercept of the original line?

૛ ૚૙ The slope is and the ࢟-intercept is ቀ૙ǡെ ቁ. ૜ ૜

b. Show that the image points satisfy the equation of another line. ࢞ ࢞ᇱ Suppose that ࢀ ቀቂ ቃቁ ൌ ൤ ൨. Then ࢞ᇱ ൌ࢞െ૛and ࢟ᇱ ൌ࢟൅૞, so ࢞ൌ࢞ᇱ ൅૛and ࢟ൌ࢟ᇱ െ૞. ܞ ࢟ ࢟Ԣ Since ૛࢞ െ ૜࢟ ൌ ૚૙, we have

૛ሺ࢞ᇱ ൅૛ሻ െ૜ሺ࢟ᇱ െ૞ሻ ൌ૚૙ ૛࢞ᇱ ൅૝െ૜࢟ᇱ ൅૚૞ൌ૚૙ ૛࢞ᇱ െ૜࢟ᇱ ൌെૢ

c. What are the slope and ࢟-intercept of this image line?

૛ The slope is and the ࢟-intercept is ሺ૙ǡ ૜ሻ. ૜

Discussion (5 minutes): Vectors in Three Dimensions A vector in two dimensions can be used to define a translation map that translates objects in the coordinate plane by the horizontal and vertical components of the vector. What types of objects could we translate by a vector in two dimensions? à Any geometric figure could be translated by a vector. This idea is easily extended to Թଷ, the Cartesian coordinate system in three dimensions. For example, the vector ܞൌ .is a translation in space ͳ unit in the ݔ direction, ͵ units in the ݕ direction and ͷ units in the ݖ direction ۄͳǡ͵ǡͷۃ

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?ۄͳǡ͵ǡͷۃൌܞ What would be the associated translation map for the vector ݔ ݔ൅ͳ .ቆቈݕ቉ቇ ൌ ൥ݕ൅͵൩ ܞà The associated translation map would be ࢀ ݖ ݖ൅ͷ What types of objects could we represent in three dimensions? à We could represent points, planes, spheres, cubes, cylinders, pyramids, or other solid figures in three dimensions. Two-dimensional figures can also be represented in three dimensions. Give students an opportunity to summarize this information in their notes or on the student pages before starting the exercises that follow.

Example 1 (8 minutes): Vectors and Translation Maps in Թ૜

If technology is available, model this example using GeoGebra by selecting 3-D Graphs Scaffolding: from the view menu. Type each figure into the input bar and then type in vector[(1,3,5)]. Help students graph in 3-D by Use the translate by vector feature in the translate menu. Students will be able to see the using 3-D or isometric graph equations updating and the image graph displayed along with the pre-image when using paper. 3-D graph paper is this feature. If technology is not available, use 3-dimensional graph paper or isometric included in Lesson 5. graph paper. 3-D graph paper is included in Lesson 5 or can be downloaded for free at http://www.waterproofpaper.com/graph-paper/isometric-graphing-paper.pdf. Take time to help students make sense of the graphs they are seeing of each object. Point out the similarities between points, planes, and spheres in three dimensions and points, lines, and circles in two dimensions. How do the coordinates of a point in two dimensions compare to coordinates in three dimensions? .à You just add another coordinate for the distance from the origin in the ݖ direction How does the equation of a line in two dimensions compare to the equation of a plane in three dimensions? à You just add another linear term using the variable ݖ. The intercepts are similar with two coordinates being Ͳ and the third coordinate giving the point where the plane crosses the given axis. How does the equation of a circle in two dimensions compare to the equation of a sphere in three dimensions? à You just add another quadratic term using the variable ݖ. The radius is the cube root of the constant and the center is the same except three coordinates.

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Example 1: Vectors and Translation Maps in Թ૜

૜ to the following objects in Թ . A sketch of the ܞby applying the translation map ࢀ ۄ૚ǡ ૜ǡ ૞ۃൌܞ Translate by the vector original object and the vector is shown. Sketch the image. ࢞ ࢞൅૚ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟൅૜൩ ࢠ ࢠ൅૞ a. The point ࡭ሺ૛ǡ െ૛ǡ ૝ሻ

The new point will be ሺ૛൅૚ǡെ૛൅૜ǡ૝൅૞ሻ or ሺ૜ǡ ૚ǡ ૢሻ.

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b. The plane ૛࢞ ൅ ૜࢟ െ ࢠ ൌ ૙

The new plane will be ૛ሺ࢞െ૚ሻ ൅૜ሺ࢟െ૜ሻ െ ሺࢠെ૞ሻ ൌ૙, which is equivalent to ૛࢞ ൅ ૜࢟ െ ࢠ ൌ ૟.

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c. The sphere ሺ࢞െ૚ሻ૛ ൅ ሺ࢟െ૜ሻ૛ ൅ࢠ૛ ൌૢ

૛ ૛ The new sphere will be ൫ሺ࢞െ૚ሻ െ૚൯ ൅ ൫ሺ࢟െ૜ሻ െ૜൯ ൅ ሺࢠെ૞ሻ૛ ൌૢ, which is equivalent to ሺ࢞െ૛ሻ૛ ൅ ሺ࢟െ૟ሻ૛ ൅ ሺࢠെ૞ሻ૛ ൌૢ.

After sharing the solutions to Example 1, conclude with a brief discussion of the results. When we translated a line by a vector, why was the image parallel to the original line? à All points in the plane were moved by the same amount in the same direction due to the translation map. Why does it make sense that, when we translate a plane by a vector, the image plane is parallel to the pre- image? à If every point moves the same distance in the same direction, then the planes will have the same orientation in space but a different location.

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When translating any geometric figure by a vector, will the image be congruent to the pre-image? Explain how we know. à Since translations are rigid transformations, the image will be congruent to the pre-image.

Exercise 4 (2 minutes) This short exercise should reveal whether or not students are able to extend their thinking about vectors to three dimensions.

Exercise 4

4. Given the sphere ሺ࢞൅૜ሻ૛ ൅ ሺ࢟െ૚ሻ૛ ൅ ሺࢠെ૜ሻ૛ ൌ૚૙. a. What are its center and radius?

The center is ሺെ૜ǡ ૚ǡ ૜ሻ, and the radius is ξ૚૙.

b. Write a vector and its associated translation map that would take this sphere to its image centered at the origin.

We need to translate the center from the point ሺെ૜ǡ ૚ǡ ૜ሻ to the point ሺ૙ǡ ૙ǡ ૙ሻ. This represents a translation of ૜ units in the ࢞ direction, െ૚ unit in the ࢟ direction, and െ૜ units in the ࢠ direction.

and the translation map would be ,ۄ૜ǡ െ૚ǡ െ૜ۃൌܞ The vector is ࢞ ࢞൅૜ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟െ૚൩. ࢠ ࢠെ૜

Example 2 (5 minutes): What is the Magnitude of a Vector in Թ૜?

Provide less scaffolding by asking students to determine an expression for the magnitude of ࢜ shown in the diagram below by simply showing the diagram on the board or document camera without the additional questions. Notice that ܞ is a vector, but that ݔ, ݕ, ݖ, and ݓ are real numbers that represent lengths. For students who need a more concrete approach, label the terminal point of the vector ܞ with an actual set of coordinates such as ሺʹǡ ͺǡ͸ሻ and then model how to find the magnitude of the vector before working through the problem in general. .Use the Pythagorean theorem to write an equation relating ݔ, ݕ, ݓ à ݔଶ ൅ݕଶ ൌݓଶ .ԡܞUse the Pythagorean theorem to write an equation relating ݓ, ݖ, ԡ ԡଶܞà ݓଶ ൅ݖଶ ൌ ԡ .ԡ, ݔ, ݕ, and ݖܞNow write an equation relating ԡ à ݔଶ ൅ݕଶ ൌݓଶ ԡଶ െݖଶܞà ݓଶ ൌ ԡ ԡଶ െݖଶܞà ݔଶ ൅ݕଶ ൌ ԡ ԡଶܞà ݔଶ ൅ݕଶ ൅ݖଶ ൌ ԡ

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Example 2: What is the Magnitude of a Vector in Թ૜?

a. Find a general formula for ԡܞԡ૛.

ԡܞԡ૛ ൌ࢞૛ ൅࢟૛ ൅ࢠ૛

b. Solve this equation for ԡܞԡ to find the magnitude of the vector.

ԡܞԡ ൌ ඥ࢞૛ ൅࢟૛ ൅ࢠ૛

Exercises 5–8 (5 minutes) In Exercise 5, students practice finding the magnitude of a vector in three dimensions. Direct students to work independently on Exercise 5 first, and then have them work with a partner to check their results.

Exercises 5–8

.Show work to support your answer ?ۄ૜ǡ െ૝ǡ ૝ۃൌܝ or ۄ૙ǡ ૞ǡ െ૝ۃൌܞ ,Which vector has greater magnitude .5

ԡܞԡ ൌ ඥ૙૛ ൅૞૛ ൅ ሺെ૝ሻ૛ ൌ ξ૝૚

ԡܝԡ ൌ ඥ૜૛ ൅ ሺെ૝ሻ૛ ൅૝૛ ൌ ξ૝૚ These vectors have equal magnitude.

6. Explain why vectors can have equal magnitude but not be the same vector.

A vector has both a magnitude and a direction. If you graphed the vectors from Exercise 5, you can see they point in different directions so they cannot be the same even though they have equal magnitude.

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Give students time to consider how the rules and representations learned in the previous lesson extend to three dimensions. If needed, scaffold this exercise by providing specific examples of vectors for students to work with (e.g., those shown in the table below) before asking them to write general rules.

Vectors in Թ૛ Vectors in Թ૜

ۄǡ͵ǡͶʹۃ ۄ͵ǡʹۃ Component Form

ʹ ʹ Column Form ቂ ቃ ൥͵൩ ͵ Ͷ

Magnitude ԡܞԡ ൌ ඥʹଶ ൅͵ଶ ൌ ξͳ͵ ԡܞԡ ൌ ඥʹଶ ൅͵ଶ ൅Ͷଶ ൌ ξʹͻ

,ۄǡ െͶǡͳʹۃൌܝ and ۄǡ͵ǡͶʹۃൌܞ If ,ۄǡ െͶʹۃൌܝ and ۄ͵ǡʹۃൌܞ If Addition Then Then ۄͶǡ െͳǡͷۃ ൌ ۄ൅ʹǡ͵൅ሺെͶሻǡͶ൅ͳʹۃൌܝ൅ܞ ۄͶǡ െͳۃ ൌ ۄ൅ʹǡ͵൅ሺെͶሻʹۃൌܝ൅ܞ ,ۄǡ െͶǡͳʹۃൌܝ and ۄǡ͵ǡͶʹۃൌܞ If ,ۄǡ െͶʹۃൌܝ and ۄ͵ǡʹۃൌܞ If Subtraction Then ۄͲǡ͹ۃ ൌ ۄെʹǡ͵െሺെͶሻʹۃൌܝ൅ܞ Then ۄ͵Ͳǡ͹ǡۃ ൌ ۄെʹǡ͵െሺെͶሻǡͶെͳʹۃൌܝ൅ܞ then ۄǡ͵ǡͶʹۃൌܞ then If ۄ͵ǡʹۃൌܞ If Scalar Multiplication ۄͶǡ͸ǡͺۃ ൌ ۄή ʹǡʹ ή ͵ǡʹ ή Ͷ ʹۃ ൌ ܞʹ ۄͶǡ͸ۃ ൌ ۄ͵ήʹǡʹήʹۃ ൌ ܞʹ

7. Vector arithmetic in Թ૜ is analogous to vector arithmetic in Թ૛. Complete the graphic organizer to illustrate these ideas.

Vectors in Թ૛ Vectors in Թ૜

ۄࢇǡ ࢈ǡ ࢉۃ ۄࢇǡ ࢈ۃ Component Form

ࢇ ࢇ Column Form ቂ ቃ ቈ࢈቉ ࢈ ࢉ

Magnitude ԡܞԡ ൌ ඥࢇ૛ ൅࢈૛ ԡܞԡ ൌ ඥࢇ૛ ൅࢈૛ ൅ࢉ૛

,ۄࢊǡ ࢋǡ ࢌۃൌܝ and ۄࢇǡ ࢈ǡ ࢉۃൌܞ If ,ۄࢉǡ ࢊۃൌܝ and ۄࢇǡ ࢈ۃൌܞ If Addition ۄࢇ൅ࢊǡ࢈൅ࢋǡࢉ൅ࢌۃൌܝ൅ܞ Then ۄࢇ൅ࢉǡ࢈൅ࢊۃൌܝ൅ܞ Then

,ۄࢊǡ ࢋǡ ࢌۃൌܝ and ۄࢇǡ ࢈ǡ ࢉۃൌܞ If ,ۄࢉǡ ࢊۃൌܝ and ۄࢇǡ ࢈ۃൌܞ If Subtraction ۄࢇെࢊǡ࢈െࢋǡࢉെࢌۃൌܝെܞ Then ۄࢇെࢉǡ࢈െࢊۃൌܝെܞ Then

and ࢑ is a real number ۄࢇǡ ࢈ǡ ࢉۃൌܞ and ࢑ is a real number If ۄࢇǡ ࢈ۃൌܞ If Scalar Multiplication ۄ࢑ࢇǡ ࢑࢈ǡ ࢑ࢉۃ ൌ ܞ࢑ ۄ࢑ࢇǡ ࢑࢈ۃ ൌ ܞ࢑

Before starting Exercise 8, make sure students have correct information in the graphic organizer. Show a completed one to the class and, if time permits, have different groups explain how they got their answers for each row. Allow students time to correct any errors they may have made.

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.ۄെ૚ǡ ૞ǡ ૜ۃൌܝ and ۄ૛ǡ ૙ǡ െ૝ۃൌܞ Given .8 a. Calculate the following. i. ܞ൅ܝ ۄ૚ǡ ૞ǡ െ૚ۃ ൌ ۄ૛൅ሺെ૚ሻǡ૙൅૞ǡെ૝൅૜ۃൌܝ൅ܞ

ii. ૛ܞ െ ܝ ۄ૞ǡ െ૞ǡ െ૚૚ۃ ൌ ۄ૛ή૛െሺെ૚ሻǡ૛ή૙െ૞ǡ૛ήሺെ૝ሻ െ૜ۃ ൌ ܝ െ ܞ૛

iii. ԡܝԡ

ԡܝԡ ൌ ඥሺെ૚ሻ૛ ൅૞૛ ൅૜૛ ൌξ૜૞

b. Suppose the point ሺ૚ǡ ૜ǡ ૞ሻ is translated by ܞ and then by ܝ. Determine a vector ܟ that would return the point back to its original location ሺ૚ǡ ૜ǡ ૞ሻ.

The point will be translated from ሺ૚ǡ ૜ǡ ૞ሻ to ሺ૛ǡ ૡǡ ૝ሻ .ۄ૚ǡ ૞ǡ െ૚ۃൌܝ൅ܞ From part (a), we have since ૚൅૚ൌ૛, ૞൅૜ൌૡ, and ૞െ૚ൌ૝. The vector that will return this point to its original location will be the opposite of ܞ൅ܝ.

ۄെ૚ǡ െ૞ǡ ૚ۃ ሻ ൌܝ൅ܞെሺ And this vector will translate ሺ૛ǡ ૡǡ ૝ሻ back to ሺ૚ǡ ૜ǡ ૞ሻ because ૛െ૚ൌ૚, ૡെ૞ൌ૜, and ૝൅૚ൌ૞.

Closing (2 minutes) Give students time to respond to the questions below individually in writing or with a partner. Why can we represent vectors as translation maps? à They mean the same thing geometrically. How are two-dimensional vectors the same as three-dimensional vectors, and how are they different? à Either dimension has a magnitude and direction and represents a translation or shift. They are different because we need one additional component to describe a vector in three dimensions. The graphic organizer in Exercise 7 can serve as a lesson summary of vector arithmetic in three dimensions along with the information about translation maps shown below.

Lesson Summary

A vector ܞ can define a translation map ࢀܞ that takes a point to its image under the translation. Applying the map to the set of all points that make up a geometric figure serves to translate the figure by the vector.

Exit Ticket (3 minutes)

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Name Date

Lesson 18: Vectors and Translation Maps

Exit Ticket

Graph the .ܞܶ find the image of the line ͵ݔ െ ʹݕ ൌ ʹ under the translation map ,ۄെʹǡͳۃൌܞ Given the vector .1 original line and its image, and explain the geometric effect of the map ܶܞ.

find the image of the circle ሺݔെʹሻଶ ൅ ሺݕ൅ͳሻଶ ൌͶ under the translation ,ۄʹെͳǡۃൌܞ Given the vector map ܶܞ. Graph the original circle and its image, and then explain the geometric effect of the map ܶܞ.

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Exit Ticket Sample Solutions

Graph the .ܞfind the image of the line ૜࢞ െ ૛࢟ ൌ ૛ under the translation map ࢀ ,ۄെ૛ǡ ૚ۃൌܞ Given the vector .1 original line and its image, and explain the geometric effect of the map ࢀܞ.

࢞ ࢞െ૛ ࢀ ቀቂ ቃቁ ൌ ൬ ൰ ૜ሺ࢞൅૛ሻ െ૛ሺ࢟െ૚ሻ ൌ૛ ܞ ࢟ ࢟൅૚ ; image of the original line: . ૜࢞ ൅ ૟ െ ૛࢟ ൅ ૛ ൌ ૛ǡ ૜࢞ െ ૛࢟ ൌ െ૟ Every point on the line is shifted ૛ units to the left and ૚ unit upward.

૛ ૛ .ܞfind the image of the circle ሺ࢞െ૛ሻ ൅ ሺ࢟൅૚ሻ ൌ૝ under the translation map ࢀ ,ۄെ૚ǡ ૛ۃൌܞ Given the vector .2 Graph the original circle and its image, and explain the geometric effect of the map ࢀܞ.

࢞ ࢞െ૚ ࢀ ቀቂ ቃቁ ൌ ൬ ൰ ሺ࢞െ૚ሻ૛ ൅ ሺ࢟െ૚ሻ૛ ൌ૝ ܞ ࢟ ࢟൅૛ ; image of original circle: .

Every point on the circle is shifted one unit to the left and two units upward. The new center is ሺ૚ǡ ૚ሻ and the radius ࢘ൌ૛ stays the same.

Problem Set Sample Solutions

࢞ ࢞൅૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ 1. Myishia says that when applying the translation map ܞ ࢟ ࢟െ૛ to a set of points given by an equation relating ࢞ and ࢟, we should replace every ࢞ that is in the equation by ࢞൅૚, and ࢟ by ࢟െ૛. For example, the equation of the parabola ࢟ൌ࢞૛ would become ࢟െ૛ൌሺ࢞൅૚ሻ૛. Is she correct? Explain your answer.

No, she is not correct. What she did translates the points on the parabola ࢟ൌ࢞૛ in the opposite direction—one unit ࢞ ࢞൅૚ ૛ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ to the left and units upward, which is not the geometric effect of ܞ ࢟ ࢟െ૛ . In order to have the correct ࢞ ࢞൅૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢞ᇱ ൌ࢞൅૚ ࢟ᇱ ൌ࢟െ૛ ࢞ൌ࢞ᇱ െ translation based on ܞ ࢟ ࢟െ૛ , we need to set and , which is equivalent to ૚ and ࢟ൌ࢟ǯ൅૛.

This process gives the transformed equation ࢟Ԣ ൅ ૛ ൌ ሺ࢞Ԣ െ ૚ሻ૛, which we write as ࢟൅૛ൌሺ࢞െ૚ሻ૛.

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Graph the original .ܞfind the image of the line ࢞൅࢟ൌ૚ under the translation map ࢀ ,ۄെ૚ǡ ૜ۃൌܞ Given the vector .2 line and its image, and explain the geometric effect of the map ࢀܞ on the line.

࢞ ࢞െ૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ൫࢞െሺെ૚ሻ൯ ൅ ൫࢟െሺ૜ሻ൯ ൌ૚࢞൅૚൅࢟െ૜ൌ૚ ࢞൅࢟ൌ૜ ܞ ࢟ ࢟൅૜ , , , Every point on the line is shifted one unit left and three units upward. The slopes of the lines remain െ૚.

૛ Draw a graph .ܞfind the image of the parabola ࢟െ૚ൌ࢞ under the translation map ࢀ ,ۄ૛ǡ ૚ۃൌܞ Given the vector .3 of the original parabola and its image, and explain the geometric effect of the map ࢀܞ on the parabola. Find the vertex and ࢞-intercepts of the graph of the image.

࢞ ࢞൅૛ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ െ ૛ ൌ ሺ࢞െ૛ሻ૛ ܞ ࢟ ࢟൅૚ , Every point on the parabola is shifted two units to the right and one unit upward.

The vertex is ሺ૛ǡ ૛ሻ, and there are no ࢞-intercepts.

૜ Draw .ܞfind the image of the graph of ࢟൅૚ൌሺ࢞൅૚ሻ under the translation map ࢀ ,ۄ૜ǡ ૛ۃൌܞ Given the vector .4 the original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the ࢞-intercepts of the graph of the image.

࢞ ࢞൅૜ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ െ ૚ ൌ ሺ࢞െ૛ሻ૜ ܞ ࢟ ࢟൅૛ , Every point on the curve is shifted three units to the right and two units upward.

The ࢞-intercept is ૚.

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Draw .ܞfind the image of the graph of ࢟൅૛ൌξ࢞൅૚ under the translation map ࢀ ,ۄ૜ǡ െ૜ۃൌܞ Given the vector .5 the original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the ࢞-intercepts of the graph of the image.

࢞ ࢞൅૜ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ ൅ ૞ ൌ ࢞െ૛ ܞ ࢟ ࢟െ૜ , ξ Every point on the curve is shifted three units to the right and three units downward.

The ࢞-intercept is ૛ૠ.

૛ Draw the .ܞfind the image of the graph of ࢟ൌξૢെ࢞ under the translation map ࢀ ,ۄെ૚ǡ െ૛ۃൌܞ Given the vector .6 original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the ࢞-intercepts of the graph of the image.

࢞ ࢞െ૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ ൅ ૛ ൌ ඥૢെሺ࢞൅૚ሻ૛ ܞ ࢟ ࢟െ૛ , Every point on the semicircle is shifted one unit to the left and two units downward.

࢞-intercepts: െ૚ ൅ ξ૞ and െ૚ െ ξ૞.

૚ ࢟ൌ ൅૚ ࢀ ۄ૚ǡ ૜ۃൌܞ 7. Given the vector , find the image of the graph of ࢞൅૛ under the translation map ܞ. Draw the original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the equations of the asymptotes of the graph of the image.

࢞ ࢞൅૚ ૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ,࢟ ൌ ൅૝ ܞ ࢟ ࢟൅૜ ࢞൅૚ Every point on the curve is shifted one unit to the right and three units upward.

The new vertical asymptote is ࢞ൌെ૚, the new horizontal asymptote is ࢟ൌ૝.

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Draw .ܞfind the image of the graph of ࢟ൌȁ࢞൅૛ȁ ൅૚ under the translation map ࢀ ,ۄെ૚ǡ ૛ۃൌܞ Given the vector .8 the original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the ࢞-intercepts of the graph of the image.

࢞ ࢞െ૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ ൌ ȁ࢞൅૜ȁ ൅૜ ܞ ࢟ ࢟൅૛ , Every point on the graph is shifted one unit to the left and two units upward.

There are no ࢞-intercepts.

࢞ Draw the original .ܞfind the image of the graph of ࢟ൌ૛ under the translation map ࢀ ,ۄ૚ǡ െ૛ۃൌܞ Given the vector .9 graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the ࢞-intercepts of the graph of the image.

࢞ ࢞൅૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ ൌ ૛࢞ି૚ െ૛ ܞ ࢟ ࢟െ૛ , Every point on the curve is shifted one unit to the right and two units downward.

࢞-intercept is ૛.

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Draw the .ܞ૛ ࢞, under the translation map ࢀ܏ܗܔfind the image of the graph of ࢟ൌ ,ۄെ૚ǡ ૜ۃൌܞ Given the vector .10 original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the ࢞-intercepts of the graph of the image.

࢞ ࢞െ૚ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ ࢟ ൌ ܔܗ܏ ሺ࢞൅૚ሻ ൅૜ ܞ ࢟ ࢟൅૜ , ૛ Every point on the curve is shifted one unit to the left and three units upward. ૠ ࢞ െ -intercept: ૡ.

࢞૛ ࢟૛ find the image of the graph of ൅ ൌ૚ under the translation map ࢀ . Draw the ,ۄ૛ǡ െ૜ۃGiven the vector ࢜ൌ .11 ૝ ૚૟ ܞ original graph and its image, and explain the geometric effect of the map ࢀܞ on the graph. Find the new center, major and minor axis of the graph of the image.

࢞ ࢞൅૛ ሺ࢞െ૛ሻ૛ ሺ࢟൅૜ሻ૛ ࢀ ቀቂ ቃቁ ൌ ൤ ൨ , ൅ ൌ૚ ܞ ࢟ ࢟െ૜ ૝ ૚૟ Every point on the ellipse is shifted two units to the right and three units downward.

The new center is ሺ૛ǡ െ૜ሻ, the major axis is ૝, and the minor axis is ૛.

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12. Given the vector ܞ, find the image of the given point ࡼ under the translation map ࢀܞ. Graph ࡼ and its image. ૚ ࡼൌ൥૛൩ ,ۄ૜ǡ ૛ǡ ૚ۃൌܞ .a ૜ ࢞ ࢞൅૜ ૝ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟൅૛൩, the new image: ൥૝൩ ࢠ ࢠ൅૚ ૝

૛ ࡼൌ൥െ૚൩ ,ۄെ૛ǡ ૚ǡ െ૚ۃൌܞ .b െ૝ ࢞ ࢞െ૛ ૙ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟൅૚൩, the new image: ൥ ૙ ൩ ࢠ ࢠെ૚ െ૞

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13. Given the vector ܞ, find the image of the given plane under the translation map ࢀܞ. Sketch the original vector and its image. ǡ૜࢞െ૛࢟െࢠൌ૙ۄ૛ǡ െ૚ǡ ૜ۃൌܞ .a ࢞ ࢞൅૛ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟െ૚൩ ,૜࢞ െ ૛࢟ െ ࢠ ൌ ૞ ࢠ ࢠ൅૜

ǡ૛࢞െ࢟൅ࢠൌ૚ۄെ૚ǡ ૛ǡ െ૚ۃൌܞ .b ࢞ ࢞െ૚ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟൅૛൩ ,᩺૛࢞ െ ࢟ ൅ ࢠ ൌ െ૝ ࢠ ࢠെ૚

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14. Given the vector ܞ, find the image of the given sphere under the translation map ࢀܞ. Sketch the original sphere and its image. ǡ࢞૛ ൅࢟૛ ൅ࢠ૛ ൌ૚ۄെ૚ǡ ૛ǡ ૜ۃൌܞ .a ࢞ ࢞െ૚ ૛ ૛ ૛ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟൅૛൩ ,ሺ࢞൅૚ሻ ൅ ሺ࢟െ૛ሻ ൅ ሺࢠെ૜ሻ ൌ૚ ࢠ ࢠ൅૜

ǡሺ࢞൅૛ሻ૛ ൅ሺ࢟െ૜ሻ૛ ൅ሺࢠ൅૚ሻ૛ ൌ૚ۄെ૜ǡ െ૛ǡ ૚ۃൌܞ .b ࢞ ࢞െ૜ ૛ ૛ ૛ ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟െ૛൩ ,ሺ࢞൅૞ሻ ൅ ሺ࢟െ૚ሻ ൅ ሺࢠሻ ൌ૚ ࢠ ࢠ൅૚

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15. Find a vector ܞ and translation map ࢀܞ that will translate the line ࢞െ࢟ൌ૚ to the line ࢞െ࢟ൌെ૜. Sketch the original vector and its image. ૙ ࢞ ࢞൅૙ Answers vary. For example, ܞൌቂ ቃ and ࢀ ቀቂ ቃቁ ൌ ൤ ൨. ૝ ܞ ࢟ ࢟൅૝

૛ ૛ 16. Find a vector ܞ and translation map ࢀܞ that will translate the parabola ࢟ൌ࢞ ൅૝࢞൅૚ to the parabola ࢟ൌ࢞ ૛ ࢞ ࢞൅૛ Because ࢟ൌ࢞૛ ൅૝࢞൅૚ can be written as ࢟൅૜ൌሺ࢞൅૛ሻ૛, ܞൌቂ ቃ and ࢀ ቀቂ ቃቁ ൌ ൤ ൨. ૜ ܞ ࢟ ࢟൅૜

૛ ૛ 17. Find a vector ܞ and translation map ࢀܞ that will translate the circle with equation ࢞ ൅࢟ െ૝࢞൅૛࢟െ૝ൌ૙ to the circle with equation ሺ࢞ ൅ ૜ሻ૛ ൅ሺ࢟െ૝ሻ૛ ൌૢ െ૞ ࢞ ࢞െ૞ Because ࢞૛ െ૝࢞൅࢟૛ ൅૛࢟ൌ૝ can be written as ሺ࢞െ૛ሻ૛ ൅ ሺ࢟൅૚ሻ૛ ൌૢǡܞൌቂ ቃ and ࢀ ቀቂ ቃቁ ൌ ൤ ൨. ૞ ܞ ࢟ ࢟൅૞

18. Find a vector ܞ and translation map ࢀܞ that will translate the graph of ࢟ൌξ࢞െ૜൅૛ to the graph of ࢟ൌξ࢞൅૛െ૜. െ૞ ࢞ ࢞െ૞ ܞൌቂ ቃ and ࢀ ቀቂ ቃቁ ൌ ൤ ൨ െ૞ ܞ ࢟ ࢟െ૞

૛ ૛ ૛ 19. Find a vector ܞ and translation map ࢀܞ that will translate the sphere ሺ࢞൅૛ሻ ൅ ሺ࢟െ૜ሻ ൅ ሺࢠ൅૚ሻ ൌ૚to the sphere ሺ࢞െ૜ሻ૛ ൅ ሺ࢟൅૚ሻ૛ ൅ ሺࢠ൅૛ሻ૛ ൌ૚ ૞ ࢞ ࢞൅૞ ܞൌ൥െ૝൩ and ࢀܞ ቆቈ࢟቉ቇ ൌ ൥࢟െ૝൩ െ૚ ࢠ ࢠെ૚

.find the following ,ۄെ૜ǡ ૟ǡ ૙ۃൌܟ and ,ۄ૛ǡ ૙ǡ െ૛ۃൌܞǡۄ૛ǡ െ૚ǡ ૜ۃൌܝ Given vectors .20 a. ૜ܝ ൅ ܞ ൅ ܟ ۄ૞ǡ ૜ǡ ૠۃ

b. ܟെ૛ܞെܝ ۄെૢǡ ૠǡ ૚ۃ

૚ ૚ ૜ቀ૛ܝെ ܞቁ െ ܟ c. ૛ ૜ ۄ૚૙ǡ െૡǡ ૛૚ۃ

d. െ૛ܝ െ ૜ሺ૞ܞ െ ૜ܟሻ ۄെ૟૚ǡ ૞૟ǡ ૛૝ۃ

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e. ԡܝԡ, ԡܞԡ, and ԡܟԡ

ԡܝԡ ൌ ξ૚૝ǡ ԡܞԡ ൌ૛ξ૛ǡ ԡܟԡ ൌξ૝૞

f. Show that ૛ԡܞԡ ൌ ԡ૛ܞԡ.

ԡ ൌ ξ૜૛ ൌ૝ξ૛ۄ૝ǡ ૙ǡ െ૝ۃԡ ൌ ԡܞԡ ൌ૛൫૛ξ૛൯ ൌ૝ξ૛ǡ᩺᩺ԡ૛ܞ૛ԡ

g. Show that ԡܝ൅ܞԡ ് ԡܝԡ ൅ ԡܞԡ.

ԡ ൌ ξ૚૝ ൅૛ξ૛ܞԡ ൅ ԡܝԡ ൌ ξ૚ૡǡ᩺᩺ԡۄ૝ǡ െ૚ǡ ૚ۃԡ ൌ ԡܞ൅ܝԡ

h. Show that ԡܞെܟԡ ് ԡܞԡ െ ԡܟԡ.

ԡ ൌ૛ξ૛ െξ૝૞ܟԡ െ ԡܞԡ ൌξ૟૞ǡ᩺᩺ԡۄ૞ǡ െ૟ǡ െ૛ۃԡ ൌ ԡܟെܞԡ

૚ ૚ i. ܝ and ቛ ܝቛ ԡܝԡ ԡܝԡ

ૢ ૛ ૚ ૜ ૚ ૛ ૛ ૚ ૛ ૜ ૛ ૝ ૚ ۄ૛ǡ െ૚ǡ ૜ۃ ૚ ฯ ൌ ඨ൬ ൰ ൅൬െ ൰ ൅൬ ൰ ൌ ඨ ൅ ൅ܝ ǡฯۄ ǡെ ǡ ۃ ൌ ൌܝ ԡܝԡ ξ૚૝ ξ૚૝ ξ૚૝ ξ૚૝ ԡܝԡ ξ૚૝ ξ૚૝ ξ૚૝ ૚૝ ૚૝ ૚૝ ൌ૚

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A STORY OF FUNCTIONS Lesson 19 M2 PRECALCULUS AND ADVANCED TOPICS

Lesson 19: Directed Line Segments and Vectors

Student Outcomes Students find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Students add vectors end-to-end using the parallelogram rule or component-wise. They understand vector arithmetic from a geometric perspective.

Lesson Notes This lesson introduces students to directed line segments and how to subtract initial point coordinates from terminal point coordinates to find the components of a vector. The lesson reviews vector arithmetic operations and introduces the parallelogram rule. Students should be able to do much of this lesson without the aid of technology. Continue to emphasize the connections between vector arithmetic and transformations.

Classwork Discussion (3 minutes) Start this lesson by introducing the new terminology relating to directed line segments. Have students read the text in the student materials and then use the discussion questions below to clarify their understanding of vectors defined by an initial point and a terminal point. Scaffolding: A vector can be used to represent a translation that takes one point to an image point. The Revise the Frayer model starting point is called the initial point, and the image point under the translation is called the terminal point. from Lesson 17 to include information about directed line segments, initial point, terminal point, and how to find the components by subtracting coordinates. Give advanced learners a vector in three dimensions and have them determine If we know the coordinates of both points, we can easily determine the horizontal and vertical an initial point, terminal components of the vector. point, and magnitude. ܣሺ͵ǡ െʹǡ ͳሻǡܤሺെͳǡͶǡെ͵ሻ

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Show the diagram below, and ask students the following questions: Which point is the initial point? Which point is the terminal point? How do you know? à The initial point is ܣ and the terminal point is ܤ. The end without the arrow tip is the initial point. The arrow indicates the direction of the translation of point ܣ to its new location at ܤ.

How many units is the initial point translated horizontally? How many units vertically? How do you know? à Ͷ units horizontally and ʹ units vertically. You can subtract corresponding coordinates or count the units on the grid. What are the components of the vector? ۄʹͶǡۃൌܞ à What is the magnitude of the vector? à The magnitude is ξʹͲ, or ʹξͷ.

Exercises 1–3 (9 minutes) Start students on the next exercise. Have them work independently for a few minutes, and then have them team up with a partner to share their work. Call on individual students to explain how they got each answer. Emphasize that we must calculate the components by starting from the initial point and ending at the terminal point. The signs of the components matter because they indicate direction.

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Exercises 1–7

1. Several vectors, represented by arrows, are shown below. For each vector, state the initial point, terminal point, component form of the vector and magnitude.

For ܞ,

ԡ ൌ ξ૚ૠܞand ԡ ۄെ૝ǡ ૚ۃൌܞ ;Initial point ሺ૛ǡ ૝ሻ and terminal point ሺെ૛ǡ ૞ሻ

For ܝ,

ԡ ൌ ξ૛૙ ൌ૛ξ૞ܝand ԡ ۄ૝ǡ ૛ۃൌܝ ;Initial point ሺ૙ǡ ૙ሻ and terminal point ሺ૝ǡ ૛ሻ For ܟ,

ԡ ൌ ξ૛૙ ൌ૛ξ૞ܟand ԡ ۄെ૛ǡ െ૝ۃൌܟ ;Initial point ሺെ૜ǡ ૛ሻ and terminal point ሺെ૞ǡ െ૛ሻ For ܉,

ԡ ൌ ξ૜૝܉and ԡ ۄ૞ǡ ૜ۃൌ܉ ;Initial point ሺെ૝ǡ െ૞ሻ and terminal point ሺ૚ǡ െ૛ሻ For ܊,

ԡ ൌ૜܊and ԡ ۄ૙ǡ െ૜ۃൌ܊ ;Initial point ሺ૜ǡ ૙ሻ and terminal point ሺ૜ǡ െ૜ሻ

If students did not struggle with Exercise 1, move them on to the next exercise which takes away their ability to count segments on the grid to determine the components of the vectors. Exercises 1–3 are designed to lead students to knowing that they can find the components by subtracting the coordinates of the initial point from the coordinates of the terminal point.

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2. Several vectors, represented by arrows, are shown below. For each vector, state the initial point, terminal point, component form of the vector, and magnitude.

For ܞ,

ԡ ൌ ξૠ૙૚ܞand ԡ ۄ૛૟ǡ ૞ۃൌܞ ;Initial point ሺെ૚૚ǡ ૚ૢሻ and terminal point ሺ૚૞ǡ ૛૝ሻ For ܝ, ԡ ൌ ξ૚૙૚૜ܝand ԡ ۄ૛૜ǡ െ૛૛ۃൌܝ ;Initial point ሺ૞ǡ ૚૞ሻ and terminal point ሺ૛ૡǡ െૠሻ For ܟ,

ԡ ൌ ξૠ૛૞ܟand ԡ ۄെ૚૙ǡ െ૛૞ۃൌܟ ;Initial point ሺ૙ǡ ૙ሻ and terminal point ሺെ૚૙ǡ െ૛૞ሻ

3. Write a rule for the component form of the vector ܞ shown in the diagram. Explain how you got your answer.

The components of the vector are the distance ࡭ is translated .ۄ࢞૛ െ࢞૚ǡ࢟૛ െ࢟૚ۃൌܞ The component form is vertically and horizontally to get to ࡮. To find the horizontal distance from ࡭ to ࡮, subtract the ࢞-coordinates. To find the vertical distance from ࡭ to ࡮, subtract the ࢟-coordinates.

Give students time to struggle with Exercise 3 before intervening. Emphasize that they must generalize the process used in Exercises 1 and 2 to write a rule in Exercise 3.

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Discussion (3 minutes) Introduce the term directed line segment before students begin Exercises 4–6. Students use the results of Exercise 3 to find the components of vectors. The next exercises use the directed line segment notation.

When we use the initial and terminal points to describe a vector, we often refer to the vector as a directed line segment. A vector or directed line segment with initial point A and terminal point ࡮ is denoted ሬ࡭࡮ሬሬሬሬሬԦ.

Exercises 4–7 (8 minutes)

4. Write each vector in component form. a. ࡭ࡱሬሬሬሬሬԦ

ۄെ૛ǡ ૟ۃ ሬሬሬሬሬԦ࡭ࡱ ൌ

b. ࡮ࡴሬሬሬሬሬሬԦ

ۄെૡǡ െ૚૛ۃ ሬሬሬሬሬሬԦ࡮ࡴ ൌ

c. ሬሬሬሬሬሬԦࡰ࡯

ۄ૝ǡ െ૛ۃ ሬሬሬሬሬሬԦࡰ࡯ ൌ

d. ࡳࡲሬሬሬሬሬԦ

ۄെ૝ǡ ૛ۃ ࡳࡲሬሬሬሬሬԦ ൌ

e. ሬሬሬԦࡵࡶ

ۄ૟ǡ ૡۃ ሬሬሬԦࡵࡶ ൌ

5. Consider points ࡼሺ૛ǡ ૚ሻ, ࡽሺെ૜ǡ ૜ሻ, and ࡾሺ૚ǡ ૝). a. Compute ࡼࡽሬሬሬሬሬሬԦ and ࡽࡼሬሬሬሬሬሬԦ and show that ࡼࡽሬሬሬሬሬሬԦ ൅ࡽࡼሬሬሬሬሬሬԦ is the zero vector. Draw a diagram to show why this makes sense geometrically.

ۄ૞ǡ െ૛ۃ and ࡽࡼሬሬሬሬሬሬറ ൌ ۄെ૞ǡ ૛ۃ ࡼࡽሬሬሬሬሬሬԦ ൌ

ۄ૙ǡ ૙ۃ ൌ ۄെ૞ ൅ ૞ǡ ૛ െ ૛ۃ ࡼࡽሬሬሬሬሬሬԦ ൅ࡽࡼሬሬሬሬሬሬԦ ൌ

This makes sense because one vector translates ࡼ to ࡽ, and then the other vector translates the point back from ࡽ to ࡼ. The sum would be the zero vector because the original point is returned to its starting location.

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b. Plot the points ࡼ, ࡽ, and ࡾ. Use the diagram to explain why ࡼࡽሬሬሬሬሬሬԦ ൅ࡽࡾሬሬሬሬሬሬԦ ൅ࡾࡼሬሬሬሬሬሬԦ is the zero vector. Show that this is true by computing the sum ሬሬሬሬሬሬԦࡼࡽ ൅ࡽࡾሬሬሬሬሬሬԦ ൅ࡾࡼሬሬሬሬሬሬԦ.

The diagram shows that the sum should be the zero vector because point ࡼ is translated back to its original coordinates when you add the three vectors.

ۄ૙ǡ ૙ۃ ൌ ۄെ૞൅૝൅૚ǡ૛൅૚െ૜ۃ ࡼࡽሬሬሬሬሬሬԦ ൅ࡽࡾሬሬሬሬሬሬԦ ൅ࡾࡼሬሬሬሬሬሬԦ ൌ

6. Show for any two points ࡭ and ࡮ that െ࡭࡮ሬሬሬሬሬሬԦ ൌ࡮࡭ሬሬሬሬሬሬԦ.

ۄ࢞૛ െ࢞૚ǡ࢟૛ െ࢟૛ۃ Consider ࡭ሺ࢞૚ǡ࢟૚ሻ and ࡮ሺ࢞૛ǡ࢟૛ሻ. Then ࡭࡮ሬሬሬሬሬሬԦ ൌ

.ۄ࢞૚ െ࢞૛ǡ࢟૚ െ࢟૛ۃ ൌ ۄെ࢞૛ ൅࢞૚ǡെ࢟૛ ൅࢟૛ۃ and െ࡭࡮ሬሬሬሬሬሬԦ ൌ

.ۄ࢞૚ െ࢞૛ǡ࢟૚ െ࢟૛ۃ This is the component form of ࡮࡭ሬሬሬሬሬሬԦ ൌ

.ۄെ૚ǡ ૝ۃൌܜ and ,ۄ૝ǡ െ૛ۃൌܝ ,ۄെ૞ǡ ૚ۃൌܟ ,ۄ૛ǡ െ૜ۃൌܞ Given the vectors .7 a. Verify that the sum of these four vectors is the zero vector.

ۄ૙ǡ ૙ۃൌܜ൅ܝ൅ܟ൅ܞ and ۄ૚ǡ െ૝ۃൌܝ൅ܟ൅ܞ and ۄെ૜ǡ െ૛ۃൌܟ൅ܞ

b. Draw a diagram representing the vectors as arrows placed end-to-end to support why their sum would be the zero vector. If we position ܞ to have an initial point at ሺ૙ǡ ૙ሻ then ܞ translates the point ሺ૙ǡ ૙ሻ to ሺ૛ǡ െ૜ሻ. Each additional vector translates the point by its components. The result of all four translations returns the point back to the origin.

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Example (5 minutes): The Parallelogram Rule for Vector Addition

This exercise introduces the parallelogram rule for vector addition. Depending on the level of students, provide more or less support with this example. Point out the use of the word position vector to denote a vector whose initial point is the origin, and then point out that the terminal point coordinates are the same as the horizontal and vertical components of the vector.

Example: The Parallelogram Rule for Vector Addition

When the initial point of a vector is the origin, then the coordinates of the terminal point will correspond to the horizontal and vertical components of the vector. This type of vector, with initial point at the origin, is often called a position vector.

.with the initial point of each vector at ሺ૙ǡ ૙ሻ ۄ૚ǡ ૠۃൌܝ and ۄ૞ǡ ૜ۃൌܞ a. Draw arrows to represent the vectors b. Add ܞ൅ܝ end-to-end. What is ܞ൅ܝ? Draw the arrow that represents ܞ൅ܝ with initial point at the origin. c. Add ܝ൅ܞ end-to-end. What is ܝ൅ܞ? Draw the arrow that represents ܝ൅ܞ with an initial point at the origin.

Lead a discussion to introduce the parallelogram rule for adding two vectors. Make sure students’ diagrams clearly show the parallelogram. Notice that the vectors lie on the sides of a parallelogram, and the sum of the two vectors is the diagonal of this parallelogram with initial point at the same location as the original two vectors. How do you know the vectors would lie on the sides of a parallelogram? à We can calculate the slope of the arrow that represents the vector. You can see that opposite sides would have the same slope and thus lie on parallel lines, so the opposite sides of the quadrilateral are parallel, which gives us a parallelogram. How does this example support the fact that the operation (i.e., vector addition) is commutative? à We got the same result regardless of the order in which we added the two vectors.

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Suppose we translated all these vectors by another vector ܜ. Would the components of ܞ൅ܝ change? Explain your reasoning. à No. Translation is a rigid transformation so the size and shape would be preserved, which means the components would remain the same. Explain how to use the parallelogram rule to add two vectors. à To use the parallelogram rule, draw both vectors with the same initial point, and then construct a parallelogram with adjacent sides corresponding to the arrows that represent each vector. The vector sum will be the diagonal from the initial point.

Exercise 8 (5 minutes)

Exercises 8–10

.ۄ૝ǡ ૜ۃൌܞ and ۄെ૛ǡ ૞ۃൌܝ Let .8 a. Draw a diagram to illustrate ܞ and ܝ, and then find ܞ൅ܝ using the parallelogram rule.

ۄ૛ǡ ૡۃൌܝ൅ܞ

b. Draw a diagram to illustrate ૛ܞ, and then find ૛ܞ ൅ ܝ using the parallelogram rule.

ۄ૟ǡ ૚૚ۃ ൌ ܝ ൅ ܞ૛

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c. Draw a diagram to illustrate െܞ, and then find ܝെܞ using the parallelogram rule.

ۄെ૟ǡ ૛ۃൌܞെܝ

d. Draw a diagram to illustrate ૜ܞ and െ૜ܞ.

i. How do the magnitudes of these vectors compare to one another and to that of ܞ?

The magnitude of ૜ܞ and െ૜ܞ are the same. The magnitude of these vectors is ૜ times the magnitude of ܞ. You can see that because scalar multiplication dilates the vector by the scalar multiple.

ii. How do the directions of ૜ܞ and െ૜ܞ compare to the direction of ܞ?

When the scalar is positive, the direction of the scalar multiple is the same going along ܞ. When the scalar is negative, the direction is opposite the direction of ܞ.

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Discussion (3 minutes) Have students read the paragraph in their student materials and summarize it with a partner in their own words. Then lead a brief discussion before starting students on the exercises that follow. How are directed line segments in Թଶ the same as directed line segments in Թଷ? à They still have an initial point and a terminal point. The components are still found by subtracting the initial point coordinates from the terminal point coordinates. How are directed line segments in Թଶ different from directed line segments in Թଷ? à Instead of a point being described by an ordered pair, we must use an ordered triple.

Directed line segments can also be represented in Թ૜. Instead of two coordinates like we use in Թ૛, we simply use three

to locate a point in space relative to the origin, denoted ሺ૙ǡ ૙ǡ ૙ሻ. Thus, the vector ሬሬሬሬሬሬԦ࡭࡮ with initial point ࡭ሺ࢞૚ǡ࢟૚ǡࢠ૚ሻ and terminal point ࡮ሺ࢞૛ǡ࢟૛ǡࢠ૛ሻ would have component form:

.ۄ࢞૛ െ࢞૚ǡ࢟૛ െ࢟૚ǡࢠ૛ െࢠ૚ۃ ሬሬሬሬሬሬԦ࡭࡮ ൌ

Exercises 9–10 (3 minutes)

9. Consider points ࡭ሺ૚ǡ ૙ǡ െ૞ሻ and ࡮ሺ૛ǡ െ૜ǡ ૟ሻ. a. What is the component form of ࡭࡮ሬሬሬሬሬሬԦ?

ۄ૚ǡ െ૜ǡ ૚૚ۃ ൌ ۄ૛െ૚ǡെ૜െ૙ǡ૟െሺെ૞ሻۃ ࡭࡮ሬሬሬሬሬሬԦ ൌ

b. What is the magnitude of ࡭࡮ሬሬሬሬሬሬԦ?

ฮሬሬሬሬሬሬԦ࡭࡮ฮ ൌ ඥ૚૛ ൅ ሺെ૜ሻ૛ ൅૚૚૛ ൌ ξ૚૜૚

10. Consider points ࡭ሺ૚ǡ ૙ǡ െ૞ሻ, ࡮ሺ૛ǡ െ૜ǡ ૟ሻ, and ࡯ሺ૜ǡ ૚ǡ െ૛ሻ. a. Show that ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡭ሬሬሬሬሬሬԦ ൌ૙. Explain your answer using geometric reasoning.

ۄെ૚ǡ ૜ǡ െ૚૚ۃ and ࡮࡭ሬሬሬሬሬሬԦ ൌ ۄ૚ǡ െ૜ǡ ૚૚ۃ ࡭࡮ሬሬሬሬሬሬԦ ൌ

ۄ૙ǡ ૙ǡ ૙ۃ ൌ ۄ૚ െ ૚ǡ െ૜ ൅ ૜ǡ ૚૚ െ ૚૚ۃ ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡭ሬሬሬሬሬሬԦ ൌ

࡭࡮ሬሬሬሬሬሬԦ translates ࡭ to ࡮ and ሬሬሬሬሬሬԦ࡮࡭ translates ࡮ back to ࡭. The net translation effect is zero.

b. Show that ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦ ൌ૙. Explain your answer using geometric reasoning.

.ۄെ૛ǡ െ૚ǡ െ૜ۃ and ࡯࡭ሬሬሬሬሬԦ ൌ ,ۄ૚ǡ ૝ǡ െૡۃ ࡮࡯ሬሬሬሬሬሬԦ ൌ ,ۄ૚ǡ െ૜ǡ ૚૚ۃ We have ࡭࡮ሬሬሬሬሬሬԦ ൌ

Adding all the respective components together gives

.ۄ૙ǡ ૙ǡ ૙ۃ ൌ ۄ૚൅૚െ૛ǡെ૜൅૝െ૚ǡ૚૚െૡെ૜ۃ ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦ ൌ

The sum of these three vectors translates ࡭ to ࡮, then ࡮ to ࡯ and then ࡯ to ࡭. The net translation effect is zero.

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Closing (3 minutes) Have students respond to these questions in writing or with a partner to summarize their learning for this lesson. How do you use the parallelogram rule to add vectors? à Construct a parallelogram with the added vectors along adjacent sides. The sum will be the diagonal from the initial points to the opposite vertex of the parallelogram. How do you find the components of a vector when you know the coordinates of the initial and terminal points? .à You subtract the initial ݔ-coordinate from the terminal ݔ-coordinate to find the horizontal components .You subtract the initial ݕ-coordinate from the terminal ݕ-coordinate to find the vertical components Why when you add the vectors that represent the directed line segments between the three vertices of a triangle do you get the zero vector? à The net effect of the transformations that result from adding these three vectors takes the initial point of the first vector back to its original location in the coordinate plane.

Lesson Summary

A vector ܞ can be used to represent a directed line segment ሬሬሬሬሬሬԦ࡭࡮. If the initial point is ࡭ሺ࢞૚ǡ࢟૚ሻ and the terminal

.ۄ࢞૛ െ࢞૚ǡ࢟૛ െ࢟૚ۃ ൌ࡭࡮ሬሬሬሬሬሬԦ ൌܞ point is ࡮ሺ࢞૛ǡ࢟૛ሻ, then the component form of the vector is

Vectors can be added end-to-end or using the parallelogram rule.

Exit Ticket (3 minutes)

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Name Date

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Exit Ticket

1. Consider vectors with their initial and terminal points as shown below. Find the components of the specified vectors and their magnitudes. a. ܝൌܧܨሬሬሬሬሬԦ and ԡܝԡ

b. ܞൌܣܤሬሬሬሬሬԦ and ԡܞԡ

c. ܟൌܥܦሬሬሬሬሬԦ and ԡܟԡ

d. ܜൌܩܨሬሬሬሬሬԦ and ԡܜԡ

2. For vectors ܝ and ܞ as in Question 1, explain how to find ܝ൅ܞ using the parallelogram rule. Support your answer graphically below.

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Exit Ticket Sample Solutions

1. Consider vectors with their initial and terminal points as shown below. Find the components of the specified vectors and their magnitudes. a. ܝൌࡱࡲሬሬሬሬሬԦ and ԡܝԡ

ԡ࢛ԡ ൌ૛ξ૞ ,ۄ૝ǡ ૛ۃ࢛ൌ

b. ܞൌ࡭࡮ሬሬሬሬሬሬԦ and ԡܞԡ

ԡ࢜ԡ ൌ ξ૚૙ ,ۄെ૚ǡ െ૜ۃ࢜ൌ

c. ܟൌ࡯ࡰሬሬሬሬሬሬԦ and ԡܟԡ

ԡ࢝ԡ ൌ૜ ,ۄ૙ǡ െ૜ۃ࢝ൌ

d. ܜൌࡳࡲሬሬሬሬሬԦ and ԡܜԡ

ԡ࢚ԡ ൌ ξ૞ ,ۄ૛ǡ ૚ۃ࢚ൌ

2. For vectors ܝ and ܞ as in Question 1, explain how to find ܝ൅ܞ using the parallelogram rule. Support your answer graphically below.

To find ࢛൅࢜ using the parallelogram rule, form a parallelogram with vectors ࢛ and ࢜ as sides. The diagonal of the parallelogram is the sum of the two vectors.

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Problem Set Sample Solutions

1. Vectors ܝ, ܞ, ܟ, ܉, and ܊ are shown at right. a. Find the component form of ܝ, ܞ, ܟ, ܉, and ܊.

ۄെ૜ǡ െ૛ۃ ࢈ ൌ,ۄ૛ǡ ૙ۃࢇൌ ,ۄ૜ǡ ૚ۃ࢝ൌ ,ۄ૚ǡ ૛ۃ࢜ൌ ,ۄെ૝ǡ ૚ۃ࢛ൌ

b. Find the magnitudes ԡܝԡ, ԡܞԡ, ԡܟԡ, ԡ܉ԡ, and ԡ܊ԡ.

ԡ࢛ԡ ൌ ξ૚ૠ,ԡ࢜ԡ ൌ ξ૞, ԡ࢝ԡ ൌ ξ૚૙, ԡࢇԡ ൌ૛, ԡ࢈ԡ ൌ ξ૚૜

c. Find the component form of ܝ൅ܞ and calculate ԡܝ൅ܞԡ.

ԡ࢛൅࢜ԡ ൌ૜ξ૛ ,ۄെ૜ǡ ૜ۃ

d. Find the component form of ܟെ܊ and calculate ԡܟെ܊ԡ.

ǡ ԡ࢝െ࢈ԡ ൌ૜ξ૞ۄ૟ǡ ૜ۃ

e. Find the component form of ૜ܝ െ ૛ܞ.

ۄെ૚૝ǡ െ૚ۃ

f. Find the component form of ܞെ૛ሺܝ൅܊ሻ.

ۄ૚૞ǡ ૝ۃ

g. Find the component form of ૛ሺܝെ૜ܞሻ െ܉.

ۄെ૚૟ǡ െ૚૙ۃ

h. Find the component form of ܝ൅ܞ൅ܟ൅܉൅܊.

ۄെ૚ǡ ૛ۃ

i. Find the component form of ܝെܞെܟെ܉൅܊.

ۄെ૚૜ǡ െ૝ۃ

j. Find the component form of ૛ሺܝ൅૝ܞሻ െ૜ሺܟെ૜܉൅૛܊ሻ.

ۄ૛ૠǡ ૛ૠۃ

2. Given points ࡭ሺ૚ǡ ૛ǡ ૜ሻ, ࡮ሺെ૜ǡ ૛ǡ െ૝ሻ, ࡯ሺെ૛ǡ૚ǡ૞ሻ, find component forms of the following vectors. a. ሬሬሬሬሬሬԦ࡭࡮ and ࡮࡭ሬሬሬሬሬሬԦ

ۄ૝ǡ ૙ǡ ૠۃ ࡮࡭ሬሬሬሬሬሬԦ ൌ ,ۄെ૝ǡ ૙ǡ െૠۃ ሬሬሬሬሬሬԦ࡭࡮ ൌ

b. ሬሬሬሬሬሬԦ࡮࡯ and ࡯࡮ሬሬሬሬሬሬԦ

ۄെ૚ǡ ૚ǡ െૢۃ ࡯࡮ሬሬሬሬሬሬԦ ൌ ,ۄૢ ૚ǡ െ૚ǡۃ ሬሬሬሬሬሬԦ࡮࡯ ൌ

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c. ሬሬሬሬሬԦ࡯࡭ and ࡭࡯ሬሬሬሬሬԦ

ۄെ૜ǡ െ૚ǡ ૛ۃ ࡭࡯ሬሬሬሬሬԦ ൌ ,ۄ૜ǡ ૚ǡ െ૛ۃ ሬሬሬሬሬԦ࡯࡭ ൌ

d. ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ െ࡭࡯ሬሬሬሬሬԦ ۄ૙ǡ ૙ǡ ૙ۃ

e. െ࡮࡯ሬሬሬሬሬሬԦ ൅࡮࡭ሬሬሬሬሬሬԦ ൅࡭࡯ሬሬሬሬሬԦ ۄ૙ǡ ૙ǡ ૙ۃ

f. ሬሬሬሬሬሬԦ࡭࡮ െ࡯࡮ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦ ۄ૙ǡ ૙ǡ ૙ۃ

3. Given points ࡭ሺ૚ǡ ૛ǡ ૜ሻ, ࡮ሺെ૜ǡ ૛ǡ െ૝ሻ, ࡯ሺെ૛ǡ૚ǡ૞ሻ, find the following magnitudes. a. ฮ࡭࡮ሬሬሬሬሬሬԦฮ

ฮ࡭࡮ሬሬሬሬሬሬԦฮ ൌ ξ૟૞

b. ฮ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ฮ.

ฮ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ฮ ൌ ξ૚૝

c. ฮሬሬሬሬሬሬԦ࡭࡮ ൅࡮࡯ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦฮ

૙

.use the parallelogram rule to graph the following vectors ,ۄ૞ǡ െ૜ۃൌܟ ,ۄ૛ǡ ૝ۃൌܞ ,ۄെ૜ǡ ૛ۃൌܝ Given vectors .4 a. ܝ൅ܞ

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b. ܞ൅ܟ

c. ܝെܞ

d. ܞെܟ

e. ૛ܟ ൅ ܞ

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f. ૜ܝ െ ૛ܞ

g. ܝ൅ܞ൅ܟ

5. Points ࡭ǡ ࡮ǡ ࡯ǡ ࡰǡ ࡱǡ ࡲǡ ࡳǡ and ࡴ and vectors ܝ,ܞ, and ܟ are shown below. Find the components of the following vectors.

a. ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦ

࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦ ൌ૙,ۄെ૚૛ǡ െ૜ۃ ࡯࡭ሬሬሬሬሬԦ ൌ ,ۄǡ െ૜ૢۃ ࡮࡯ሬሬሬሬሬሬԦ ൌ ,ۄ૜ǡ ૟ۃ ሬሬሬሬሬሬԦ࡭࡮ ൌ

b. ܝ൅ܞ൅ܟ ൌ૙ܟ൅ܞ൅ܝ ,ۄെ૝ǡ െ૚ۃൌܟ ,ۄ૜ǡ െ૚ۃ ൌ ܞ,ۄ૚ǡ ૛ۃൌܝ

c. ࡭ࡰሬሬሬሬሬሬԦ ൅࡮ࡱሬሬሬሬሬሬԦ ൅࡯ࡳሬሬሬሬሬԦ

ۄ૚ǡ ૛ۃ ǡ࡭࡮ሬሬሬሬሬሬԦ ൅࡮࡯ሬሬሬሬሬሬԦ ൅࡯࡭ሬሬሬሬሬԦ ൌۄെ૝ǡ െ૚ۃ ǡ࡯ࡳሬሬሬሬሬԦ ൌۄ૜ǡ െ૚ۃ ǡ࡮ࡱሬሬሬሬሬሬԦ ൌۄ૛ǡ ૝ۃ ࡭ࡰሬሬሬሬሬሬԦ ൌ

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6. Consider Example 5, part (b) in the lesson and Problem 5, part (a) above. What can you conclude about three vectors that form a triangle when placed tip-to-tail? Explain by graphing.

If three vectors form a triangle when placed tip-to-tail, then the sum of those three vectors is zero.

7. Consider the vectors shown below.

a. Find the components of vectors ܝൌ࡭࡯ሬሬሬሬሬԦ , ܟൌ࡭ࡰሬሬሬሬሬሬԦ , ܞൌ࡭࡮ሬሬሬሬሬሬԦ, and ܋ൌࡱࡲሬሬሬሬሬԦ.

ۄ૛ǡ ૚ۃ ൌࡱࡲሬሬሬሬሬԦ ൌ܋ ,ۄૡǡ ૝ۃ ൌ࡭࡮ሬሬሬሬሬሬԦ ൌܞ ,ۄ૝ǡ ૛ۃ ൌ࡭ࡰሬሬሬሬሬሬԦ ൌܟ ,ۄ૛ǡ ૚ۃ ൌ࡭࡯ሬሬሬሬሬԦ ൌܝ

b. Is vector ܝ equal to vector ࢉ?

.ൌࢉ, they have the same components and direction ۄ૛ǡ ૚ۃൌܝ ,Yes

c. Jens says that if two vectors ܝ and ܞ have the same initial point ࡭ and lie on the same line, then one vector is a scalar multiple of the other. Do you agree with him? Explain how you know. Give an example to support your answer.

Yes. Suppose that the terminal point of ܝ is ࡯ and the terminal point of ܞ is ࡮. Then ܝ has magnitude ԡ࡭࡯ԡ, where ԡ࡭࡯ԡ ്૙, and ܞ has magnitude ԡ࡭࡮ԡ, whereԡ࡭࡮ԡ ്૙. If ܝ and ܞ point in the same direction, then ฮ࡭࡮ฮ ฮ࡭࡮ฮ ܞൌ൬ ൰ܝ. If ܝ and ܞ point in opposite directions, then ܞൌെ൬ ൰ܝ. For example in the diagram ฮ࡭࡯ฮ ฮ࡭࡯ฮ ඥૡ૙ .ܝൌ૝ܝ ൌܞ and we have ,ۄૡǡ ૝ۃ ൌ࡭࡮ሬሬሬሬሬሬԦ ൌܞ and vector ۄ૛ǡ ૚ۃ ൌ࡭࡯ሬሬሬሬሬԦ ൌܝ above, vector ඥ૞

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Lesson 20: Vectors and Stone Bridges

Student Outcomes Students understand the forces involved in constructing a stone arch. Students add and subtract vectors given in magnitude and direction form. Students solve problems that can be represented by vectors.

Lesson Notes Lesson 19 specified vectors by using either the initial and terminal point, such as ሬሬሬሬሬԦܣܤ, or by the components, such as In this lesson, students apply the vector and magnitude form of a vector as they explore how tall a stone arch .ۄʹͳǡۃൌܞ can be built without the thrust forces causing the structure to collapse. Magnitude and direction are both properties of vectors and can be used together to describe a vector. There is no universal way to represent a vector using its magnitude and direction. In this lesson and the ensuing problem set, two different but equivalent methods are used. The magnitude of a vector is clearly defined and described as a positive number that measures the length of the directed line segment that defines the vector. The direction can be specified either by using a geographical description, or by measuring the amount of rotation ߠ that the positive ݔ-axis undergoes to align with the vector when its tail is placed at the origin. For example, consider the vector ܞ shown to the right. We can use magnitude and direction to describe ܞ in the following ways: 1. Magnitude 2 and direction ͵ʹι north of west. .Magnitude 2 and direction ͳͶͺι from the positive ݔ-axis .2 There is some ambiguity in using the geographical description of direction, because ͵ʹι north of west is equivalent to ͷͺι west of north. Thus, there is more than one valid way to use a geographic description to specify a vector. However, in application problems in physics, this type of geographic description of direction is the norm, so students should learn both approaches, and see that they are equivalent. The construction of a stone arch is illustrated in the interactive app, Physics of Stone Arches, available from PBS/Nova here: http://www.pbslearningmedia.org/resource/nv37.sci.engin.design.arches/physics-of-arches/. If students have access to computers for classroom use, then adapt the Opening to let them experiment with the software themselves. Otherwise, demonstrate this interactive app at the front of the classroom. The Physics of Stone Arches app emulates the construction of a stone arch on top of a stack of blocks called the base column. The goal of the app is to construct an arch on as tall a base column as possible. Although the app allows users to experiment with different fortification strategies, this lesson only addresses buttressing.

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There is also a GeoGebra app available, NineStoneArch (http://eureka-math.org/G12M2L20/geogebra-NineStoneArch), that may be used for demonstrations at the front of the room or for students to experiment with during the lesson. This app shows a simplified version of the force vectors in a stone arch with nine stones, allowing users to increase the weight of the individual stones as well as the height of the base columns. To extend this lesson, consider asking students to construct physical arch bridges from a substance such as floral foam or Styrofoam. Consider extending this lesson to two days, allowing a full day for the Exploratory Challenge.

Materials needed: Protractor marked in degrees, a ruler marked in centimeters, and a computer with access to GeoGebra and the Internet.

Classwork Opening (3 minutes) The ancient Romans were the first to recognize the potential of arches for bridge construction. In 1994, Vittorio Galliazzo counted ͻ͵ͳ surviving ancient Roman bridges scattered throughout ʹ͸ different countries. Most of these bridges were made of stone, and many have survived for more than ʹǡͲͲͲ years. Roman arch bridges were usually constructed using semicircular arcs, although some used arcs less than a semicircle. The Pont Julien (French for Julien Bridge) in southeast France, built in the year 3 B.C.E., is based on semicircular arcs. It was used for all traffic, including car traffic, until 2005, when a replacement bridge was constructed to preserve the old bridge. This bridge was used for more than ʹǡͲͲͲ years!

Photograph by Veronique Pagnier, courtesy of Wikimedia Commons/Public domain

Stone arch bridge construction centers on the idea that the arch distributes the force of gravity acting on the stone (i.e., the thrust) through the curve of the arch and into the column of stones beneath it, which we will call the base columns. If the resultant forces were not contained in the stones or the ground, the structure would collapse.

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Demonstrate the Physics of Stone Arches app. Let students guide the construction of an arch and watch it inevitably collapse as more and more blocks are added to the base columns. Experiment with the different types of fortification: buttresses, pinnacles, and flying buttresses. Although there is an option to select a pointed or rounded arch, this lesson only addresses the rounded arch and simple buttresses, so use discretion whether to explore the other possibilities. The Physics of Stone Arches app demonstrates an arch with nine stones. The Pont Julien contains many more stones in its three arches. To keep things simple, experiment with arches made of nine or fewer stones. Additionally, our work will not completely align with the Physics of Stone Arches app because we need to simplify the model to make it accessible to students. Our biggest simplification is in assuming that each stone pushes on another with a force of equal magnitude. In reality, the angle of the stone affects the magnitude of these force vectors.

Discussion (12 minutes) This discussion relies on the GeoGebra app, NineStoneArch previously mentioned in this lesson. Ideally, students will be able to access this app directly, but if there are not enough available computers, then demonstrate at the front of the classroom. What forces are acting on the stones in the bridge? à Gravity, friction, compression Note to teacher: It is likely that students will be able to name gravity as a force acting on the stone but possibly not friction or compression. Friction is the attractive force between stones caused by the roughness of the surface that keeps the stones from sliding away from each other. Compression is the force of one stone pushing on the one next to it. We will disregard the force of friction in our model, which is common with mathematical models. Have you heard the saying, “For every action, there is an equal and opposite reaction?” à Students answer yes or no. Note to teacher: Continue to explain the meaning of this statement either way. “For every action there is an equal and opposite reaction” is a restatement of Newton’s third law of motion, which says “When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.” What does this mean for the stones in our bridge? à If a stone pushes down on the stone under it, the lower stone also pushes up on the stone above it in the opposite direction but with the same magnitude. Let’s consider the keystone, that is, the stone at the top of the bridge. This stone is often shaped differently than the others for aesthetic reasons, but for our purposes, all the stones in the arch are the same shape and size. If the bridge is standing, is the keystone moving? à No. If the keystone is moving, then the bridge is falling down. What forces are acting on the keystone, and in what direction? à Gravity is pulling the keystone straight downward. The stone on the left is pushing upward on the left edge of the keystone in the direction perpendicular to the edge between the stones. The stone on the right is pushing upward on the right edge of the keystone in the direction perpendicular to the edge between the stones.

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Draw or display a figure of the trapezoidal stone shown on the right, marking the three force vectors acting on the stone. There is no way to accurately represent the magnitude of these vectors, but the two vectors on the sides should mirror each other. Some assumptions are made later in the lesson about these vectors, which enables some calculations with their magnitude and direction.

What do we know about the sum of these three vectors acting on the keystone? Why? à Because the keystone is not moving, the net force acting on it is zero. That is, the three force vectors sum to zero. This means that, if placed tip-to-tail, the three vectors would form a triangle. We are going to greatly simplify our model by assuming that all of the stones except the keystone push on each other with compression forces of equal magnitude. The weight of the keystone is carried by both the left and right sides of the arch, so it pushes on the stones to the left and right with compression forces of half of the magnitude of the other stones. Thus, the force vectors for each stone push downward in a direction perpendicular to the joint between the stones and, except for the forces from the keystone, the force vectors are assumed to have the same magnitude. Display the GeoGebra app, NineStoneArch, and show the force vectors. Except for the one from the keystone, the blue vectors all have the same magnitude, and each is perpendicular to the edge that it crosses. In the diagram, the blue vectors have an initial point at the center of mass of each stone. The green vector shows the result of adding up all of the blue force vectors. The bridge stands or collapses based on whether or not the tip of this resultant vector is contained either in the arch itself or in the ground. The arch stands if the vector is green and falls if it is red.

Figure 1: The resultant force is contained within the Figure 2: The resultant force is not contained within arch. The arch stands. the arch. The arch collapses.

Use the GeoGebra app to explore what happens as the height ݄ of the base columns changes using the slider on the left.

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If there is time, ask students what should happen if they use the slider on the right to change the weight ݓ of the stones. à The resultant force is stronger but points in the same direction, so the weight of the stones in this model does not affect the outcome. Check the box to add buttresses to the model and explore the result. When buttresses are added, the width of each base column doubles. Does that double the maximum height of the base columns before the structure collapses? à No. Without the buttresses, the structure stands with base columns of height up to Ͷ units. With the buttresses, the structure stands with base columns of height up to ͸Ǥͺ units. Adding buttresses makes the arch stronger, but does not double the maximum height of the base columns.

Discussion (3 minutes)

This discussion describes the magnitude and direction form of a vector. Highlight this Scaffolding: description before having students start the challenge. Use a visual approach to magnitude and direction form. In Lesson 19, we described vectors by either specifying the coordinates of their Show a vector and ask students endpoints or by specifying their components. We can also describe vectors to describe it in magnitude and using magnitude and direction: if we know the length of a vector and the direction form. direction in which it points, then we have uniquely identified that vector in magnitude and direction form. However, there are two acceptable ways for us to identify the direction of a vector. a. We can describe the direction of the vector relative to the compass points north, east, south, and west. Then we can describe the direction of the vector .as Ͷͷι north of west ۄെͳǡͳۃൌܞ b. We can describe the direction of the vector as the amount of rotation, ߠ, measured in degrees, that the positive horizontal axis must undergo to align with the vector when its tail is placed at the origin, for .is ͳ͵ͷι from the positive horizontal axis ۄെͳǡͳۃൌܞ Ȃ ͳͺͲ ൏ ߠ ൑ ͳͺͲ. Then the direction of the vector :can be described in any of these ways ۄെͳǡͳۃൌܞ Then the magnitude and direction form of the vector a. Magnitude ξʹ, direction Ͷͷι north of west. b. Magnitude ξʹ, direction Ͷͷι west of north. c. Magnitude ξʹ, direction ͳ͵ͷι from the positive horizontal axis.

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Exploratory Challenge (20 minutes) In these exercises, students calculate the magnitude and direction of the force vectors acting on the stones in the arch for a bridge with five arch stones. They can then test the stability of the arch by finding the sum of the three vectors acting on one side of it. To test their understanding of magnitude and direction form of vectors, this challenge does not ask students to find the component form of the vectors; the arch stability is tested geometrically.

Exploratory Challenge

For this Exploratory Challenge, we consider an arch made with five trapezoidal stones on top of the base columns as shown. We focus only on the stones labeled 1, 2, and 3.

a. Study the force vectors acting on the keystone (stone 1) and stones 2 and 3 on the left side of the arch. Why is it acceptable for us to disregard the forces on the right side of the arch?

Due to symmetry, the forces on the right side of the arch are the same magnitude as the forces on the left, but with directions reflected across the vertical line through the center of the keystone.

b. First, focus on the forces acting on the keystone. Stone 2 pushes on the left side of the keystone with force

.܀૚ܘ The stone to the right of the keystone pushes on the right of the keystone with force vector .ۺ૚ܘ vector We know that these vectors push perpendicular to the sides of the stone, but we do not know their

.have the same magnitude ܀૚ܘ and ۺ૚ܘ magnitude. All we know is that vectors

i. Find the measure of the acute angle formed by ࢖૚ࡸ and the horizontal.

We need to consider the angles in the triangles formed by the trapezoidal stones. Since there are five stones that form the arch, each trapezoid creates a triangle with angles that measure

is shown in green. Looking more closely at just the keystone, we see ۺ૚ܘ ૜૟ι,ૠ૛ι,andૠ૛ι. The vector that the ૜૟ι angle is bisected by the vertical line through the center of the keystone. Thus, the angle

makes an ૚ૡι angle with the ۺ૚ܘ makes with the vertical direction is ૢ૙ι ൅ ૚ૡι. Therefore, vector ۺ૚ܘ horizontal axis.

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ii. Find the measure of the acute angle formed by ܘ૚܀ and the horizontal.

Due to symmetry, the acute angle formed by ܘ૚܀ and the horizontal is congruent to the acute angle .and the horizontal. Thus, this angle measures ૚ૡι ۺ૚ܘ formed by

and ܏ tip-to-tail. (g is the force due to gravity and should be vertical.) Why must these܀૚ܘ,ۺ૚ܘ c. Move vectors vectors form a triangle?

Because the keystone does not move, we know that the forces acting on the stone sum to zero. Thus, the vectors that represent these forces will form a triangle when placed tip-to-tail, as we saw in the previous lesson.

d. Suppose that vector ܏ has magnitude 1. Use triangle trigonometry together with the measure of the angles

.to the nearest tenth of a unit ܀૚ܘ and ۺ૚ܘ you found in part (b) to find the magnitudes of vectors i. Find the magnitude and direction form of ܏.

We know that ܏ has magnitude ૚ and direction Ȃૢ૙ι from the positive horizontal axis.

.ۺ૚ܘ ii. Find the magnitude and direction form of Using the figure at right, we see that there are two right triangles with angles ૚ૡι and ૠ૛ι. Since ܏ has magnitude ૚, these triangles have a short leg of length ૚Ȁ૛. We will

Using triangle .ۺ૚ܘ use ࢞ to represent the magnitude of trigonometry, we see that ૚Ȁ૛ ૚ ܛܑܖሺ૚ૡιሻ ൌ and then ࢞ൌ ൎ૚Ǥ૟. ࢞ ૛ܛܑܖ൫૚ૡι൯

.has magnitude ૚Ǥ ૟ and direction ૚ૡι from the positive horizontal axis ۺ૚ܘ Thus, vector

iii. Find the magnitude and direction form of ܘ૚܀.

ԡ so the magnitude of ࢖૚ࡾ is also ૚Ǥ ૟. Then the vectorۺ૚ܘԡ ൌ ԡ܀૚ܘDue to symmetry, we know that ԡ ܘ૚܀has magnitude૚Ǥ ૟and direction૚૟૛ι from the positive horizontal axis.

,represents the force of stone ૚ pushing on the keystone, and by Newton’s third law of motion ۺ૚ܘ e. Vector there is an equal and opposite reaction. Thus, there is a force of the keystone acting on stone ૛ that has the

.ۺ૚ܞ and the opposite direction. Call this vector ۺ૚ܘ same magnitude as

.ۺ૚ܞ i. Find the magnitude and direction form of

.has magnitude ૚Ǥ ૟ and direction െ૚૟૛ι from the positive horizontal axis ۺ૚ܞ The vector

on the arch on the next page, with initial point at the point marked ࡻ, which ۺ૚ܞ ii. Carefully draw vector is the center of mass of the keystone. Use a protractor measured in degrees and a ruler measured in centimeters.

See the final drawing in part (i).

of stone 3 acting on the base column have the ۺ૜ܞ of stone 2 acting on stone 3 and ۺ૛ܞ f. Assume that the forces Why does it make sense that the .ۺ૚ܞ same magnitude as each other, and twice the magnitude as the force ?is significantly shorter than the other two force vectors ۺ૚ܞ force vector The keystone compresses the stones on both the left and right sides of the arch equally, so the gravitational pull on that stone is split in half down the right and left sides. Thus, it is reasonable to assume that the stones 2 and 3 act with twice the compressive force as stone 1.

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g. Find the magnitude and direction form of vector ܞ , the force of stone 2 pressing on :Scaffolding ۺ૛ on the arch on the next page, placing its initial ۺ૛ܞ stone 3. Carefully draw vector Challenge advanced .ۺ૚ܞ point at the terminal point of

so its magnitude is ૜Ǥ ૛. Because students or rapid finishers ,ۺ૚ܞ has twice the magnitude of vector ۺ૛ܞ Vector is rotated െ૚૟૛ι ൅ to calculate the maximumۺ૛ܞ each stone is rotated ૜૟ι from the neighboring stones, vector has magnitude ૜Ǥ ૛ and direction height of the base columns ۺ૛ܞ ૜૟ι ൌ െ૚૛૟ι from the horizontal. Thus, vector െ૚૛૟ι from the positive horizontal axis. before the arch collapses. Also, challenge these the force of stone ૜pressing on students to consider the ,ۺ૜ܞ h. Find the magnitude and direction form of vector the base column. Carefully draw vector ܞ on the arch on the next page, placing its effect of adding buttresses ۺ૜ .ۺ૛ܞ initial point at the terminal point of to the arch; with Vector ܞ has the same magnitude as vector ܞ and it is rotated an additional ૜૟ι buttresses, what is the ۺ૛ ۺ૜ has magnitude ૜Ǥ ૛ and direction ۺ૜ܞ counterclockwise. Thus, vector points maximum height of theۺ૜ܞ െ૚૛૟ι ൅ ૜૟ι ൌ െૢ૙ι from the positive horizontal axis. Thus, vector straight downward. base columns before the arch collapses?

ۺ૜ܞ and ,ۺ૛ܞ,ۺ૚ܞ i. Use the parallelogram method to find the sum of the force vectors on the left side of the arch.

(The diagram of the force vectors is shown below.)

j. Will the arch stand or fall? Explain how you know.

The forces are all contained within the arch, so the arch will stand.

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Plot the force vectors acting on the arch on this diagram to determine whether or not this arch is able to stand or if it collapses.

Closing (3 minutes) Ask students to turn to a partner and explain how they found the following forces in the arch. Partner 1 can explain (a), and partner 2 can explain (b).

ۺ૚ܘ a. The magnitude of the force of stone 2 pushing on stone 1, which we called b. The total force acting on the left side of the arch

Lesson Summary

A vector can be described using its magnitude and direction.

The direction of a vector ܞcan be described either using geographical description, such as ૜૛ι north of west, or by the amount of rotation the positive ࢞-axis must undergo to align with the vector ܞ, such as rotation by ૚૝ૡι from the positive ࢞-axis.

Exit Ticket (4 minutes)

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Name Date

Lesson 20: Vectors and Stone Bridges

Exit Ticket

We saw in the lesson that the forces acting on a stone in a stable arch must sum to zero since the stones do not move.

ۺܘ ,Consider the upper-left stone in a stable arch made of six stones. Denote this stone by ܵ. In the image below represents the force acting on stone ܵfrom the stone on the left. Vector ܘ܀ represents the force acting on stone ܵ from the stone on the right. Vector ܏ represents the downward force of gravity.

in terms of rotation from the positive ݔ-axis by ߠ degrees, for ܀ܘ and ,ۺܘ,܏ a. Describe the directions of vectors െͳͺͲ ൏ ߠ ൏ ͳͺͲ.

.܀ܘ and ۺܘ b. Suppose that vector ܏ has a magnitude of ͳ. Find the magnitude of vectors

.in magnitude and direction form ܀ܘ and ,ۺܘ,܏ c. Write vectors

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Exit Ticket Sample Solutions

We saw in the lesson that the forces acting on a stone in a stable arch must sum to zero since the stones do not move.

ۺܘ ,Consider the upper-left stone in a stable arch made of six stones. Denote this stone by ࡿ. In the image below represents the force acting on stone ࡿfrom the stone on the left. Vector ܘ܀ represents the force acting on stone ࡿ from the stone on the right. Vector ܏ represents the downward force of gravity.

in terms of rotation from the positive ࢞-axis by ࣂ degrees, for ܀ܘ and ,ۺܘ,܏ a. Describe the directions of vectors െ૚ૡ૙ ൏ ࣂ ൑ ૚ૡ૙.

Vector ܏ is vertical, pointing downward, so ܏ aligns with the terminal ray of rotation by Ȃૢ૙ι.

Vector ܘ܀ is perpendicular to the vertical edge of the stone at point ࡺ, so it is horizontal and

pointing to the left. Thus, ܘ܀ aligns with the terminal ray of rotation by ૚ૡ૙ι. ശሬሬሬሬሬሬԦ is perpendicular to line ࡹࡻ shownۺܘ Vector is rotated ૚૛૙ι ۺܘ on the left. Thus, vector aligns with the ۺܘ ,from vertical. Hence terminal ray of rotation by ૜૙ι from horizontal.

.܀ܘ and ۺܘ b. Suppose that vector ܏ has a magnitude of ૚. Find the magnitude of vectors We need to move the vectors tip-to-tail, and since the forces sum to zero, the vectors should make a triangle. Since ܏ is vertical and

ܘ܀ is horizontal, the vectors make a right triangle.

.measures ૜૙ι ۺܘ and ܀ܘ By part (a), the angle made by vectors ૚ We are given that ԡ܏ԡ ൌ૚. Then we know that ܛܑܖሺ૜૙ιሻ ൌ , ฮۺܘฮ ૚ ૚ .ԡ ൌ૛ۺܘso ൌ and then ԡ ૛ ฮ࢖ࡸฮ ԡ ԡ ฮܘ܀ฮ ξ૜ ܘ܀ Then ܋ܗܛሺ૜૙ιሻ ൌ , so ൌ and ԡܘ܀ԡ ൌ ξ૜. ฮ ૛ ૛ۺܘฮ

.in magnitude and direction form ܀ܘ and ,ۺܘ,܏ c. Write vectors Vector ܏ has magnitude૚ and direction െૢ૙ι from the positive horizontal axis.

.has magnitude ૛ and direction ૜૙ι from the positive horizontal axis ۺܘ Vector

Vector ܘ܀ has magnitude ξ૜ and direction ૚ૡ૙ιfrom the positive horizontal axis.

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Problem Set Sample Solutions

1. Vectors ܞ and ܟ are given in magnitude and direction form. Find the coordinate representation of the sum ܞ൅ܟ and the difference ܞെܟ. Give coordinates to the nearest tenth of a unit. a. ܞǣ magnitude ૚૛, direction ૞૙ι east of north ܟǣ magnitude ૡ, direction ૜૙ι north of east

ۄሺ૜૙ιሻܖܑܛ ሺ૜૙ιሻ ,ૡܛܗ܋ૡۃ ൌ ܟ;ۄሺ૝૙ιሻܖܑܛ ሺ૝૙ιሻ ,૚૛ܛܗ܋ ૚૛ۃൌܞ ۄ૚૟Ǥ ૚ǡ ૚૚Ǥ ૠۃൎܟ൅ܞ ۄൎ ૛Ǥ ૜ǡ ૜Ǥ ૠ ܟ െ ܞ

b. ܞǣ magnitude ૛૙, direction ૞૝ι south of east ܟǣ magnitude ૜૙, direction ૚ૡι west of south

ۄሺെ૚૙ૡιሻܖܑܛ ሺെ૚૙ૡιሻ ,૜૙ܛܗ܋ ૜૙ۃ ൌ ܟ;ۄሺെ૞૝ιሻܖܑܛ ሺെ૞૝ιሻ ,૛૙ܛܗ܋ ૛૙ۃൌܞ ۄ૛Ǥ ૞ǡ െ૝૝Ǥ ૠۃ ൅ൎ ۄ૛૚Ǥ૙ǡ૚૛Ǥ૝ۃ െൎ

2. Vectors ܞ and ܟ are given by specifying the length ࢘ and the amount of rotation from the positive ࢞-axis. Find the coordinate representation of the sum ܞ൅ܟ and the difference ܞെܟ. Give coordinates to the nearest tenth of a unit. a. ܞǣ length ࢘ൌ૜, rotated ૚૛ι from the positive ࢞-axis ܟǣ length ࢘ൌ૝, rotated ૚ૡι from the positive ࢞-axis

ۄሺ૚ૡιሻܖܑܛ ሺ૚ૡιሻ ,૝ܛܗ܋૝ۃൌܟ ;ۄሺ૚૛ιሻܖܑܛ ሺ૚૛ιሻ ,૜ܛܗ܋૜ۃൌܞ ۄૢ ૟Ǥ ૠǡ ૚Ǥۃ ൅ൎ ۄെ૙Ǥ ૢǡ െ૙Ǥ ૟ۃ ൎ ܟ െ ܞ

b. ܞǣ length ࢘ൌ૚૟, rotated ૚૟૛ι from the positive ࢞-axis ܟǣ length ࢘ൌ૝૝, rotated െ૚ૡι from the positive ࢞-axis

ۄሺെ૚ૡιሻܖܑܛ ሺെ૚ૡιሻ ,૝૝ܛܗ܋ ૝૝ۃ ൌ ܟ;ۄሺ૚૟૛ιሻܖܑܛ ሺ૚૟૛ιሻ ,૚૟ܛܗ܋ ૚૟ۃൌܞ ۄ૛૟Ǥ ૟ǡ െૡǤ ૠۃ ൅ൎ ۄെ૞ૠǤ ૚ǡ ૚ૡǤ ૞ۃ ൎ ܟ െ ܞ

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3. Vectors ܞ and ܟ are given in magnitude and direction form. Find the magnitude and direction of the sum ܞ൅ܟ and the difference ܞെܟ. Give the magnitude to the nearest tenth of a unit and the direction to the nearest tenth of a degree. a. ܞǣmagnitude ૛૙, direction ૝૞ι north of east ܟǣ magnitude ૡ, direction ૝૞ι west of north The tip of the vector ܞ has coordinates ൫૚૙ξ૛ǡ૚૙ξ૛൯ and tip of vector ܟ has coordinates ൫െ૝ξ૛ǡ૝ξ૛൯. Then the sum has tip ܞ൅ܟൌ൫૟ξ૛ǡ૚૝ξ૛൯. The rotation of ܞ൅ ૚૝ඥ૛ ૠ ܟ is ࣂ ൌ ܉ܚ܋ܜ܉ܖ ൬ ൰ ൌ ܉ܚ܋ܜ܉ܖ ቀ ቁ, so ࣂൎ ૟ඥ૛ ૜ ૟૟Ǥ ૡι. The length of ܞ൅ܟ is ȁܞ൅ܟȁ ൌ ૛ ૛ ට൫૟ξ૛൯ ൅ ൫૚૝ξ૛൯ ൌ ξ૝૟૝ ൎ ૛૚Ǥ ૞. Thus, ܞ൅ܟ has magnitude approximately ૛૚Ǥ ૞ and direction ૟૟Ǥ ૡι north of east. The difference has tip ܞെܟൌ൫૚૝ξ૛ǡ૟ξ૛൯. ૟ඥ૛ The rotation of ܞെܟ is ࣂ ൌ ܉ܚ܋ܜ܉ܖ ൬ ൰ൌ ૚૝ඥ૛ ૜ ܉ܚ܋ܜ܉ܖ ቀ ቁ ࣂൎ૛૜Ǥ૛ι ૠ , so . The length of ૛ ૛ ܞെܟ is ȁܞെܟȁ ൌ ට൫૚૝ξ૛൯ ൅ ൫૟ξ૛൯ ൌ ξ૝૟૝ ൎ ૛૚Ǥ ૞. Thus, ܞെܟ has magnitude approximately ૛૚Ǥ ૞ and direction ૛૜Ǥ ૛ι north of east.

b. ܞǣ magnitude ૚૛Ǥ ૝, direction ૞૝ι south of west ܟǣ magnitude ૚૟Ǥ ૙, direction ૜૟ι west of south

Since ૞૝ι south of west and ૜૟ι west of south are the same direction, vectors and are collinear. Thus, the vector ൅ has length ૚૛Ǥ ૝ ൅ ૚૟ ൌ ૛ૡǤ ૝ and direction ૞૝ιsouth of west.

The vector െ has length ȁ૚૛Ǥ ૝ െ ૚૟ȁ ൌ ૜Ǥ ૟ and direction ૞૝ι north of east.

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4. Vectors ܞ and ܟ are given by specifying the length ࢘ and the amount of rotation from the positive ࢞-axis. Find the length and direction of the sum ܞ൅ܟ and the difference ܞെܟ. Give the magnitude to the nearest tenth of a unit and the direction to the nearest tenth of a degree. a. ܞǣ magnitude ࢘ൌ૚, rotated ૚૙૛ι from the positive ࢞-axis ૚ ܟǣ ࢘ൌ ૚ૡι ࢞ magnitude ૛, rotated from the positive -axis ૚ ૚ ۄሺ૚ૡιሻܖܑܛ , ሺ૚ૡιሻܛܗ܋ ۃൌܟ ;ۄሺ૚૙૛ιሻܖܑܛ, ሺ૚૙૛ιሻܛܗ܋ۃൌܞ ૛ ૛ ۄ૙Ǥ ૜ǡ ૚Ǥ ૚ۃൎܟ൅ܞ

ԡܞ൅ܟԡ ൎ ඥ૙Ǥ ૜૛ ൅૚Ǥ૚૛ ൎ૚Ǥ૛ ૚Ǥ ૚ ܉ܚ܋ܜ܉ܖ ൬ ൰ ൎ ૠ૝Ǥ ૠι ૙Ǥ ૜ The sum ࢜൅࢝ has magnitude approximately ૚Ǥ ૛ and direction ૠ૝Ǥ ૠι from the positive ࢞-axis.

ۄെ૙Ǥૠǡ૙Ǥૡۃെൎ ԡ െԡ ൎ ඥ૙Ǥ ૠ૛ ൅૙Ǥૡ૛ ൎ૚Ǥ૚ ૙Ǥ ૡ ܉ܚ܋ܜ܉ܖ ൬െ ൰ ൎ െ૝ૡǤ ૡι ૙Ǥ ૠ Since ܞെܟ lies in the second quadrant, it aligns with the terminal ray of rotation by ૚ૡ૙ι െ ૝ૡǤ ૡι ൌ ૚૜૚Ǥ ૛ι. The vector ܞെܟ has magnitude approximately ૚Ǥ ૚and direction ૚૜૚Ǥ ૛ι from the positive ࢞-axis.

b. ܞǣ magnitude ࢘ ൌ ૚૙૙૙, rotated െ૚૛૟ι from the positive ࢞-axis ܟǣ magnitude ࢘ ൌ ૞૙૙, rotated െ૚ૡι from the positive ࢞-axis

ۄሺെ૚ૡιሻܖܑܛ ሺെ૚ૡሻι , ૞૙૙ܛܗ܋ ૞૙૙ۃൌܟ ;ۄሺെ૚૛૟ιሻܖܑܛ ሺെ૚૛૟ιሻ ,૚૙૙૙ܛܗ܋ ૚૙૙૙ۃൌܞ ۄെ૚૚૛Ǥ ૜ǡ െૢ૟૜Ǥ ૞ۃൎܟ൅ܞ

ԡܞ൅ܟԡ ൎ ඥ૚૚૛Ǥ ૜૛ ൅ ૢ૟૜Ǥ ૞૛ ൎ ૢૠ૙Ǥ ૙ െૢ૟૜Ǥ ૞ ܉ܚ܋ܜ܉ܖ ൬ ൰ ൎ ૡ૜Ǥ ૝ι െ૚૚૛Ǥ ૜ Since ܞ൅ܟ lies in the third quadrant, it aligns with the terminal ray of rotation by െሺ૚ૡ૙ι െ ૡ૜Ǥ ૝ιሻ ൌ െૢ૟Ǥ ૟ι. The sum ܞ൅ܟ has magnitude approximately ૢૠ૙ and direction െૢ૟Ǥ ૟ι from the positive ࢞-axis.

ۄെ૚૙૟૜Ǥ ૜ǡ െ૟૞૝Ǥ ૞ۃ ൎ ܟ െ ܞ ԡܞെܟԡ ൎ ඥ૚૙૟૜Ǥ ૜૛ ൅ ૟૞૝Ǥ ૞૛ ൎ ૚૛૝ૡǤ ૟ െ૟૞૝Ǥ ૞ ܉ܚ܋ܜ܉ܖ ൬ ൰ ൎ ૜૚Ǥ ૟ι െ૚૙૟૜Ǥ ૜ Since ܞെܟ lies in the third quadrant, it aligns with the terminal ray of rotation by െሺ૚ૡ૙ι െ ૜૚Ǥ ૟ιሻ ൌ െ૚૝ૡǤ ૝ι. The difference ܞെܟ has magnitude approximately ૚૛૝ૡǤ ૟ and directionെ૚૝ૡǤ ૝ι from the positive ࢞-axis.

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5. You hear a rattlesnake while out on a hike. You abruptly stop hiking at point ࡿ and take eight steps. Then you take another six steps. For each distance below, draw a sketch to show how the sum of your two displacements might add so that you find yourself that distance from point ࡿ. Assume that your steps are a uniform size. a. ૚૝ steps

b. ૚૙ steps

c. ૛ steps

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6. A delivery driver travels૛Ǥ ૟ܓܕ due north, then ૞Ǥ ૙ܓܕ due west, and then ૝Ǥ ૛ܓܕ૝૞ι north of west. How far is he from his starting location? Include a sketch with your answer.

He is about ૢǤ ૠ૛ܓܕ from his starting location.

7. Morgan wants to swim directly across a river, from the east to the west side. She swims at a rate of ૚ܕȀܛ. The current in the river is flowing due north at a rate of ૜ܕȀܛ. Which direction should she swim so that she travels due west across the river?

There is no way for Morgan to swim due west if she can only swim at a rate of ૚ܕȀܛ. She would need to cancel out the north vector by swimming south at an equal rate (૜ܕȀܛ). If she swims due south, she will still be swept downstream at a rate of૛ܕȀܛ.

8. A motorboat traveling at a speed of ૝Ǥ ૙ܕȀܛ pointed east encounters a current flowing at a speed ૜Ǥ ૙ܕȀܛ north.

a. What is the speed and direction that the motorboat travels?

The motorboat is traveling ૞ܕȀܛ at a direction ૜૟Ǥ ૡૠι N of E.

b. What distance downstream does the boat reach the opposite shore?

Since the vectors are perpendicular, the north flowing current does not affect the easterly direction and vice ૛૙ versa. It takes ൌ૞ seconds to reach the other side, and in that time the boat will have moved ૝ .downstreamܕ૞ൌ૚૞ڄ૜

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experiences a force ۴due to gravity of ૝Ǥ ૢnewtons directed vertically downward. If this ܏ܓA ball with mass ૙Ǥ ૞ .9 ball is rolling down a ramp that is ૜૙ι inclined from the horizontal, what is the magnitude of the force that is directed parallel to the ramp? Assume that the ball is small enough so that all forces are acting at the point of contact of the ball and the ramp.

The diagram at right shows the forces that arise in this situation. We know that the magnitude of ۴ is ૝Ǥ ૢ N, and we need to find the magnitude of vector ܞǤ The three force vectors form a ૜૙ι-૟૙ι-ૢ૙ι triangle, so we know that ԡܞԡ ܛܑܖሺ૜૙ιሻ ൌ . ԡ۴ԡ Thus, ሺ૜૙ιሻܖܑܛൌԡ۴ԡܞ ሺ૜૙ιሻܖܑܛሻۼૢ ൌ ሺ૝Ǥ .ۼൌ૛Ǥ૝૞

Therefore, the force along the ramp has magnitude ૛Ǥ ૝૞newtons.

10. The stars in the Big Dipper may all appear to be the same distance from Earth, but they are, in fact, very far from each other. Distances between stars are measured in light years, the distance that light travels in one year. The star Alkaid at one end of the Big Dipper is ૚૜ૡ light years from Earth, and the star Dubhe at the other end of the Big Dipper is ૚૙૞ light years from Earth. From Earth, it appears that Alkaid and Dubhe are ૛૞Ǥ ૠι apart. Find the distance in light years between stars Alkaid and Dubhe.

The angle of elevation does not matter and the only thing that does is the angle between them. The distance between the two stars is the same no matter what angle we observe them, so we can treat the Alkaid star as being at a direction of ૙ι and the Dubhe star as being at a direction of ૛૞Ǥ ૠι. Then we find the difference between the two vectors, and we find that the distance between them is about ૟૛Ǥ ૢ light years.

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11. A radio station has selected three listeners to compete for a prize buried in a large, flat field. Starting in the center, the contestants are given a meter stick, a compass, a calculator, and a shovel. Each contestant is given the following three vectors, in a different order for each contestant. ૟૝Ǥ ૛ܕ, ૜૟ι east of north ૝૛Ǥ ૞ܕ, ૛૙ι south of west ૚ૡǤ ૛ܕ due south

The three displacements led to the point where the prize was buried. The contestant that found the prize first won. Instead of measuring immediately, the winner began by doing calculations on paper. What did she calculate?

The winner calculated the sum of the three vectors. The prize is െ૛Ǥ ૛ܕ to the west and ૚ૢǤ ૛ܕ to the north. This is only about૚ૢǤ ૜ܕ away from the starting position, while someone following the directions blindly would travel a total of ૚૛૝Ǥ ૢܕ.

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Lesson 21: Vectors and the Equation of a Line

Student Outcomes Students write the equation for a line in Թଶ or Թଷ using vectors. Students write the parametric equations for a line in Թଶ or Թଷ. Students convert between parametric equations and the slope-intercept form of a line in Թଶ.

Lesson Notes In Algebra I, students wrote equations in the following forms to represent lines in the plane: Slope-intercept form: ݕൌ݉ݔ൅ܾ

Point-slope form: ݕെݕଵ ൌ݉ሺݔെݔଵሻ .ܿ Standard form: ܽݔ ൅ ܾݕ ൌ In Algebra II, students studied functions. This lesson introduces students to parametric equations, showing the connection of functions to vectors. Students look for ways to describe a line in Թଷso they arenot restricted to two- dimensions and can model real-life scenarios. This requires using vectors and parameters and writing parametric equations, which give a way to move and model three-dimensional models with a two-dimensional system. Thus, the lesson starts by reconsidering how to describe a line in the plane using vectors and parameters and then applies this description to lines in Թଷ. The shift to describing a line using vectors to indicate its direction requires that students think geometrically about lines in the plane instead of algebraically. Lessons 21 and 22 are important because they set the mathematical foundation for students to understand the definition of vectors.

Classwork Opening Exercise (3 minutes) The purpose of the second exercise below is to remind students how to graph a line by using the slope to generate points on the line. Encourage students to think geometrically, not algebraically, for this exercise.

Opening Exercise

a. Find three different ways to write the equation that represents the line in the plane that passes through points ሺ૚ǡ ૛ሻ and ሺ૛ǡ െ૚ሻ.

The following four equations show different forms of the equation that represents the line through ሺ૚ǡ ૛ሻ and ሺ૛ǡ െ૚ሻ. ሺ࢟െ૛ሻ ൌെ૜ሺ࢞െ૚ሻ ࢟൅૚ൌെ૜ሺ࢞െ૛ሻ ࢟ ൌ െ૜࢞ ൅ ૞ ૜࢞ ൅ ࢟ ൌ ૞

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b. Graph the line through point ሺ૚ǡ ૚ሻ with slope ૛.

Discussion (12 minutes) In the Opening Exercise, you found three different equations to represent a specific line in the plane.

Consider the point-slope form of a line: ݕെݕଵ ൌ݉ሺݔെݔଵሻ. With the equation in this form, we know that the line passes through point ሺݔଵǡݕଵሻ and has slope ݉. Using this information, we can draw the line as we did in Opening Exercise 2.

ͳ Suppose that the equation of a line κ is ݕെ͵ൌ ሺݔെͶሻ. Then we know that κ passes through the point ʹ ଵ ሺͶǡ͵ሻ and has slope . This means that if we start at point ሺͶǡ͵ሻ and move ݐ units horizontally, we need to ଶ ଵ .move ݐ units vertically to arrive at a new point on the line ଶ ଵ That is, all points on line κ can be found by moving ݐ units right and ݐ units up from ሺͶǡ͵ሻ or by moving ݐ units ଶ ଵ .left and ݐ units down from ሺͶǡ͵ሻ ଶ How could the process of finding a new point on line κ be found using vectors? ͳ Ͷ à A new point on line κ can be found by adding a multiple of the vector ܞൌቈଵ቉ to the vector ቂ ቃ that ଶ ͵ represents point ሺͶǡ͵ሻ.

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?What point do we find when we let ݐൌെͳ? When we let ݐൌ͸? When we let ݐൌͲ? When we let ݐൌെͶ ͵ ݔ Ͷ ͳ ଵ ଵ .à If ݐൌെͳ, then the new point is represented by ቂݕቃൌቂ ቃെቈ ቉ൌቈʹ ቉ ͵ ଶ ଶ ݔ Ͷ ͳ Ͷ ͸ ͳͲ .͸ൌቂ ቃ൅ቂ ቃൌቂ ቃڄà If ݐൌ͸, then the new point is represented by ቂݕቃൌቂ ቃ൅ቈଵ቉ ͵ ଶ ͵ ͵ ͸ ݔ Ͷ ͳ Ͷ .Ͳൌቂ ቃڄà If ݐൌͲ, then the new point is represented by ቂݕቃൌቂ ቃ൅ቈଵ቉ ͵ ଶ ͵ ݔ Ͷ ͳ Ͳ .ሺെͶሻൌቂ ቃڄà If ݐൌെͶ, then the new point is represented by ቂݕቃൌቂ ቃ൅ቈଵ቉ ͵ ଶ ͳ

Based on the calculations of different values of ݐ, what is an equation that uses vectors to represent the points on line κ? Explain your reasoning. ݔ Ͷ ͳ à The points on line κ can be represented by ቂݕቃൌቂ ቃ൅ቈଵ቉ݐ for real numbers ݐ. This is a vector ͵ ଶ Ͷ equation of the line κ. The vector ܞൌቂ ቃ is a direction vector for the line. There are many different ͵ ways to choose the starting point and the direction vector, so the vector form of a line is not unique. Both ݔ and ݕ are both functions of the real number ݐ, which is called a Scaffolding: parameter. We can rewrite Have students create a Frayer ݔ Ͷ ͳ ݔ Ͷ൅ݐ ቂ ቃൌቂ ቃ൅ቈଵ቉ݐ, as ቂ ቃൌቈ ଵ ቉, which means that as functions of ݐ, we diagram for parametric ݕ ͵ ݕ ͵൅ ݐ ଶ ଶ equations. (See Module 1 have Lesson 5 for an example.) ݔሺݐሻ ൌͶ൅ݐ ͳ ݕሺݐሻ ൌ͵൅ ݐ ʹ .These equations for ݔ and ݕ as functions of ݐ are parametric equations of line κ

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Do these parametric equations agree with our original equation in point-slope form that represents line κ? ͳ :That equation is ݕെ͵ൌ ሺݔെͶሻ. Let’s see, using our parameterized equations ʹ ͳ ͵ ݕെ͵ൌ൬͵൅ ݐ൰ െ ʹ ͳ ൌ ݐ ʹ ͳ ͳ ሺݔെͶሻ ൌ ൫ሺͶ൅ݐሻ െͶ൯ ʹ ʹ ͳ ൌ ݐ ʹ So, we see that for any point ൫ݔሺݐሻǡݕሺݐሻ൯ that satisfies the parametric equations ݔሺݐሻ ൌͶ൅ݐ ͳ ͳ .and ݕሺݐሻ ൌ͵൅ ݐ, we have ݕെ͵ൌ ሺݔ െ Ͷሻ, so the point ൫ݔሺݐሻǡݕሺݐሻ൯ is on line κ ʹ ʹ Take a few minutes and explain to your neighbor what you have learned about parametric equations. Use the summary box below to debrief parametric equations as a class and use this as an informal assessment of student knowledge.

ܽ ሺ ሻ ܞ ቂ ቃ Let κ be a line in the plane that contains point ݔଵǡݕଵ and has direction vector ൌ ܾ . If the slope of line κ is ܾ defined, then ݉ൌ . ܽ x A vector form of the equation that represents line κ is ܽ ݔ ݔ .ቂ ቃൌቂ ଵቃ൅ቂ ቃݐ ܾ ݕ ݕଵ x Parametric equations that represent line κ are

ݔሺݐሻ ൌݔଵ ൅ܽݐ .ݕሺݐሻ ൌݕଵ ൅ܾݐ

Exercises 1–3 (10 minutes) Have students work on this exercise in pairs or small groups. Take the time to debrief Exercise 2 and emphasize that ௕ choosing a different starting point ሺݔ ǡݕ ሻ on the line κ or different values ܽ and ܾ so that is the slope of line κ ଵ ଵ ௔ produces equivalent equations of the line that look significantly different. That is, there are multiple correct forms of the vector and parametric equations of a line.

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Exercises

1. Consider the line र in the plane given by the equation ૜࢞ െ ૛࢟ ൌ ૟. a. Sketch a graph of line र on the axes provided.

b. Find a point on line र and the slope of line र. ૜ Student responses for the point will vary; common choices include ሺ૙ǡ െ૜ሻ or ሺ૛ǡ ૙ሻ. The slope of the line is . ૛

c. Write a vector equation for line र using the information you found in part (b).

Student responses will vary. Sample responses are ࢞ ૙ ૚ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ ࢟ െ૜ ૚Ǥ ૞ ࢞ ૛ ૚ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. ࢟ ૙ ૚Ǥ ૞

d. Write parametric equations for line र.

Student responses will vary. Sample responses are

࢞ሺ࢚ሻ ൌ࢚ ࢟ሺ࢚ሻ ൌെ૜൅૚Ǥ૞࢚

or ࢞ሺ࢚ሻ ൌ૛൅࢚ ࢟ሺ࢚ሻ ൌ૚Ǥ૞࢚.

e. Verify algebraically that your parametric equations produce points on line र. ૜࢞ െ ૛࢟ ൌ ૜ሺ࢚ሻ െ૛ሺെ૜ ൅ ૚Ǥ ૞࢚ሻ ൌ૜࢚൅૟െ૜࢚ ൌ૟

Thus, the parametric equations ࢞ሺ࢚ሻ ൌ࢚ and ࢟ሺ࢚ሻ ൌ െ૜ ൅ ૚Ǥ ૞࢚ produce points on line र.

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2. Olivia wrote parametric equations ࢞ሺ࢚ሻ ൌ૝൅૛࢚ and ࢟ሺ࢚ሻ ൌ૜൅૜࢚. Are her equations correct? What did she do differently from you?

Her equations are also correct: ૜࢞ െ ૛࢟ ൌ ૜ሺ૝൅૛࢚ሻ െ૛ሺ૜൅૜࢚ሻ ൌ૚૛൅૟࢚െ૟െ૟࢚ ൌ૟.

૛ She chose the point ሺ૝ǡ ૜ሻ on the line and used the vector ቂ ቃ. ૜

3. Convert the parametric equations ࢞ሺ࢚ሻ ൌ૛െ૜࢚ and ࢟ሺ࢚ሻ ൌ૝൅࢚ into slope-intercept form. ࢞ ૛ െ૜ ૚ One vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the line passes through ሺ૛ǡ ૝ሻ with slope ࢓ൌെ . Then the line has ࢟ ૝ ૚ ૜ equation ૚ ࢟െ૝ൌെ ሺ࢞െ૛ሻ ૜ ૚ ૛ ࢟ൌെ ࢞൅ ൅૝ ૜ ૜ ૚ ૚૝ ࢟ൌെ ࢞൅ . ૜ ૜

Discussion (4 minutes) A line is uniquely determined in the plane if we know a point that it passes through and its slope. Do a point and a slope provide enough to uniquely identify a line in Թଷ? à No, we have no sense of a slope in space, so a point and a slope won’t clearly identify a line in Թଷ. How can we uniquely specify a line κ in space? à If we know a point that κ passes through and the direction in which the line points, then we can uniquely specify that line.

.That is, we need to know a point ሺݔଵǡݕଵǡݖଵሻ on line κ and a vector that is pointed in the same direction as κ .Then we can start at that point ሺݔଵǡݕଵǡݖଵሻ, and move in the direction of the vector to find new points on κ

Example (5 minutes)

In this Example, the process is extended from lines in the plane to lines in space. Consider the line κ in space that passes through point ሺͳǡͳǡʹሻ and ͳ has direction vector ܞൌ൥െ͵൩, as shown to the right. ʹ Can you find another point on κ by moving ݐ steps in the direction of ܞ from point ሺͳǡͳǡʹሻ? ݔ ͳ ͳ .à ቈݕ቉ൌ൥ͳ൩൅൥െ͵൩ݐ ʹ ʹ ݖ

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What are the parametric equations of κ? ݔൌͳ൅ݐ ݕൌͳെ͵ݐ .ݖൌʹ൅ʹݐ Debrief this activity in class using the summary box below.

ܽ .ൌቈܾ቉ܞ Let κ be a line in space that contains point ሺݔଵǡݕଵǡݖଵሻ and has direction vector ܿ x A vector form of the equation that represents line κ is ܽ ݔ ݔଵ .ቈݕ቉ൌ൥ݕଵ൩൅ቈܾ቉ݐ ܿ ݖ ݖଵ x Parametric equations that represent line κ are

ݔሺݐሻ ൌݔଵ ൅ܽݐ ݕሺݐሻ ൌݕଵ ൅ܾݐ .ݖሺݐሻ ൌݖଵ ൅ܿݐ

Exercises 4–5 (4 minutes) Keep students working in the same pairs or small groups as in the previous exercise.

4. Find parametric equations to represent the line that passes through point ሺ૝ǡ ૛ǡ ૢሻ and has direction vector ૛ ܞൌ൥െ૚൩ . െ૜ ࢞ ૝ ૛ ቈ࢟቉ ൌ ൥૛൩ ൅ ൥െ૚൩ ࢚ ࢠ ૢ െ૜

࢞ሺ࢚ሻൌ૝൅૛࢚ ࢟ሺ࢚ሻ ൌ૛െ࢚ ࢠሺ࢚ሻ ൌૢെ૜࢚

5. Find a vector form of the equation of the line given by the parametric equations ࢞ሺ࢚ሻ ൌ૜࢚ ࢟ሺ࢚ሻ ൌെ૝െ૛࢚ ࢠሺ࢚ሻ ൌ૜െ࢚.

࢞ ૙ ૜ ቈ࢟቉ ൌ ൥െ૝൩ ൅ ൥െ૛൩ ࢚ ࢠ ૜ െ૚

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Closing (3 minutes) Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements are listed below.

Lesson Summary

Lines in the plane and lines in space can be described by either a vector equation or a set of parametric equations. ࢇ Let र be a line in the plane that contains point ሺ࢞ ǡ࢟ ሻ and has direction vector ܞൌቂ ቃ. If the slope of ૚ ૚ ࢈ ࢈ र ࢓ൌ line is defined, then ࢇ. A vector form of the equation that represents line र is ࢞ ࢞ ࢇ ቂ ቃൌቂ ૚ቃ൅ቂ ቃ࢚. ࢟ ࢟૚ ࢈ Parametric equations that represent line र are

࢞ሺ࢚ሻ ൌ࢞૚ ൅ࢇ࢚

࢟ሺ࢚ሻ ൌ࢟૚ ൅࢈࢚. ࢇ Let र be a line in space that contains point ሺ࢞૚ǡ࢟૚ǡࢠ૚ሻ and has direction vector ܞൌቈ࢈቉. ࢉ A vector form of the equation that represents line र is

࢞ ࢞૚ ࢇ ቈ࢟቉ ൌ ൥࢟૚൩ ൅ ቈ࢈቉ ࢚. ࢠ ࢠ૚ ࢉ Parametric equations that represent line र are

࢞ሺ࢚ሻ ൌ࢞૚ ൅ࢇ࢚

࢟ሺ࢚ሻ ൌ࢟૚ ൅࢈࢚

ࢠሺ࢚ሻ ൌࢠ૚ ൅ࢉ࢚.

Exit Ticket (4 minutes)

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Name Date

Lesson 21: Vectors and the Equation of a Line

Exit Ticket

.͵Find parametric equations for the line in the plane given by ݕൌʹݔ൅ .1

.Do ݕൌ͹െݔ and ݔሺݐሻ ൌെͳ and ݕሺݐሻ ൌͳ൅͹ݐ represent the same line? Explain why or why not .2

͵ 3. Find parametric equations for the line in space that passes through point ሺͳǡͲǡͶሻ with direction vector ܞൌ൥ʹ൩. ͳ

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Exit Ticket Sample Solutions

1. Find parametric equations for the line in the plane given by ࢟ൌ૛࢞൅૜.

This line passes through point ሺ૙ǡ ૜ሻ with slope ࢓ൌ૛. Then the vector form of the line is ࢞ ૙ ૚ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, ࢟ ૜ ૛ so the parametric equations are ࢞ሺ࢚ሻ ൌ࢚ ࢟ሺ࢚ሻ ൌ૜൅૛࢚.

2. Do ࢟ൌૠെ࢞ and ࢞ሺ࢚ሻ ൌെ૚ and ࢟ሺ࢚ሻ ൌ૚൅ૠ࢚ represent the same line? Explain why or why not.

We can see that if ࢞ሺ࢚ሻ ൌെ૚ and ࢟ሺ࢚ሻ ൌૠെ࢚, then ૠെ࢞ൌૠെሺെ૚ሻൌૡ so ്࢟ૠെ࢞, and thus the equations are not the same line.

૜ 3. Find parametric equations for the line in space that passes through point ሺ૚ǡ ૙ǡ ૝ሻ with direction vector ܞൌ൥૛൩. ૚ ࢞ሺ࢚ሻ ൌ૚൅૜࢚ ࢟ሺ࢚ሻ ൌ૙൅૛࢚ ࢠሺ࢚ሻ ൌ૝൅࢚.

Problem Set Sample Solutions The vector and parametric forms of equations in the plane and in space are not unique. There are many different forms of correct answers to these questions. One correct sample response is included, but there are many other correct responses students could provide. Problems 1–6 address lines in the plane, and Problems 7–11 address lines in space.

૚ ࢞ሺ࢚ሻ ൌ૝െ૜࢚ ࢟ሺ࢚ሻ ൌ૚൅ ࢚ 1. Find three points on the line in the plane with parametric equations and ૜ . Student responses will vary. Using ࢚ൌ૙, ࢚ൌ૜, and ࢚ൌെ૜ gives the three points ሺ૝ǡ ૚ሻ, ሺെ૞ǡ ૝ሻ and ሺ૚૜ǡ ૙ሻ.

2. Find vector and parametric equations to represent the line in the plane with the given equation. a. ࢟ൌ૜࢞െ૝ ૜ ࢞ ૙ ૚ Since the slope is ૜ൌ and a point on the line is ሺ૙ǡ െ૝ሻ, a vector form of the equation is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. ૚ ࢟ െ૝ ૜ Then the parametric equations are ࢞ሺ࢚ሻ ൌ࢚ and ࢟ൌെ૝൅૜࢚.

b. ૛࢞ െ ૞࢟ ൌ ૚૙

૛ ૛ First, we rewrite the equation of the line in slope-intercept form: ࢟ൌ ࢞െ૛. Since the slope is and a point ૞ ૞ ࢞ ૙ ૞ on the line is ሺ૙ǡ െ૛ሻ, a vector form of the equation is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. Then the parametric equations are ࢟ െ૛ ૛ ࢞ሺ࢚ሻ ൌ૞࢚ and ࢟ൌെ૛൅૛࢚.

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c. ࢟ൌെ࢞ ࢞ ૙ ૚ Since the slope is െ૚ and a point on the line is ሺ૙ǡ ૙ሻ, a vector form of the equation is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. ࢟ ૙ െ૚ Then the parametric equations are ࢞ሺ࢚ሻ ൌ࢚ and ࢟ൌെ࢚.

d. ࢟െ૛ൌ૜ሺ࢞൅૚ሻ First, we rewrite the equation of the line in slope-intercept form: ࢟ൌ૜࢞൅૞. Since the slope is ૜ and a point ࢞ ૙ ૚ on the line is ሺ૙ǡ ૞ሻ, a vector form of the equation is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. Then the parametric equations are ࢟ ૞ ૜ ࢞ሺ࢚ሻ ൌ࢚ and ࢟ൌ૞൅૜࢚.

3. Find vector and parametric equations to represent the following lines in the plane. a. the ࢞-axis ૚ A vector in the direction of the ࢞-axis is ቂ ቃ and the line passes through the origin ሺ૙ǡ ૙ሻ, so a vector form is ૙ ࢞ ૚ ቂ ቃൌቂ ቃ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ࢚ and ࢟ሺ࢚ሻ ൌ૙. ࢟ ૙

b. the ࢟-axis ૙ A vector in the direction of the ࢟-axis is ቂ ቃ, and the line passes through the origin ሺ૙ǡ ૙ሻ, so a vector form is ૚ ࢞ ૙ ቂ ቃൌቂ ቃ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ૙ and ࢟ሺ࢚ሻ ൌ࢚. ࢟ ૚

c. the horizontal line with equation ࢟ൌ૝ ૚ A vector in the direction of the line is ቂ ቃ and the line passes through ሺ૙ǡ ૝ሻ, so a vector form is ૙ ࢞ ૙ ૚ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ࢚ and ࢟ሺ࢚ሻ ൌ૝. ࢟ ૝ ૙

d. the vertical line with equation ࢞ൌെ૛ ૙ A vector in the direction of the line is ቂ ቃ and the line passes through ሺെ૛ǡ ૙ሻ, so a vector form is ૚ ࢞ െ૛ ૙ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌെ૛ and ࢟ሺ࢚ሻ ൌ࢚. ࢟ ૙ ૚

e. the horizontal line with equation ࢟ൌ࢑, for a real number ࢑ ૚ A vector in the direction of the line is ቂ ቃ and the line passes through ሺ૙ǡ ࢑ሻ, so a vector form is ૙ ࢞ ૙ ૚ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ࢚ and ࢟ሺ࢚ሻ ൌ࢑. ࢟ ࢑ ૙

f. the vertical line with equation ࢞ൌࢎ, for a real number ࢎ ૙ A vector in the direction of the line is ቂ ቃ and the line passes through ሺࢎǡ ૙ሻ, so a vector form is ૚ ࢞ ࢎ ૙ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌࢎ and ࢟ሺ࢚ሻ ൌ࢚. ࢟ ૙ ૚

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4. Find the point-slope form of the line in the plane with the given parametric equations. a. ࢞ሺ࢚ሻ ൌ૛െ૝࢚, ࢟ሺ࢚ሻ ൌ૜െૠ࢚ ࢞ ૛ െ૝ െૠ ૠ The vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the line passes through the point ሺ૛ǡ ૜ሻ with slope ࢓ൌ ൌ . ࢟ ૜ െૠ െ૝ ૝ ૠ ࢟െ૜ൌ ሺ࢞െ૛ሻ Thus, the point-slope form of the line is ૝ .

૛ ࢞ሺ࢚ሻ ൌ૛െ ࢚ ࢟ሺ࢚ሻ ൌ૟൅૟࢚ b. ૜ , The vector form is ૛ ࢞ ૛ ቂ ቃൌቂ ቃ൅൥െ ൩ ࢚, ࢟ ૟ ૜ ૟ so the line passes through the point ሺ૛ǡ ૟ሻ with slope ૟ ࢓ൌ ൌെૢ. ૛ െ ૜ Thus, the point-slope form of the line is ࢟െ૟ൌെૢሺ࢞െ૛ሻ.

c. ࢞ሺ࢚ሻ ൌ૜െ࢚, ࢟ሺ࢚ሻ ൌ૜ ࢞ ૜ െ૚ The vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the line passes through the point ሺ૜ǡ ૜ሻ with slope ࢓ൌ૙. Thus, the ࢟ ૜ ૙ point-slope form of the line is ࢟െ૜ൌ૙ሺ࢞െ૜ሻ, which is equivalent to ࢟ൌ૜.

d. ࢞ሺ࢚ሻ ൌ࢚, ࢟ሺ࢚ሻ ൌ࢚ ࢞ ૙ ૚ The vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the line passes through the point ሺ૙ǡ ૙ሻ࢝ith slope ࢓ൌ૚. Thus, the ࢟ ૙ ૚ point-slope form of the line is ࢟ൌ࢞.

5. Find vector and parametric equations for the line in the plane through point ࡼ in the direction of vector ܞ. ૛ a. ࡼൌሺ૚ǡ૞ሻ, ܞൌቂ ቃ െ૚ ࢞ ૚ ૛ The vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the parametric equations are ࢞ሺ࢚ሻ ൌ૚൅૛࢚ and ࢟ሺ࢚ሻ ൌ૞െ࢚. ࢟ ૞ െ૚

૝ b. ࡼൌሺ૙ǡ૙ሻ, ܞൌቂ ቃ ૝ ࢞ ૙ ૝ The vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the parametric equations are ࢞ሺ࢚ሻ ൌ૝࢚ and ࢟ሺ࢚ሻ ൌ૝࢚. ࢟ ૙ ૝

૚ c. ࡼൌሺെ૜ǡ െ૚ሻ, ܞൌቂ ቃ ૛ ࢞ െ૜ ૚ The vector form is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, so the parametric equations are ࢞ሺ࢚ሻ ൌെ૜൅࢚ and ࢟ሺ࢚ሻ ൌെ૚൅૛࢚. ࢟ െ૚ ૛

6. Determine if the point ࡭ is on the line र represented by the given parametric equations. a. ࡭ൌሺ૜ǡ ૚ሻ, ࢞ሺ࢚ሻ ൌ૚൅૛࢚ and ࢟ሺ࢚ሻ ൌ૜െ૛࢚

Point ࡭ is on the line if there is a single value of ࢚ so that ૜ൌ૚൅૛࢚ and ૚ൌ૜െ૛࢚. If ૚൅૛࢚ൌ૜, then ࢚ൌ૚. If ૚ൌ૜െ૛࢚, then ࢚ൌ૚. Thus, ࡭ is on the line given by these parametric equations.

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b. ࡭ൌሺ૙ǡ ૙ሻ, ࢞ሺ࢚ሻ ൌ૜൅૟࢚ and ࢟ሺ࢚ሻ ൌ૛൅૝࢚

Point ࡭ is on the line if there is a single value of ࢚ so that ૙ൌ૜൅૟࢚ and ૙ൌ૛൅૝࢚. If ૜൅૟࢚ൌ૙, then ૚ ૚ ࢚ൌെ ૙ൌ૛൅૝࢚ ࢚ൌെ ࡭ ૛. If , then ૛. Thus, is on the line given by these parametric equations.

c. ࡭ൌሺ૛ǡ ૜ሻ, ࢞ሺ࢚ሻ ൌ૝െ૛࢚ and ࢟ሺ࢚ሻ ൌ૝൅࢚

Point ࡭ is on the line if there is a single value of ࢚ so that ૛ൌ૝െ૛࢚ and ૜ൌ૝൅࢚. If ૝െ૛࢚ൌ૛, then ࢚ൌ૚. If ૜ൌ૝൅࢚, then ࢚ൌെ૚. Since there is no value of ࢚ that gives ૝െ૛࢚ൌ૛ and ૜ൌ૝൅࢚, point ࡭ is not on the line.

d. ࡭ൌሺ૛ǡ ૞ሻ, ࢞ሺ࢚ሻ ൌ૚૛൅૛࢚ and ࢟ሺ࢚ሻ ൌ૚૞൅૛࢚

Point ࡭ is on the line if there is a single value of ࢚ so that ૚૛൅૛࢚ൌ૛ and ૚૞ ൅ ૛࢚ ൌ ૞. If ૚૛ ൅ ૛࢚ ൌ ૛, then ࢚ൌെ૞. If ૚૞ ൅ ૛࢚ ൌ ૞, then ࢚ൌെ૞. Thus, ࡭ is on the line given by these parametric equations.

7. Find three points on the line in space with parametric equations ࢞ሺ࢚ሻ ൌ૝൅૛࢚, ࢟ሺ࢚ሻ ൌ૟െ࢚, and ࢠሺ࢚ሻ ൌ࢚.

Student responses will vary. Using ࢚ൌ૙, ࢚ൌ૚, and ࢚ൌെ૚ gives the three points ሺ૝ǡ ૟ǡ ૙ሻ, ሺ૟ǡ ૞ǡ ૚ሻ, and ሺ૛ǡ ૠǡ െ૚ሻ.

8. Find vector and parametric equations to represent the following lines in space. a. The ࢞-axis ૚ A vector in the direction of the ࢞-axis is ൥૙൩ and the line passes through the origin ሺ૙ǡ ૙ǡ ૙ሻ, so a vector form is ૙ ࢞ ૚ ቈ࢟቉ ൌ ൥૙൩ ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ࢚, ࢟ሺ࢚ሻ ൌ૙, and ࢠሺ࢚ሻ ൌ૙. ࢠ ૙

b. The ࢟-axis ૙ A vector in the direction of the ࢟-axis is ൥૚൩ and the line passes through the origin ሺ૙ǡ ૙ǡ ૙ሻ, so a vector form is ૙ ࢞ ૙ ቈ࢟቉ ൌ ൥૚൩ ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ૙, ࢟ሺ࢚ሻ ൌ૚, and ࢠሺ࢚ሻ ൌ૙. ࢠ ૙

c. The ࢠ-axis ૙ A vector in the direction of the ࢠ-axis is ൥૙൩ and the line passes through the origin ሺ૙ǡ ૙ǡ ૙ሻ, so a vector form is ૚ ࢞ ૙ ቈ࢟቉ ൌ ൥૙൩ ࢚. Thus, the parametric equations are ࢞ሺ࢚ሻ ൌ૙, ࢟ሺ࢚ሻ ൌ૙, and ࢠሺ࢚ሻ ൌ૚. ࢠ ૚

9. Convert the equation given in vector form to a set of parametric equations for the line र. ࢞ ૚ ૛ a. ቈ࢟቉ ൌ ൥૚൩ ൅ ൥૜൩ ࢚ ࢠ ૚ ૝ ࢞ሺ࢚ሻ ൌ૚൅૛࢚, ࢟ሺ࢚ሻ ൌ૚൅૜࢚, and ࢠሺ࢚ሻ ൌ૚൅૝࢚

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࢞ ૜ ૙ b. ቈ࢟቉ ൌ ൥૙൩ ൅ ൥ ૚ ൩ ࢚ ࢠ ૙ െ૛ ࢞ሺ࢚ሻ ൌ૜,࢟ሺ࢚ሻ ൌ࢚, and ࢠሺ࢚ሻ ൌ െ૛࢚

࢞ ૞ ૝ c. ቈ࢟቉ ൌ ൥૙൩ ൅ ൥െ૜൩ ࢚ ࢠ ૛ െૡ ࢞ሺ࢚ሻ ൌ૞൅૝࢚, ࢟ሺ࢚ሻ ൌ െ૜࢚, and ࢠሺ࢚ሻ ൌ૛െૡ࢚

10. Find vector and parametric equations for the line in space through point ࡼ in the direction of vector ܞ. ૜ a. ࡼൌሺ૚ǡ૝ǡ૜ሻ, ܞൌ൥ ૟ ൩ െ૛ ࢞ ૚ ૜ ቈ࢟቉ ൌ ൥૝൩ ൅ ൥ ૟ ൩ ࢚;࢞ሺ࢚ሻ ൌ૚൅૜࢚,࢟ሺ࢚ሻ ൌ૝൅૟࢚, and ࢠሺ࢚ሻ ൌ૜െ૛࢚ ࢠ ૜ െ૛

૚ b. ࡼൌሺ૛ǡ૛ǡ૛ሻ, ܞൌ൥૚൩ ૚ ࢞ ૛ ૚ ቈ࢟቉ ൌ ൥૛൩ ൅ ൥૚൩ ࢚;࢞ሺ࢚ሻ ൌ૛൅࢚,࢟ሺ࢚ሻ ൌ૛൅࢚, and ࢠሺ࢚ሻ ൌ૛൅࢚ ࢠ ૛ ૚

૝ c. ࡼൌሺ૙ǡ ૙ǡ ૙ሻ, ܞൌ൥ ૝ ൩ െ૛ ࢞ ૙ ૝ ቈ࢟቉ ൌ ൥૙൩ ൅ ൥ ૝ ൩ ࢚;࢞ሺ࢚ሻ ൌ૝࢚,࢟ሺ࢚ሻ ൌ૝࢚, and ࢠሺ࢚ሻ ൌ െ૛࢚ ࢠ ૙ െ૛

11. Determine if the point ࡭ is on the line र represented by the given parametric equations. a. ࡭ൌሺ૜ǡ૚ǡ૚ሻ, ࢞ሺ࢚ሻ ൌ૞െ࢚, ࢟ሺ࢚ሻ ൌെ૞൅૜࢚, and ࢠሺ࢚ሻ ൌૢെ૝࢚

If ࡭ is on line र, then there is a single value of ࢚ so that ૞െ࢚ൌ૜, െ૞൅૜࢚ൌ૚, and ૢെ૝࢚ൌ૚. If ૞െ࢚ൌ ૜, then ࢚ൌ૛. If Ȃ૞൅૜࢚ൌ૚, then ࢚ൌ૛. If ૢെ૝࢚ൌ૚, then ࢚ൌ૛. Thus, ࡭ lies on line र.

b. ࡭ൌሺ૚ǡ ૙ǡ ૛ሻ, ࢞ሺ࢚ሻ ൌૠെ૛࢚, ࢟ሺ࢚ሻ ൌ૜െ࢚, and ࢠሺ࢚ሻ ൌ૝െ࢚

If ࡭ is on line र, then there is a single value of ࢚ so that ૠെ૛࢚ൌ૚, ૜െ࢚ൌ૙, and ૝െ࢚ൌ૛. If ૠെ૛࢚ൌ૚, then ࢚ൌ૜. If ૜െ࢚ൌ૙, then ࢚ൌ૜. If ૝െ࢚ൌ૛, then ࢚ൌ૛. Thus, there is no value of ࢚ that satisfies all three equations, so point ࡭ is not on line र.

c. ࡭ൌሺ૞ǡ૜ǡ૛ሻ, ࢞ሺ࢚ሻ ൌૡ൅࢚, ࢟ሺ࢚ሻ ൌെ࢚, and ࢠሺ࢚ሻ ൌെ૝െ૛࢚

If ࡭ is on line र, then there is a single value of ࢚ so that ૡ൅࢚ൌ૞, െ࢚ ൌ ૜, and െ૝ െ ૛࢚ ൌ ૛. If ૡ൅࢚ൌ૞, then ࢚ൌെ૜. If Ȃ࢚ൌ૜, then ࢚ൌെ૜. If െ૝ െ ૛࢚ ൌ ૛, then ࢚ൌെ૜. Thus, ࡭ lies on line र.

Lesson 21: Vectors and the Equation of a Line 375

Lesson 22: Linear Transformations of Lines

Student Outcomes Students write parametric equations for a line through two points in Թଶ and Թଷ and for a line segment between two points in Թଶ and Թଷ. Students write parametric equations for the image of a line under a given linear transformation in Թଶ and Թଷ and for the image of a line segment between two points under a given linear transformation in Թଶ and Թଷ.

Lesson Notes In this lesson, students continue their work with parametric equations to see the relationship between their work with functions and vectors. This lesson continues the work of understanding the definition of a vector. The main question of this lesson is whether the image of a line under a linear transformation is again a line. Before answering this, the process of finding parametric equations for a line in Թଶ and Թଷ introduced in Lesson 21 needs to be extended. In the previous lesson, students found vector and parametric equations for a line given a point and a vector; in this lesson, they extend the process to finding parametric equations for the line given two points on the line. They also consider the question of how to parameterize a line segment. In Topic E, students use linear transformations to emulate 3-dimensional motion on a 2-dimensional screen and learn that one of the fundamental qualities of linear transformations is that they preserve lines.

Classwork Opening Exercise (3 minutes) The Opening Exercise reviews the process from Lesson 21 of finding parametric equations of lines in Թଶ and Թଷ given a point and a vector. This lesson extends this process to find parametric equations of lines through two given points and to find parametric equations of line segments.

Opening Exercise െ૛ a. Find parametric equations of the line through point ࡼሺ૚ǡ ૚ሻ in the direction of vector ቂ ቃ. ૜ ࢞ ૚ െ૛ A vector form of the equation is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚, which gives parametric equations ࢞ሺ࢚ሻ ൌ૚െ૛࢚ and ࢟ ૚ ૜ ࢟ሺ࢚ሻ ൌ૚൅૜࢚ for any real number ࢚.

૝ b. Find parametric equations of the line through point ࡼሺ૛ǡ ૜ǡ ૚ሻ in the direction of vector ൥ ૚ ൩. െ૚ ࢞ ૛ ૝ A vector form of the equation is ቈ࢟቉ ൌ ൥૜൩ ൅ ൥ ૚ ൩ ࢚, which gives parametric equations ࢞ሺ࢚ሻ ൌ૛൅૝࢚ and ࢠ ૚ െ૚ ࢟ሺ࢚ሻ ൌ૜൅࢚ and ࢠሺ࢚ሻ ൌ૚െ࢚ for any real number ࢚.

376 Lesson 22: Linear Transformations of Lines

A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

Discussion (5 minutes) In the Opening Exercise, we found parametric equations for the line κ through ܲሺͳǡͳሻ with direction vector െʹ ቂ ቃ. How can we find parametric equations for this line if all we know is that points ܲሺͳǡͳሻ and ܳሺെͳǡͶሻ are ͵ on line κ? à First, we find the vector that points from ܲ to ܳ, and then we apply the process from the last lesson. What is this direction vector? à The direction vector ܞ is the difference between the vectors representing points ܳ and ܲ: െͳ ͳ െʹ ܞൌቂ ቃെቂ ቃ so ܞൌቂ ቃ. Ͷ ͳ ͵ What are parametric equations for the line κ? à This is the same direction vector as we had in Problem 1 of the Opening Exercise, so parametric .equations are ݔሺݐሻ ൌͳെʹݐ, ݕሺݐሻ ൌͳ൅͵ݐ for any real number ݐ Scaffolding: What would happen if we swapped ܲ and ܳ? Do we get parametric equations For struggling learners, for a different line? display an image of the ʹ à No. If we interchange ܲ and ܳ, then we get a direction vector ቂ ቃ, and point ܲሺͳǡͳሻ and the line െ͵ ͵ ͷ the parametric equations are ݔሺݐሻ ൌͳ൅ʹݐ, ݕሺݐሻ ൌͳെ͵ݐ for any real ݕൌെ ݔ൅ . Place a ʹ ʹ number ݐ. This describes the same line, but it is being traversed marker on ܲ to indicate backward. Instead of moving from ܲ to ܳ as ݐ increases, this new line when ݐൌͲ, and slide the locates points from ܳ to ܲ as ݐ increases. marker upward to the left to illustrate the point on the line corresponding to .Example (8 minutes) increasing values of ݐ This example is analogous to the one in the previous discussion but in Թଷ instead of Թଶ. It Ask advanced learners to then proceeds to describe how to use parametric equations to describe a line segment find the parametric ܲܳ. equations described in the Example in pairs without What if we had a line in Թଷ? Suppose we want to find parametric equations of the guiding questions and the line through points ܲሺͳǡʹǡ͵ሻ and ܳሺെͶǡͳǡͲሻ. How do we find these then present their work to equations? the class. à First, we need the vector that points from ܲ to ܳ: െͶ ͳ െͷ ݔ ͳ െͷ ൌ൥ͳ ൩െ൥ʹ൩ൌ൥െͳ൩. Then we have the vector form of the equation ቈݕ቉ൌ൥ʹ൩൅൥െͳ൩ݐ, whichܞ ͵Ͳ ͵ െ͵ ݖ ͵ െ .gives three parametric equationsݔሺݐሻ ൌͳെͷݐ, ݕሺݐሻ ൌʹെݐ, and ݖሺݐሻ ൌ͵െ͵ݐ for real numbers ݐ Is there a way to use parametric equations to describe just the line segment ܲܳ instead of the entire line ܲܳ? Give students time to figure this out on their own or with a partner, and then discuss it later with the class as shown below. ?ܲ What is the value of ݐ in ݔሺݐሻ ൌͳെͷݐ, ݕሺݐሻ ൌʹെݐ, and ݖሺݐሻ ൌ͵െ͵ݐ that produces point ݔሺͲሻ ͳ .à If ݐൌͲ, then ቎ݕሺͲሻ቏ൌ൥ʹ൩ൌܲ ͵ ݖሺͲሻ

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A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

?ܳ What value of ݐ in ݔሺݐሻ ൌͳെͷݐ, ݕሺݐሻ ൌʹെݐ, and ݖሺݐሻ ൌ͵െ͵ݐ produces point ݔሺͳሻ െͶ .à If ݐൌͳ, then ቎ݕሺͳሻ቏ൌ൥ͳ ൩ൌܳ ݖሺͳሻ Ͳ So, how can we use parametric equations to describe just തതതതܲܳ? .à Use the same parametric equations as for ശሬሬሬሬԦܲܳ, but restrict Ͳ൑ݐ൑ͳ In general, describe the process for finding parametric equations of the line through ܲ and ܳ. à First, find the direction vector ܞ by subtracting the vector associated with ܲ from the vector associated with ܳ. Then, find the vector form of the equation of the line and the parametric form. Let ݐ take on any real number value. In general, describe the process for finding parametric equations of തതതതܲܳ. à First, find the direction vector ܞ by subtracting the vector associated with ܲ from the vector associated with ܳ. Then, find the vector form of the equation of the line and the parametric form. തതതതܲܳ .corresponds to the part of the line with Ͳ൑ݐ൑ͳ

Discussion (12 minutes) This discussion starts with an example that shows that the image of a particular line in Թଷ under a given linear transformation is again a line in Թଷ. Once this example has been established, the discussion proceeds to establish this fact for any line in Թଶ or Թଷ and any linear transformation ܮ. Now, we want to explore what happens when we transform a line using a linear transformation. What do you expect the image of a line to be under a linear transformation? Why? à I don’t know. Linear transformations include things like rotation, dilation, and reflection. All of these operations transform a line into another line. But, there might be a linear transformation that does something else that might distort or bend a line. Suppose that the line passes through points ܲሺͳǡͲǡͳሻ and ܳሺ͵ǡ െ͵ǡʹሻ and we have a linear transformation ݔ ͳʹͳݔ ?ሺܳሻܮ ሺܲሻ andܮ ቆቈݕ቉ቇ ൌ ൥ʹͳ͵൩ቈݕ቉. Then what are the transformed pointsܮ ݖ ͳͳʹݖ ͳ ͳʹͳͳ ʹ ͵ ͳʹͳ ͵ െͳ à ܮሺܲሻ ൌܮ൭൥Ͳ൩൱ ൌ ൥ʹͳ͵൩൥Ͳ൩ൌ൥ͷ൩ and ܮሺܳሻ ൌܮ൭൥െ͵൩൱ ൌ ൥ʹͳ͵൩൥െ͵൩ൌ൥ͻ ൩ ͳ ͳͳʹͳ ͵ ʹ ͳͳʹ ʹ Ͷ How can we describe a point on ശሬሬሬሬԦܲܳ? ͵ ͳ ʹ à We can use the parametric equations for ശሬሬሬሬԦܲܳ: First, a direction vector is ൥െ͵൩െ൥Ͳ൩ൌ൥െ͵൩. Then, a ʹ ͳ ͳ ʹ ݔ ͳ vector form of the equation of ശሬሬሬሬԦܲܳ is ቈݕ቉ൌ൥Ͳ൩൅൥െ͵൩ݐ. Finally, parametric equations for ശሬሬሬሬԦܲܳ are ݖ ͳ ͳ .ݔሺݐሻ ൌͳ൅ʹݐ, ݕሺݐሻ ൌെ͵ݐ, and ݖሺݐሻ ൌͳ൅ݐ for all real numbers ݐ

378 Lesson 22: Linear Transformations of Lines

A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

ͳ൅ʹݐ :ܮ Since we know that ൥ െ͵ݐ ൩ is a generic point on ശܲܳሬሬሬሬԦ , we can transform this point under ͳ൅ݐ ͳ൅ʹݐ ͳʹͳͳ൅ʹݐ ൭൥ െ͵ݐ ൩൱ ൌ ൥ʹͳ͵൩൥ െ͵ݐ ൩ܮ ͳ൅ݐ ͳͳʹ ͳ൅ݐ ͳሺͳ൅ʹݐሻ ൅ʹሺെ͵ݐሻ ൅ͳሺͳ൅ݐሻ ൌ቎ʹሺͳ൅ʹݐሻ ൅ͳሺെ͵ݐሻ ൅͵ሺͳ൅ݐሻ቏ ͳሺͳ൅ʹݐሻ ൅ͳሺെ͵ݐሻ ൅ʹሺͳ൅ݐሻ െ͵ݐʹ ൌ൥ͷ൅Ͷݐ൩ ൅ݐ͵ But, this is how we express a line in vector form. So, any point on the line ܲܳ is transformed into a point on ʹ ͵ݔ ʹ െ ?ሺܲሻ ൌ൥ͷ൩. Is this a coincidenceܮ the line ቈݕ቉ൌ൥ͷ൩൅൥ Ͷ ൩ݐ. We saw earlier that ͵ ݖ ͵ ͳ à No, it’s probably not a coincidence because the starting point is when ݐൌͲ, and when ݐൌͲ in our parametric equation, we get the initial point. Now, let’s generalize this result to any transformation ܮ and any line κ through points ܲ and ܳ in Թଷ. Let κ be a line in either Թଶ or Թଷ, and let ܮ be a linear transformation on that space that can be represented by

݌ଵ multiplication by matrix ܣ. Let point ܲ be represented by vector ൥݌ଶ൩ and let point ܳ be represented by vector ݌ଷ ଵ െ݌ଵ ݔݍ ଵݍ ଶ െ݌ଶ൩. Any point ቈݕ቉ on ശሬሬሬሬԦܲܳ is given byݍൌ൥ܞ by ܞ ଶ൩. Then, we can find the direction vectorݍ൥ ଷ െ݌ଷ ݖݍ ଷݍ ଵ െ݌ଵݍ ݔ ݌ଵ ଶ െ݌ଶ൩ݐ for some real number ݐ. Then, the transformed point is given byݍቈݕ቉ൌ൥݌ଶ൩൅൥ ଷ െ݌ଷݍ ݖ ݌ଷ ݔ ݔ ቆቈݕ቉ቇ ܣ ቆቈݕ቉ቇ ൌܮ ݖ ݖ ݌ଵ ݍଵ െ݌ଵ ଶ െ݌ଶ൩ݐ൱ݍ൭൥݌ଶ൩൅൥ܣൌ ݌ଷ ݍଷ െ݌ଷ ݌ଵ ݍଵ െ݌ଵ ଶ െ݌ଶ൩൱ ݐݍ൥ܣ൥݌ଶ൩൅൭ܣൌ ݌ଷ ݍଷ െ݌ଷ ݌ଵ ݍଵ ݌ଵ ൥݌ଶ൩൱ ݐܣଶ൩െݍ൥ܣ൥݌ଶ൩൅൭ܣൌ ݌ଷ ݍଷ ݌ଷ ሺܲሻሻݐܮሺܳሻ െܮሺܲሻ ൅ሺܮൌ Since ܮሺܲሻ and ሺܮሺܳሻ െܮሺܲሻሻ are vectors that represent points in space, this is the vector form of a line that passes through ܮሺܲሻ and has direction vector ൫ܮሺܳሻ െܮሺܲሻ൯. Therefore, the image of any line in Թଷ under a linear transformation ܮ is again a line.

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A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

Exercises (8 minutes) Have students work in pairs or small groups on these exercises.

Exercises ࢞ ૚૛െ૚࢞ 1. Consider points ࡼሺ૛ǡ ૚ǡ ૝ሻ and ࡽሺ૜ǡ െ૚ǡ ૛ሻ, and define a linear transformation by ࡸ ቆቈ࢟቉ቇ ൌ ൥૙૚ ૛൩ቈ࢟቉. ࢠ ૜െ૚૚ ࢠ Find parametric equations to describe the image of ࡼࡽശሬሬሬሬԦ under the transformation ࡸ. ૜ ૛ ૚ Direction vector: ൥െ૚൩ െ ൥૚൩ ൌ ൥െ૛൩ ૛ ૝ െ૛ ࢞ ૛ ૚ Vector equation: ቈ࢟቉ ൌ ൥૚൩ ൅ ൥െ૛൩ ࢚ for all real numbers t ࢠ ૝ െ૛ Parametric Equations: ࢞ሺ࢚ሻ ൌ૛൅࢚, ࢟ሺ࢚ሻ ൌ૚െ૛࢚, and ࢠሺ࢚ሻ ൌ૝െ૛࢚ for all real numbers t

2. The process that we developed for images of lines in Թ૜ also applies to lines in Թ૛. Consider points ࡼሺ૛ǡ ૜ሻ and ࢞ ૚૛࢞ ࡽሺെ૚ǡ ૝ሻ. Define a linear transformation by ࡸቀቂ ቃቁ ൌ ቂ ቃቂ ቃ. Find parametric equations to describe the ࢟ ૜െ૚࢟ image of ࡼࡽശሬሬሬሬԦ under the transformation ࡸ. െ૚ ૛ െ૜ Direction vector: ቂ ቃെቂ ቃൌቂ ቃ ૝ ૜ ૚ ࢞ ૛ െ૜ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ ૜ ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૛െ૜࢚ and ࢟ሺ࢚ሻ ൌ૜൅࢚ for all real numbers ࢚

3. Not only is the image of a line under a linear transformation another line, but the image of a line segment under a linear transformation is another line segment. Let ࡼ, ࡽ, and ࡸ be as specified in Exercise 2. Find parametric equations to describe the image of ࡼࡽതതതത under the transformation ࡸ. െ૚ ૛ െ૜ Direction vector: ቂ ቃെቂ ቃൌቂ ቃ ૝ ૜ ૚ ࢞ ૛ െ૜ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for 0 ൑࢚൑૚ ࢟ ૜ ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૛െ૜࢚ and ࢟ሺ࢚ሻ ൌ૜൅࢚ for ૙൑࢚൑૚

380 Lesson 22: Linear Transformations of Lines

A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

Closing (4 minutes) Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements are listed below.

Lesson Summary

Vector and parametric equations of a line in the plane or in space can be found if two points that the line passes through are known, and parametric equations of a line segment in the plane or in space can be found by restricting the values of ࢚ in the parametric equations for the line.

Let र be a line in the plane that contains point࢙ ࡼሺ࢞૚ǡ࢟૚ሻ and ࡽሺ࢞૛ǡ࢟૛ሻ. Then a direction vector is ࢞ െ࢞ given by ቂ ૛ ૚ቃ, and an equation in vector form that represents line र is ࢟૛ െ࢟૚ ࢞ ࢞ ࢞ െ࢞ ቂ ቃൌቂ ૚ቃ൅ቂ ૛ ૚ቃ࢚ǡ for all real numbers ࢚. ࢟ ࢟૚ ࢟૛ െ࢟૚ Parametric equations that represent line र are

࢞ሺ࢚ሻ ൌ࢞૚ ൅ ሺ࢞૛ െ࢞૚ሻ࢚ ࢟ሺ࢚ሻ ൌ ࢟૚ ൅ሺ࢟૛ െ࢟૚ሻ࢚for all real numbers࢚. Parametric equations that represent ࡼࡽതതതത are

࢞ሺ࢚ሻ ൌ࢞૚ ൅ ሺ࢞૛ െ࢞૚ሻ࢚ ࢟ሺ࢚ሻ ൌ࢟૚ ൅ ሺ࢟૛ െ࢟૚ሻ࢚for ૙൑࢚൑૚.

Let र be a line in space that contains pointsࡼሺ࢞૚ǡ࢟૚ǡࢠ૚ሻ and ࡽሺ࢞૛ǡ࢟૛ǡࢠ૛ሻ. Then a direction vector is ࢞૛ െ࢞૚ given by ൥࢟૛ െ࢟૚൩, and an equation in vector form that represents line र is ࢠ૛ െࢠ૚

࢞ ࢞૚ ࢞૛ െ࢞૚ ቈ࢟቉ ൌ ൥࢟૚൩ ൅ ൥࢟૛ െ࢟૚൩ ࢚, for all real numbers ࢚. ࢠ ࢠ૚ ࢠ૛ െࢠ૚ Parametric equations that represent line र are

࢞ሺ࢚ሻ ൌ࢞૚ ൅ ሺ࢞૛ െ࢞૚ሻ࢚ ࢟ሺ࢚ሻ ൌ ࢟૚ ൅ሺ࢟૛ െ࢟૚ሻ࢚ ࢠሺ࢚ሻ ൌ ࢠ૚ ൅ሺࢠ૛ െࢠ૚ሻ࢚for all real numbers࢚. Parametric equations that represent ࡼࡽതതതത are

࢞ሺ࢚ሻ ൌ࢞૚ ൅ ሺ࢞૛ െ࢞૚ሻ࢚ ࢟ሺ࢚ሻ ൌ ࢟૚ ൅ሺ࢟૛ െ࢟૚ሻ࢚ ࢠሺ࢚ሻ ൌࢠ૚ ൅ ሺࢠ૛ െࢠ૚ሻ࢚for ૙൑࢚൑૚.

The image of ࡼࡽശሬሬሬሬԦ in the plane under a linear transformation ࡸ is given by ࢞ ቂ࢟ቃൌࡸሺࡼሻ ൅ ൫ࡸሺࡽሻ െࡸሺࡼሻ൯࢚, for all real numbers ࢚.

The image of ࡼࡽശሬሬሬሬԦ in space under a linear transformation ࡸ is given by ࢞ ቈ࢟቉ ൌࡸሺࡼሻ ൅ ൫ࡸሺࡽሻ െࡸሺࡼሻ൯࢚, for all real numbers ࢚. ࢠ

Exit Ticket (5 minutes)

Lesson 22: Linear Transformations of Lines 381

A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

Name Date

Lesson 22: Linear Transformations of Lines

Exit Ticket

1. Consider points ܲሺʹǡͳሻ and ܳሺʹǡͷሻ. Find parametric equations that describe points on തതതതܲܳ.

ݔ ͳെ͵ݔ 2. Suppose that points ܲሺʹǡͳሻ and ܳሺʹǡͷሻ are transformed under the linear transformation ܮቀቂ ቃቁ ൌ ቂ ቃቂ ቃ. ݕ Ͳͳݕ Find parametric equations that describe the image of ശሬሬሬሬԦܲܳ under this transformation.

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A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

Exit Ticket Sample Solutions

1. Consider points ࡼሺ૛ǡ ૚ሻ and ࡽሺ૛ǡ ૞ሻ. Find parametric equations that describe points on ࡼࡽതതതത . ૛ ૛ ૙ ࢞ ૛ ૙ A direction vector ܞ is given by ܞൌቂ ቃെቂ ቃൌቂ ቃ, so a vector form of ࡼࡽതതതത is ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ ૞ ૚ ૝ ࢟ ૚ ૝ for ૙൑࢚൑૚. This gives the parametric equations ࢞ሺ࢚ሻ ൌ૛ and ࢟ሺ࢚ሻ ൌ૚൅૝࢚ for ૙൑࢚൑૚.

࢞ ૚െ૜࢞ 2. Suppose that points ࡼሺ૛ǡ ૚ሻ and ࡽሺ૛ǡ ૞ሻ are transformed under the linear transformation ࡸቀቂ ቃቁ ൌ ቂ ቃቂ ቃ. ࢟ ૙૚࢟ Find parametric equations that describe the image of ࡼࡽശሬሬሬሬԦ under this transformation.

The images of ࡼ and ࡽ are ૚െ૜૛ െ૚ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ ૙૚૚ ૚ ૚െ૜૛ െ૚૜ ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ . ૙૚૞ ૞ െ૚૜ െ૚ െ૚૛ The direction vector ܞ is then ܞൌቂ ቃെቂ ቃൌቂ ቃ, so the vector form of the image of ࡼࡽശሬሬሬሬԦ is ૞ ૚ ૝ ࢞ െ૚ െ૚૛ ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚. ࢟ ૚ ૝ Parametric equations that represent the limit of ࡼࡽശሬሬሬሬԦ are ࢞ሺ࢚ሻ ൌ െ૚ െ ૚૛࢚ and ࢟ሺ࢚ሻ ൌ૚൅૝࢚ for all real numbers ࢚.

Problem Set Sample Solutions

1. Find parametric equations of ࡼࡽശሬሬሬሬԦ through points ࡼ and ࡽ in the plane. a. ࡼሺ૚ǡ ૜ሻ, ࡽሺ૛ǡ െ૞ሻ ૛ ૚ ૚ Direction vector: ܞൌቂ ቃെቂ ቃൌቂ ቃ െ૞ ૜ െૡ ࢞ ૚ ૚ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ ૜ െૡ Parametric Equations: ࢞ሺ࢚ሻ ൌ૚൅࢚ and ࢟ሺ࢚ሻ ൌ૜െૡ࢚ for all real numbers ࢚

b. ࡼሺ૜ǡ ૚ሻ, ࡽሺ૙ǡ ૛ሻ ૙ ૜ െ૜ Direction vector: ܞൌቂ ቃെቂ ቃൌቂ ቃ ૛ ૚ ૚ ࢞ ૜ െ૜ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ ૚ ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૜െ૜࢚ and ࢟ሺ࢚ሻ ൌ૚൅࢚ for all real numbers ࢚

c. ࡼሺെ૛ǡ ૛ሻ, ࡽሺെ૜ǡ െ૝ሻ െ૜ െ૛ െ૚ Direction vector: ܞൌቂ ቃെቂ ቃൌቂ ቃ െ૝ ૛ െ૟ ࢞ െ૛ െ૚ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ ૛ െ૟ Parametric Equations: ࢞ሺ࢚ሻ ൌെ૛െ࢚ and ࢟ሺ࢚ሻ ൌ૛െ૟࢚ for all real numbers ࢚

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A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

2. Find parametric equations of ശሬሬሬሬԦࡼࡽ through points ࡼ and ࡽ in space. a. ࡼሺ૚ǡ ૙ǡ ૛ሻ, ࡽሺ૝ǡ ૜ǡ ૚ሻ ૝ ૚ ૜ Direction vector: ܞൌ൥૜൩ െ ൥૙൩ ൌ ൥ ૜ ൩ ૚ ૛ െ૚ ࢞ ૚ ૜ Vector equation: ቈ࢟቉ ൌ ൥૙൩ ൅ ൥ ૜ ൩ ࢚ for all real numbers ࢚ ࢠ ૛ െ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૚൅૜࢚, ࢟ሺ࢚ሻ ൌ૜࢚, and ࢠሺ࢚ሻ ൌ૛െ࢚ for all real numbers ࢚

b. ࡼሺ૜ǡ ૚ǡ ૛ሻ, ࡽሺ૛ǡ ૡǡ ૜ሻ ૛ ૜ െ૚ Direction vector: ܞൌ൥ૡ൩ െ ൥૚൩ ൌ ൥ ૠ ൩ ૜ ૛ ૚ ࢞ ૜ െ૚ Vector equation: ቈ࢟቉ ൌ ൥૚൩ ൅ ൥ ૠ ൩ ࢚ for all real numbers ࢚ ࢠ ૛ ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૜െ࢚, ࢟ሺ࢚ሻ ൌ૚൅ૠ࢚, and ࢠሺ࢚ሻ ൌ૛൅࢚ for all real numbers ࢚

c. ࡼሺ૚ǡ ૝ǡ ૙ሻ, ࡽሺെ૛ǡ ૚ǡ െ૚ሻ െ૛ ૚ െ૜ Direction vector: ܞൌ൥ ૚ ൩ െ ൥૝൩ ൌ ൥െ૜൩ െ૚ ૙ െ૚ ࢞ ૚ െ૜ Vector equation: ቈ࢟቉ ൌ ൥૝൩ ൅ ൥െ૜൩ ࢚ for all real numbers ࢚. ࢠ ૙ െ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૚െ૜࢚, ࢟ሺ࢚ሻ ൌ૝െ૜࢚, and ࢠሺ࢚ሻ ൌെ࢚ for all real numbers ࢚.

3. Find parametric equations of ࡼࡽതതതത through points ࡼ and ࡽ in the plane. a. ࡼሺ૛ǡ ૙ሻ, ࡽሺ૛ǡ ૚૙ሻ ૛ ૛ ૙ Direction vector: ܞൌቂ ቃെቂ ቃൌቂ ቃ ૚૙ ૙ ૚૙ ࢞ ૛ ૙ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for ૙൑࢚൑૚ ࢟ ૙ ૚૙ Parametric Equations: ࢞ሺ࢚ሻ ൌ૛ and ࢟ሺ࢚ሻ ൌ ૚૙࢚ for ૙൑࢚൑૚

b. ࡼሺ૚ǡ ૟ሻ, ࡽሺെ૜ǡ ૞ሻ െ૜ ૚ െ૝ Direction vector: ܞൌቂ ቃെቂ ቃൌቂ ቃ ૞ ૟ െ૚ ࢞ ૚ െ૝ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for ૙൑࢚൑૚ ࢟ ૟ െ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૚െ૝࢚ and ࢟ሺ࢚ሻ ൌ૟െ࢚ for ૙൑࢚൑૚

384 Lesson 22: Linear Transformations of Lines

A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

c. ࡼሺെ૛ǡ ૝ሻ, ࡽሺ૟ǡ ૢሻ ૟ െ૛ ૡ Direction vector: ܞൌቂ ቃെቂ ቃൌቂ ቃ ૢ ૝ ૞ ࢞ െ૛ ૡ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for ૙൑࢚൑૚ ࢟ ૝ ૞ Parametric Equations: ࢞ሺ࢚ሻ ൌെ૛൅ૡ࢚ and ࢟ሺ࢚ሻ ൌ૝൅૞࢚ for ૙൑࢚൑૚

4. Find parametric equations of ࡼࡽതതതത through points ࡼ and ࡽ in space. a. ࡼሺ૚ǡ ૚ǡ ૚ሻ, ࡽሺ૙ǡ ૙ǡ ૙ሻ ૙ ૚ െ૚ Direction vector: ܞൌ൥૙൩ െ ൥૚൩ ൌ ൥െ૚൩ ૙ ૚ െ૚ ࢞ ૚ െ૚ Vector equation: ቈ࢟቉ ൌ ൥૚൩ ൅ ൥െ૚൩ ࢚ for ૙൑࢚൑૚ ࢠ ૚ െ૚ Parametric Equations: ࢞ሺ࢚ሻ ൌ૚െ࢚, ࢟ሺ࢚ሻ ൌ૚െ࢚ and ࢠሺ࢚ሻ ൌ૚െ࢚ for ૙൑࢚൑૚

b. ࡼሺ૛ǡ ૚ǡ െ૜ሻ, ࡽሺ૚ǡ ૚ǡ ૝ሻ ૚ ૛ െ૚ Direction vector: ܞൌ൥૚൩ െ ൥ ૚ ൩ ൌ ൥ ૙ ൩ ૝ െ૜ ૠ ࢞ ૛ െ૚ Vector equation: ቈ࢟቉ ൌ ൥ ૚ ൩ ൅ ൥ ૙ ൩ ࢚ for ૙൑࢚൑૚ ࢠ െ૜ ૠ Parametric Equations: ࢞ሺ࢚ሻ ൌ૛െ࢚, ࢟ሺ࢚ሻ ൌ࢚ and ࢠሺ࢚ሻ ൌെ૜൅ૠ࢚ for ૙൑࢚൑૚

c. ࡼሺ૜ǡ ૛ǡ ૚ሻ, ࡽሺ૚ǡ ૛ǡ ૜ሻ ૚ ૜ െ૛ Direction vector: ܞൌ൥૛൩ െ ൥૛൩ ൌ ൥ ૙ ൩ ૜ ૚ ૛ ࢞ ૜ െ૛ Vector equation: ቈ࢟቉ ൌ ൥૛൩ ൅ ൥ ૙ ൩ ࢚ for ૙൑࢚൑૚ ࢠ ૚ ૛ Parametric Equations: ࢞ሺ࢚ሻ ൌ૜െ૛࢚, ࢟ሺ࢚ሻ ൌ૛, and ࢠሺ࢚ሻ ൌ૚൅૛࢚ for ૙൑࢚൑૚

5. Jeanine claims that the parametric equations ࢞ሺ࢚ሻ ൌ૜െ࢚ and ࢟ሺ࢚ሻ ൌ૝െ૜࢚ describe the line through points ࡼሺ૛ǡ ૚ሻ and ࡽሺ૜ǡ ૝ሻ. Is she correct? Explain how you know.

Yes, she is correct. If ࢚ൌ૚, then ࢞ሺ࢚ሻ ൌ૛ and ࢟ሺ࢚ሻ ൌ૚, so the line passes through point ࡼ. If ࢚ൌ૙, then ࢞ሺ࢚ሻ ൌ૜ and ࢟ሺ࢚ሻ ൌ૝, so the line passes through point ࡽ.

6. Kelvin claims that the parametric equations ࢞ሺ࢚ሻ ൌ૜൅࢚ and ࢟ሺ࢚ሻ ൌ૝൅૜࢚ describe the line through points ࡼሺ૛ǡ ૚ሻ and ࡽሺ૜ǡ ૝ሻ. Is he correct? Explain how you know.

Yes, he is correct. If ࢚ൌെ૚, then ࢞ሺ࢚ሻ ൌ૛ and ࢟ሺ࢚ሻ ൌ૚, so the line passes through point ࡼ. If ࢚ൌ૙, then ࢞ሺ࢚ሻ ൌ૜ and ࢟ሺ࢚ሻ ൌ૝, so the line passes through point ࡽ.

Lesson 22: Linear Transformations of Lines 385

A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

7. LeRoy claims that the parametric equations ࢞ሺ࢚ሻ ൌ૚൅૜࢚ and ࢟ሺ࢚ሻ ൌെ૛൅ૢ࢚ describe the line through points ࡼሺ૛ǡ ૚ሻ and ࡽሺ૜ǡ ૝ሻ. Is he correct? Explain how you know. ૚ ૛ ൌ ࢞ሺ࢚ሻ ൌ૛ ࢟ሺ࢚ሻ ൌ૚ ࡼ ൌ ࢞ሺ࢚ሻ ൌ૜ Yes, he is correct. If ૜ , then and , so the line passes through point . If ૜ , then and ࢟ሺ࢚ሻ ൌ૝, so the line passes through point ࡽ.

8. Miranda claims that the parametric equations ࢞ሺ࢚ሻ ൌെ૛൅૛࢚ and ࢟ሺ࢚ሻ ൌ૜െ࢚ describe the line through points ࡼሺ૛ǡ ૚ሻ and ࡽሺ૜ǡ ૝ሻ. Is she correct? Explain how you know.

No, she is not correct. If ࢚ൌ૛, then ࢞ሺ࢚ሻ ൌ૛ and ࢟ሺ࢚ሻ ൌ૚, so the line passes through point ࡼ. However, when we ૞ Ȃ૛൅૛࢚ൌ૜ ࢚ൌ ૜െ࢚ൌ૝ ࢚ൌെ૚ ࢚ solve , we find ૛, and when we solve , we find that . Thus, there is no value of so that ൫࢞ሺ࢚ሻǡ࢟ሺ࢚ሻ൯ ൌ ሺ૜ǡ ૝ሻ, so this line does not pass through point ࡽ.

࢞ ࢞ ശሬሬሬሬԦ 9. Find parametric equations of the image of ࡼࡽ under the transformation ࡸቀቂ࢟ቃቁ ൌ ࡭ ቂ࢟ቃ for the given points ࡼ, ࡽ, and matrix ࡭. ૚૜ a. ࡼሺ૛ǡ ૝ሻ, ࡽሺ૞ǡ െ૚ሻ, ࡭ൌቂ ቃ ૚૛ ૚૜૛ ૚૝ ૚૜ ૞ ૛ ૛ ૚૝ െ૚૛ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ and ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ so ܞൌቂ ቃെቂ ቃൌቂ ቃ ૚૛૝ ૚૙ ૚૛െ૚ ૜ ૜ ૚૙ െૠ ࢞ ૚૝ െ૚૛ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ ૚૙ െૠ Parametric equations: ࢞ሺ࢚ሻ ൌ ૚૝ െ ૚૛࢚ and ࢟ሺ࢚ሻ ൌ૚૙െૠ࢚ for all real numbers ࢚

૚૛ b. ࡼሺ૚ǡ െ૛ሻ, ࡽሺ૙ǡ ૙ሻ, ࡭ൌቂ ቃ െ૛ ૚ ૚૛૚ െ૜ ૚૛૙ ૙ ૙ െ૜ ૜ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ and ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ so ܞൌቂ ቃെቂ ቃൌቂ ቃ െ૛ ૚ െ૛ െ૝ െ૛ ૚ ૙ ૙ ૙ െ૝ ૝ ࢞ െ૜ ૜ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ െ૝ ૝ Parametric equations: ࢞ሺ࢚ሻ ൌെ૜൅૜࢚ and ࢟ሺ࢚ሻ ൌെ૝൅૝࢚ for all real numbers ࢚

૚૝ c. ࡼሺ૛ǡ ૜ሻ, ࡽሺ૚ǡ ૚૙ሻ, ࡭ൌቂ ቃ ૙૚ ૚૝૛ ૚૝ ૚૝ ૚ ૝૚ ૝૚ ૚૝ ૛ૠ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ and ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ so ܞൌቂ ቃെቂ ቃൌቂ ቃ ૙૚૜ ૜ ૙૚૚૙ ૚૙ ૚૙ ૜ ૠ ࢞ ૚૝ ૛ૠ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for all real numbers ࢚ ࢟ ૜ ૠ Parametric equations: ࢞ሺ࢚ሻ ൌ ૚૝ ൅ ૛ૠ࢚ and ࢟ሺ࢚ሻ ൌ૜൅ૠ࢚ for all real numbers ࢚

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A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

࢞ ࢞ 10. Find parametric equations of the image of ࡼࡽശሬሬሬሬԦ under the transformation ࡸ ቆቈ࢟቉ቇ ൌ࡭ቈ࢟቉ for the given points ࡼ, ࡽ, ࢠ ࢠ and matrix ࡭. ૛૚૙ a. ࡼሺ૚ǡ െ૛ǡ ૚ሻ, ࡽሺെ૚ǡ ૚ǡ ૜ሻ, ࡭ൌ൥૙૚૚൩ ૚૛૜ ૛૚૙ ૚ ૙ ૛૚૙െ૚ െ૚ െ૚ ૙ െ૚ ࡸሺࡼሻ ൌ ൥૙૚૚൩൥െ૛൩ ൌ ൥െ૚൩ and ࡸሺࡽሻ ൌ ൥૙૚૚൩൥ ૚ ൩ ൌ ൥ ૝ ൩ so ܞ ൌ ൥ ૝ ൩ െ ൥െ૚൩ ൌ ൥ ૞ ൩ ૚૛૜ ૚ ૙ ૚૛૜ ૜ ૚૙ ૚૙ ૙ ૚૙ ࢞ ૙ െ૚ Vector equation: ቈ࢟቉ ൌ ൥െ૚൩ ൅ ൥ ૞ ൩ ࢚ for all real numbers ࢚ ࢠ ૙ ૚૙ Parametric equations: ࢞ሺ࢚ሻ ൌെ࢚ and ࢟ሺ࢚ሻ ൌെ૚൅૞࢚ and ࢠሺ࢚ሻ ൌ ૚૙࢚ for all real numbers ࢚

૚૚૚ b. ࡼሺ૛ǡ ૚ǡ ૝ሻ, ࡽሺ૚ǡ െ૚ǡ െ૜ሻ, ࡭ൌ൥૚૚૛൩ ૚૙૚ ૚૚૚૛ ૠ ૚૚૚ ૚ െ૜ െ૜ ૠ െ૚૙ ࡸሺࡼሻ ൌ ൥૚૚૛൩൥૚൩ ൌ ൥૚૚൩ and ࡸሺࡽሻ ൌ ൥૚૚૛൩൥െ૚൩ ൌ ൥െ૟൩ so ܞ ൌ ൥െ૟൩ െ ൥૚૚൩ ൌ ൥െ૚ૠ൩ ૚૙૚૝ ૟ ૚૙૚െ૜ െ૛ െ૛ ૟ െૡ ࢞ ૠ െ૚૙ Vector equation: ቈ࢟቉ ൌ ൥૚૚൩ ൅ ൥െ૚ૠ൩ ࢚ for all real numbers ࢚ ࢠ ૟ െૡ Parametric equations: ࢞ሺ࢚ሻ ൌ ૠ െ ૚૙࢚ and ࢟ሺ࢚ሻ ൌ ૚૚ െ ૚ૠ࢚ and ࢠሺ࢚ሻ ൌ૟െૡ࢚ for all real numbers ࢚

૚૜૙ c. ࡼሺ૙ǡ ૙ǡ ૚ሻ, ࡽሺ૝ǡ ૛ǡ ૜ሻ, ࡭ൌ൥૚૚૚൩ ૙૛૚ ૚૜૙૙ ૙ ૚૜૙૝ ૚૙ ૚૙ ૙ ૢ ࡸሺࡼሻ ൌ ൥૚૚૚൩൥૙൩ ൌ ൥૚൩ and ࡸሺࡽሻ ൌ ൥૚૚૚൩൥૛൩ ൌ ൥ ૢ ൩ so ܞ ൌ ൥ ૢ ൩ െ ൥૚൩ ൌ ൥ૡ൩ ૙૛૚૚ ૚ ૙૛૚૜ ૠ ૠ ૚ ૟ ࢞ ૙ ૢ Vector equation: ቈ࢟቉ ൌ ൥૚൩ ൅ ൥ૡ൩ ࢚ for all real numbers ࢚ ࢠ ૚ ૟ Parametric equations: ࢞ሺ࢚ሻ ൌૢ࢚, ࢟ሺ࢚ሻ ൌ૚൅ૡ࢚ and ࢠሺ࢚ሻ ൌ૚൅૟࢚ for all real numbers ࢚

࢞ ࢞ തതതത 11. Find parametric equations of the image of ࡼࡽ under the transformation ࡸቀቂ࢟ቃቁ ൌ ࡭ ቂ࢟ቃ for the given points ࡼ, ࡽ, and matrix ࡭. ૚૜ a. ࡼሺ૛ǡ ૚ሻ, ࡽሺെ૚ǡ െ૚ሻ, ࡭ൌቂ ቃ ૚૛ ૚૜૛ ૞ ૚૜െ૚ െ૝ െ૝ ૞ െૢ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ and ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ so ܞ ൌቂ ቃെቂ ቃൌቂ ቃ ૚૛૚ ૝ ૚૛െ૚ െ૜ െ૜ ૝ െૠ ࢞ ૞ െૢ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for ૙൑࢚൑૚ ࢟ ૝ െૠ Parametric equations: ࢞ሺ࢚ሻ ൌ૞െૢ࢚ and ࢟ሺ࢚ሻ ൌ૝െૠ࢚ for ૙൑࢚൑૚

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A STORY OF FUNCTIONS Lesson 22 M2 PRECALCULUS AND ADVANCED TOPICS

૚૛ b. ࡼሺ૙ǡ ૙ሻ, ࡽሺ૝ǡ ૛ሻ, ࡭ൌቂ ቃ െ૛ ૚ ૚૛૙ ૙ ૚૛૝ ૡ ૡ ૙ ૡ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ and ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ so ܞ ൌቂ ቃെቂ ቃൌቂ ቃ െ૛ ૚ ૙ ૙ െ૛ ૚ ૛ െ૟ െ૟ ૙ െ૟ ࢞ ૙ ૡ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for ૙൑࢚൑૚ ࢟ ૙ െ૟ Parametric equations: ࢞ሺ࢚ሻ ൌૡ࢚ and ࢟ሺ࢚ሻ ൌ െ૟࢚ for ૙൑࢚൑૚

૚૝ c. ࡼሺ૜ǡ ૚ሻ, ࡽሺ૚ǡ െ૛ሻ, ࡭ൌቂ ቃ ૙૚ ૚૝૜ ૠ ૚૝ ૚ െૠ െૠ ૠ െ૚૝ ࡸሺࡼሻ ൌቂ ቃቂ ቃൌቂ ቃ and ࡸሺࡽሻ ൌቂ ቃቂ ቃൌቂ ቃ so ܞ ൌቂ ቃെቂ ቃൌቂ ቃ ૙૚૚ ૚ ૙૚െ૛ െ૛ െ૛ ૚ െ૜ ࢞ ૠ െ૚૝ Vector equation: ቂ ቃൌቂ ቃ൅ቂ ቃ࢚ for ૙൑࢚൑૚ ࢟ ૚ െ૜ Parametric equations: ࢞ሺ࢚ሻ ൌ ૠ െ ૚૝࢚ and ࢟ሺ࢚ሻ ൌ૚െ૜࢚ for ૙൑࢚൑૚

࢞ ࢞ 12. Find parametric equations of the image of തതതതࡼࡽ under the transformation ࡸ ቆቈ࢟቉ቇ ൌ࡭ቈ࢟቉ for the given points ࡼ, ࡽ, ࢠ ࢠ and matrix ࡭. ૛૚૙ a. ࡼሺ૙ǡ ૚ǡ ૚ሻ, ࡽሺെ૚ǡ ૚ǡ ૛ሻ, ࡭ൌ൥૙૚૚൩ ૚૛૜ ૛૚૙૙ ૚ ૛૚૙െ૚ െ૚ െ૚ ૚ െ૛ ࡸሺࡼሻ ൌ ൥૙૚૚൩൥૚൩ ൌ ൥૛൩ and ࡸሺࡽሻ ൌ ൥૙૚૚൩൥ ૚ ൩ ൌ ൥ ૜ ൩ so ܞ ൌ ൥ ૜ ൩ െ ൥૛൩ ൌ ൥ ૚ ൩ ૚૛૜૚ ૞ ૚૛૜ ૛ ૠ ૠ ૞ ૛ ࢞ ૚ െ૛ Vector equation: ቈ࢟቉ ൌ ൥૛൩ ൅ ൥ ૚ ൩ ࢚ for ૙൑࢚൑૚ ࢠ ૞ ૛ Parametric equations: ࢞ሺ࢚ሻ ൌ૚െ૛࢚, ࢟ሺ࢚ሻ ൌ૛൅࢚, and ࢠሺ࢚ሻ ൌ૞൅૛࢚ for ૙൑࢚൑૚

૚૚૚ b. ࡼሺ૛ǡ ૚ǡ ૚ሻ, ࡽሺ૚ǡ ૚ǡ ૛ሻ, ࡭ൌ൥૚૚૛൩ ૚૙૚ ૚૚૚૛ ૝ ૚૚૚૚ ૝ ૝ ૝ ૙ ࡸሺࡼሻ ൌ ൥૚૚૛൩൥૚൩ ൌ ൥૞൩ and ࡸሺࡽሻ ൌ ൥૚૚૛൩൥૚൩ ൌ ൥૟൩ so ܞ ൌ ൥૟൩ െ ൥૞൩ ൌ ൥૚൩ ૚૙૚૚ ૜ ૚૙૚૛ ૜ ૜ ૜ ૙ ࢞ ૝ ૙ Vector equation: ቈ࢟቉ ൌ ൥૞൩ ൅ ൥૚൩ ࢚ for ૙൑࢚൑૚ ࢠ ૜ ૙ Parametric equations: ࢞ሺ࢚ሻ ൌ૝, ࢟ሺ࢚ሻ ൌ૞൅࢚, and ࢠሺ࢚ሻ ൌ૜ for ૙൑࢚൑૚

૚૜૙ c. ࡼሺ૙ǡ ૙ǡ ૚ሻ, ࡽሺ૚ǡ ૙ǡ ૙ሻ, ࡭ൌ൥૚૚૚൩ ૙૛૚ ૚૜૙૙ ૙ ૚૜૙૚ ૚ ૚ ૙ ૚ ࡸሺࡼሻ ൌ ൥૚૚૚൩൥૙൩ ൌ ൥૚൩ and ࡸሺࡽሻ ൌ ൥૚૚૚൩൥૙൩ ൌ ൥૚൩ so ܞ ൌ ൥૚൩ െ ൥૚൩ ൌ ൥ ૙ ൩ ૙૛૚૚ ૚ ૙૛૚૙ ૙ ૙ ૚ െ૚ ࢞ ૙ ૚ Vector equation: ቈ࢟቉ ൌ ൥૚൩ ൅ ൥ ૙ ൩ ࢚ for ૙൑࢚൑૚ ࢠ ૚ െ૚ Parametric equations: ࢞ሺ࢚ሻ ൌ࢚, ࢟ሺ࢚ሻ ൌ૚, and ࢠሺ࢚ሻ ൌ૚െ࢚ for ૙൑࢚൑૚

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Lesson 23: Why Are Vectors Useful?

Student Outcomes Students solve problems involving physical phenomena that can be represented by vectors.

Lesson Notes Vectors are generally described as a quantity that has both a magnitude and a direction. In the next two lessons, students work on examples that give a context to that description. The most basic definition of a vector is that it is a description of a shift or translation. Students see that any physical operation that induces a shift of some kind is often thought of as a vector. Hence, vectors are prevalent in mathematics, science, and engineering. For example, force is often interpreted as a vector in physics because a force exerted on an object is a push of some magnitude that causes the object to shift in some direction. In this lesson, students solve problems involving velocity as well as other quantities, such as force, that can be represented by a vector. Students make use of the law of cosines and the law of sines when working with non-right triangles. Students continue to work on adding and subtracting vectors but interpret the resulting magnitude and direction within a context. The focus of this lesson is using vectors to model real-world phenomena, in particular focusing on relating abstract representations to real-world aspects.

Classwork Opening (5 minutes) Ask students to brainstorm real-world situations where vectors might be useful. Engage them by showing the vector video from NBC Learn on the science of NFL football (http://www.nbclearn.com/nfl/cuecard/51220). Remind students that vectors are used to describe anything that has both a direction and a magnitude. List these phenomena on the board, an interactive board, or index cards. Have students sort the phenomena into two categories—those that can be described by vectors and those that cannot—and then explain the rationale behind their sorting. Discuss student choices and rationales as a class. à Position of a moving object à Wind à Position of a ball that has been thrown à Temperature à Mass à Velocity of a ball that has been thrown à Volume à Water current Phenomena that could be described using a vector: à Wind, position of a ball that has been thrown, velocity of a ball that has been thrown, water current; they all have a direction and a magnitude.

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Phenomena that could not be described using a vector: à Temperature, mass, and volume; these are just measurements with no magnitude and direction. Why would the position of a moving object need to be described as a vector rather than a scalar? à Just knowing how far an object moved would not be enough information to locate it. You would also need to know the direction.

Opening Exercise (5 minutes) Give students time to work on the Opening Exercise independently, and then discuss as a group.

Opening Exercise

Suppose a person walking through South Boston, MA travels from point ࡭ due east along E 3rd Street for ૙Ǥ ૜ miles and then due south along K Street for ૙Ǥ ૝ miles to end on point ࡮ (as shown on the map).

a. Find the magnitude and direction of vector ࡭࡮ሬሬሬሬሬሬԦ.

ȁ࡭࡮ȁ ൌ ξ૙Ǥ ૜૛ ൅૙Ǥ૝૛ ൌ૙Ǥ૞ The magnitude is ૙Ǥ ૞ miles. ૙Ǥ૝ ࣂൌܜ܉ܖି૚ ቀ ቁൌ૞૜Ǥ૚ι ૙Ǥ૜

b. What information does vector ࡭࡮ሬሬሬሬሬሬԦ provide?

The magnitude tells us that the person’s displacement was ૙Ǥ ૞ miles. In other words, the person ended up ૙Ǥ ૞ miles from the point where he or she started. The angle tells us that the person traveled ૞૜Ǥ ૚ι south of east.

Why is a vector a useful way to map a person’s location? à It tells us the actual distance between the starting and ending point and also the direction in which the person traveled. If I knew just the direction, could I map the person’s position? à No. We could draw an arrow in the correct direction but would not know where to put the endpoint. If I knew just the magnitude, could I map the person’s position? à No. The person could have traveled in any direction.

Example (10 minutes)

Discuss the example as a class before giving students time to work on solving the problem. Provide students with grid paper as needed. Debrief as a class, and guide students as necessary.

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Example

An airplane flying from Dallas-Fort Worth to Atlanta veers off course to avoid a storm. The plane leaves Dallas-Fort Worth traveling ૞૙ι east of north and flies for ૝૞૙ miles before turning to travel ૠ૙ι east of south for ૜૞૙ miles. What is the resultant displacement of the airplane? Include both the magnitude and direction of the displacement.

࢞૚ ൌ ૝૞૙ ܋ܗܛሺ૝૙ιሻ ൌ ૜૝૝Ǥ ૠ૛૙

࢟૚ ൌ ૝૞૙ ܛܑܖሺ૝૙ιሻ ൌ ૛ૡૢǤ ૛૞૝

࢞૛ ൌ ૜૝૝Ǥ ૠ૛૙ ൅ ૜૛ૡǤ ૡૢ૛ ൌ ૟ૠ૜Ǥ ૟૚૛

࢟૛ ൌ ૛ૡૢǤ ૛૞૝ െ ૚૚ૢǤ ૠ૙ૠ ൌ ૚૟ૢǤ ૞૝ૠ

ࢊൌξ૟ૠ૜Ǥ ૟૚૛૛ ൅ ૚૟ૢǤ ૞૝ૠ૛ ൌ ૟ૢ૝Ǥ ૟૛૛ ૚૟ૢǤ૞૝ૠ ࣂൌܜ܉ܖ-1 ቀ ቁൌ૚૝Ǥ૚ι ૟ૠ૜Ǥ૟૚૛

The resultant displacement is ૟ૢ૝Ǥ ૟૛૛ miles at ૠ૞Ǥ ૢι east of north.

Exercises (17 minutes) Allow students time to work in groups, stopping to debrief as students complete the exercises.

Exercises ࢞ሺ࢚ሻ 1. A motorized robot moves across the coordinate plane. Its position ൬ ൰ at time ࢚ seconds ࢟ሺ࢚ሻ Scaffolding: ࢞ሺ࢚ሻ ૝ െ૝ is given by ൬ ൰ൌ܉൅࢚ܞ where ܉ൌቀ ቁ and ൌቀ ቁ. The units of distance are ࢟ሺ࢚ሻ െ૚૙ ૜ For students who are measured in meters. struggling, use a graphing a. Where is the robot at time ࢚ൌ૙? utility to reinforce the concept. ૝ Have them graph ቀ ቁ െ૚૙ ݔሺݐሻ ൌͶെͶݐ and ݕሺݐሻ ൌെͳͲ൅͵ݐ in parametric mode on the graphing calculator.

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b. Plot the path of the robot.

c. Describe the path of the robot. ૜ ࢟ൌെ ࢞െૠ The robot is moving along the line ૝ .

d. Where is the robot ૚૙ seconds after it starts moving? െ૜૟ ቀ ቁ ૛૙

e. Where is the robot when it is ૚૙ meters from where it started? െ૝ ቀ ቁ െ૝

f. Is the robot traveling at a constant speed? Explain, and if the speed is constant, state the robot’s speed.

Yes, the robot is traveling at a constant speed of ૞ܕȀܛ. For every one second that elapses, the robot moves ૜ units up and ૝ units to the left, which means the robot moves ૞ meters every second.

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What is the speed of the robot? à The speed is ͷȀ. How did you determine the speed? à I knew the robot moved ͵ units up and Ͷ units to the left each second. I used the Pythagorean theorem to find the actual distance the robot traveled. What is the relationship between the speed and vector ܞ? à The magnitude of a vector is its length. The length can be determined by the product of the constant speed for a given number of time units.

Make the point that vector ܞ is the velocity of the robot, and the magnitude of the velocity vector is the speed of the robot. Velocity is a vector because it has both a magnitude and a direction. Speed and time results in the magnitude of a vector. How could we interpret vector ܞ in terms of the velocity of the robot? à Vector ܞ is the velocity of the robot. It tells us both the speed and the direction of the robot. How could we interpret vector ܞ in terms of the speed of the robot? à The magnitude of vector ܞ is the distance traveled or the length of the vector. It tells us how fast the robot is moving but not the direction in which the robot moves.

2. A rowboat is crossing a river that is ૞૙૙ܕ wide and traveling due east at a speed of ૛Ǥ ૛ܕȀܛ. The river’s current is ૙Ǥ ૡܕȀܛ due south. a. What is the resultant velocity of the boat?

ȁܞȁ ൌ ξ૛Ǥ ૛૛ ൅૙Ǥૡ૛ ൌ૛Ǥ૜૝ The velocity is ૛Ǥ ૜૝ܕȀܛ. ૙Ǥૡ ࣂൌܜ܉ܖି૚ ቀ ቁൌ૛૙Ǥ૙ι ૛Ǥ૛

b. How long does it take for the boat to cross the river? ૞૙૙ ࢚ൌ ൌ ૛૛ૠǤ ૛ૠ૜ ૛Ǥ૛ The time is ૛૛ૠǤ ૛ૠ૜ seconds.

c. How far downstream is the boat when it reaches the other side?

ࢊൌ૙Ǥૡሺ૛૛ૠǤ ૛ૠ૜ሻ ൌ ૚ૡ૚Ǥ ૡ૚ૡܕ The distance is ૚ૡ૚Ǥ ૡ૚ૡ meters.

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3. Consider the airplane from the Example that leaves Dallas-Fort Worth with a bearing of ૞૙ι. (Note that the bearing is the number of degrees east of north.) The plane is traveling at a speed of ૞૞૙ܕܘܐ. There is a crosswind of ૝૙ܕܘܐ due east. What is the resultant velocity of the airplane?

ȁܞȁ ൌ ඥ૞૞૙૛ ൅૝૙૛ െ૛ሺ૞૞૙ሻሺ૝૙ሻ܋ܗܛሺ૚૝૙ιሻ ൌ ૞ૡ૚Ǥ ૛૚૚

The velocity is ૞ૡ૚Ǥ ૛૚૚ܕܘܐ.

Another physical operation often interpreted in physics as a vector is force. A force exerted on an object is a push of some magnitude in some direction that causes the object to have a tendency to shift. Force is often defined in units of newtons (N). One newton is defined as the force required to accelerate an object with a mass of one kilogram (ͳ) one meter per second (ͳȀ ଶ).

due to the ۼA raft floating in the water experiences an eastward force of ૚૙૙ .4 .due to wind ۼcurrent of the water and a southeast force of ૝૙૙ a. In what direction will the boat move?

૜૟Ǥ ૞ι south of east

b. What is the magnitude of the resultant force on the boat?

ۼ૝ૠ૞Ǥ ૢૢ૛

c. If the force due to the wind doubles, does the resultant force on the boat double? Explain or show work that supports your answer.

.ۼNo. If the force of the wind doubles, the resultant force on the boat is ૡૠ૜Ǥ ૞ૠૠ

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Closing (3 minutes) Ask students to write a brief answer to the question, “Why are vectors useful?” and share responses with a partner. Then, share responses as a class. Why are vectors useful? à Vectors can be used to describe any sort of physical phenomena that have both a magnitude and a direction. They are useful for describing a moving object’s displacement or velocity where just a single number would not provide an adequate description.

Exit Ticket (5 minutes)

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Name Date

Lesson 23: Why Are Vectors Useful?

Exit Ticket

͵ ݔሺݐሻ ݔሺݐሻ Ͳ A hailstone is traveling through the sky. Its position ቌݕሺݐሻቍ in meters is given by ቌݕሺݐሻቍൌ൭ Ͳ ൱൅൭െʹ൱ݐ where ݐ ݖሺݐሻ ݖሺݐሻ ʹͳ͸Ͳ െͻ is the time in seconds since the hailstone began being tracked. ?a. If ݔሺݐሻ represents an east-west location, how quickly is the hailstone moving to the east

?b. If ݕሺݐሻ represents a north-south location, how quickly is the hailstone moving to the south

c. What could be causing the east-west and north-south velocities for the hailstone?

?d. If ݖሺݐሻ represents the height of the hailstone, how quickly is the hailstone falling

?e. At what location will the hailstone hit the ground (assume ݖൌͲ is the ground)? How long will this take

f. What is the overall speed of the hailstone? To the nearest meter, how far did the hailstone travel from ݐൌͲ to when it hit the ground?

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Exit Ticket Sample Solutions

࢞ሺ࢚ሻ ࢞ሺ࢚ሻ ૙ ૜ A hailstone is traveling through the sky. Its position ቌ࢟ሺ࢚ሻቍ in meters is given by ቌ࢟ሺ࢚ሻቍ ൌ൭ ૙ ൱൅൭െ૛൱࢚ where ࢚ ࢠሺ࢚ሻ ࢠሺ࢚ሻ ૛૚૟૙ െૢ is the time in seconds since the hailstone began being tracked.

a. If ࢞ሺ࢚ሻ represents its east-west location, how quickly is the hailstone moving to the east?

૜ܕȀܛ

b. If ࢟ሺ࢚ሻ represents its north-south location, how quickly is the hailstone moving to the south?

૛ܕȀܛ

c. What could be causing the east-west and north-south velocities for the hailstone?

Wind is probably the number one factor affecting the speed of the hailstone.

d. If ࢠሺ࢚ሻ represents the height of the hailstone, how quickly is the hailstone falling?

ૢܕȀܛ

e. At what location will the hailstone hit the ground (assume ࢠൌ૙ is the ground)? How long will this take?

࢚ ൌ ૛૚૟૙ ൊ ૢ ൌ ૛૝૙ The hailstone will hit the ground at ૛૝૙ seconds, which is ૝ minutes. In ૝ minutes it will move ૠ૛૙ meters to ૠ૛૙ the east and ૝ૡ૙ meters to the south to impact at ൭૝ૡ૙൱. ૙

f. What is the overall speed of the hailstone? To the nearest meter, how far did the hailstone travel from ࢚ൌ૙ to when it hit the ground?

ξ૜૛ ൅૛૛ ൅ૢ૛ ൌ ξૢ૝

૛૝૙ξૢ૝ ൎ ૛૜૛ૠ

The hailstone is traveling at ξૢ૝ meters per second. It traveled approximately ૛૜૛ૠ meters.

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Problem Set Sample Solutions

1. Suppose Madison is traveling due west for ૙Ǥ ૞ miles and then due south for ૚Ǥ ૛ miles. a. Draw a picture of this scenario with her starting point labeled ࡭, ending point ࡮, and include the vector ࡭࡮ሬሬሬሬሬሬԦ.

b. State the value of ࡭࡮ሬሬሬሬሬሬԦ. െ૙Ǥ ૞ ࡭࡮ሬሬሬሬሬሬԦ ൌቀ ቁ െ૚Ǥ ૛

c. What is the magnitude and direction of ሬሬሬሬሬሬԦ࡭࡮? െ૙Ǥ ૞ ቛቀ ቁቛ ൌ ඥ૙Ǥ ૞૛ ൅૚Ǥ૛૛ ൌ ξ૚Ǥ ૟ૢ ൌ૚Ǥ૜ െ૚Ǥ ૛ ૚૛ ૚ ൬ ൰ ൎ ૟ૠǤ ૜ૡ૙ι ֜ ૛૝ૠǤ ૜ૡ૙ιିܖ܉ܜ ૞ The vector has a magnitude of ૚Ǥ ૜ and a direction of ૛૝ૠǤ ૜ૡ૙ι from east, or ૟ૠǤ ૜ૡ૙ι south of west.

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2. An object’s azimuth is the angle of rotation of its path measured clockwise from due north. For instance, an object traveling due north would have an azimuth of ૙ι, and due east would have an azimuth of ૢ૙ι. a. What are the azimuths for due south and due west?

૚ૡ૙ι and ૛ૠ૙ι

b. Consider a craft on an azimuth of ૛૚૞ι traveling ૜૙ knots. i. Draw a picture representing the situation.

ii. Find the vector representing this craft’s speed and direction.

This vector is ૛૚૞ι from north, which is the same as saying ૞૞ι south of west. Thus, the ࢞-coordinate of the vector will be ૜૙܋ܗܛሺ૞૞ιሻ ൎ ૚ૠǤ ૛૙ૠ, and the ࢟-coordinate will be ૜૙ܛܑܖሺ૞૞ιሻ ൎ૛૝Ǥ૞ૠ૞. The െ૚ૠǤ ૛૙ૠ vector is represented by ቀ ቁ. െ૛૝Ǥ ૞ૠ૞

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3. Bearings can be given from any direction, not just due north. For bearings, like azimuths, clockwise angles are represented by positive degrees and counterclockwise angles are represented by negative degrees. A ship is traveling ૜૙ι east of north at ૚ૡܓܖ, then turns ૛૙ι, maintaining its speed. a. Draw a picture representing the situation.

b. Find vectors ܞ and ܟ representing the first and second bearing.

ૢ ૢ ૚ૡܛܑܖሺ૞૙ιሻ ૚૜Ǥ ૠૡૢ ܞൌ൬ ൰ൎቀ ቁ,ܟ ൌ ൬ ൰ൎቀ ቁ ૢξ૜ ૚૞Ǥ ૞ૡૡ ૚ૡ܋ܗܛሺ૞૙ιሻ ૚૚Ǥ ૞ૠ૙

c. Find the sum of ܞ and ܟ. What does ܞ൅ܟ represent? ૛૛Ǥ ૠૡૢ ܞ൅ܟൎቀ ቁ ૛ૠǤ ૚૞ૢ The sum represents the ship’s position relative to its starting point.

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d. If the ship travels for one hour along each bearing, then how far north of its starting position has it traveled? How far east has it traveled?

The ship has traveled ૛૛Ǥ ૠૡૢnautical miles north and ૛ૠǤ ૚૞ૢnautical miles east.

૝ 4. A turtle starts out on a grid with coordinates ቀ ቁ where each unit is one furlong. Its horizontal location is given െ૟Ǥ ૞ by the function ࢞ሺ࢚ሻ ൌ ૝ ൅ െ૛࢚, and its vertical location is given by ࢟ሺ࢚ሻ ൌെ૟Ǥ૞൅૜࢚ for ࢚ in hours. a. Write the turtle’s location using vectors.

࢞ሺ࢚ሻ ૝ െ૛ ൬ ൰ൌቀ ቁ൅ቀ ቁ࢚ ࢟ሺ࢚ሻ െ૟Ǥ ૞ ૜

b. What is the speed of the turtle?

ξ૛૛ ൅૜૛ ൌ ξ૚૜

The turtle is traveling at ξ૚૜furlongs per hour.

࢞ ሺ࢚ሻ ૛૜ െૡ c. If a hare’s location is given as ൬ ࢎ ൰ൌ൅࢚ where ൌቀ ቁ and ൌቀ ቁ, then what is the speed of ࢟ࢎሺ࢚ሻ െ૜૞ ૚૛ the hare? How much faster is the hare traveling than the turtle?

ξૡ૛ ൅૚૛૛ ൌ ξ૛૙ૡ

ξ૛૙ૡ ൌ૝ξ૚૜

The hare is traveling at a speed of ξ૛૙ૡfurlongs per hour, which is ૝ times faster than the turtle.

െ૚ d. Which creature will reach ቀ ቁ first? ૚ െ૚ The turtle will reach ቀ ቁ at ࢚ൌ૛Ǥ૞, and the hare will reach it at ࢚ൌ૜, so the turtle will beat the hare by ૚ half an hour.

5. A rocket is launched at an angle of ૜૜ι from the ground at a rate of ૞૙ܕȀܛ. a. How fast is the rocket traveling up, to the nearest m/s?

ሺ૜૜ιሻ ൎ૛ૠܖܑܛ ڄ ૞૙ The rocket is traveling up at a rate of approximately ૛ૠܕȀܛ.

b. How fast is the rocket traveling to the right, to the nearest m/s?

ሺ૜૜ιሻ ൎ૝૛ܛܗ܋ ڄ ૞૙ The rocket is traveling to the right at a rate of approximately ૝૛ܕȀܛ.

c. What is the rocket’s velocity vector? ૝૛ The velocity vector is ቀ ቁ. ૛ૠ

d. Does the magnitude of the velocity vector agree with the setup of the problem? Why or why not? ૝૛ ቛቀ ቁቛ ൌ ξ૝૛૛ ൅૛ૠ૛ ൌ ξ૛૝ૢ૜ ൎ ૝ૢǤ ૢ૛ૢૢ૞ ૛ૠ This is effectively the original speed and is only different because of rounding errors.

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e. If a laser is in the path of the rocket and would like to strike the rocket, in what direction should the laser be aimed? Express your answer as a vector.

If the laser is in the path of the rocket, then it should be aimed in the opposite direction the rocket is traveling, െ૝૛ which is ቀ ቁ. െ૛ૠ

6. A boat is drifting downriver at a rate of ૞nautical miles per hour. If the occupants of the boat want to travel to shore, do they need to overcome the current downriver? Use vectors to explain why or why not.

The occupants do not need to overcome the current since the boat is moving downriver, and they only want to move perpendicular to the direction the stream is moving. Since they will be moving perpendicular, their motions will have no impact on the downriver rate, nor should they. Any effort fighting the current only wastes resources. Orthogonal vectors do not affect the position.

7. A group of friends moored their boats together and fell asleep on the lake. Unfortunately, their lashings came undone in the night, and they drifted apart. Gerald’s boat traveled due west along with the current of the lake ૚ which moves at a rate of ܕܑȀܐܚ, and Helena’s boat was pulled southeast by some pranksters and set drifting at a ૛ rate of ૛ܕܑȀܐܚ. a. If the boats came untied three hours ago, how far apart are the boats?

ሺ૚૜૞ιሻ ൌ ඥ૛Ǥ ૛૞ ൅ ૜૟ ൅ ૢξ૛ ൎૠǤ૚૜ܛܗ܋ڄ૟ڄ૚Ǥ૞ڄඥሺ૚Ǥ ૞ሻ૛ ൅૟૛ െ૛ Gerald traveled ૚Ǥ ૞miles to the west, and Helena traveled ૟miles to the southeast. Using the law of cosines, we see that they are approximately ૠǤ ૚૜miles away from each other.

b. If Gerald drops anchor, then in what direction should Helena travel in order to reunite with Gerald?

૚Ǥ૞൅૜ξ૛ ൬ ൰ ૜ξ૛

8. Consider any two vectors in space, ܝ and ܞ with ࣂ the angle between them. a. Use the law of cosines to find the value of ԡ െԡ.

ԡܝെܞԡ૛ ൌ ԡܝԡ૛ ൅ ԡܞԡ૛ െ૛ԡܝԡԡܞԡ܋ܗܛሺࣂሻ

ԡܝെܞԡ ൌඥԡܝԡ૛ ൅ ԡܞԡ૛ െ૛ԡܝԡԡܞԡ܋ܗܛሺࣂሻ

b. Use the law of sines to find the value of ࣒, the angle between ܝെܞ and ܝ. State any restrictions on the variables. ܛܑܖሺࣂሻ ܛܑܖሺ࣒ሻ ൌ ԡܝെܞԡ ԡܞԡ ԡܞԡܛܑܖሺࣂሻ ࣒ൌܛܑܖି૚ ቆ ቇ ԡܝെܞԡ

The inverse sine only returns values between െૢ૙ι and ૢ૙ι, so if the angle is greater than this, then trig identities should be used to help find the value.

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Student Outcomes Students apply linear transformations and vectors to understand the conditions required for a sequence of transformations to preserve the solution set to the system of equations.

Lesson Notes Vectors are useful for representing systems of equations. Vectors (and matrices) are essential to the representation of systems of equations in higher dimensions that students encounter in more advanced mathematics classes. In this lesson, students work with systems of linear equations in Թଶ, but the ideas can be extended to higher-order linear systems. Vectors are extremely useful in a wide variety of settings as a means of representing and manipulating geometric objects as well as real-world quantities. In this lesson, students see how vectors can be applied to solving a system of equations in two variables by exploring the geometry of solving systems of equations. Students are guided to the understanding (using pictures only) that the method of transforming systems of linear equations while preserving the solution can be rephrased in terms of a series of linear transformations and translations. Again, students write vectors as parametric equations to understand the coherence between their work with functions, linear transformations, and vectors. The work in this lesson helps set the stage for further application of vectors and matrices in the last three lessons of this module.

Classwork Opening Exercise (7 minutes) Give students time to work on the problem individually, and then compare work with a partner. Encourage students to analyze the problem in a variety of ways: algebraically, graphically, and numerically. A graphing utility could also be used.

Opening Exercise ૚ െ૛ Two particles are moving in a coordinate plane. Particle 1 is at the point ቀ ቁ and moving along the velocity vector ቀ ቁ. ૚ ૚ െ૚ ૚ Particle 2 is at the point ቀ ቁ and moving along the velocity vector ቀ ቁ. Are the two particles going to collide? If so, at ૚ ૛ what point, and at what time? Assume that time is measured in seconds.

࢞ሺ࢚ሻ ૚ െ૛ ૜ ૚ Particle 1: ൬ ൰ൌቀ ቁ൅ቀ ቁ࢚ ࢞ሺ࢚ሻ ൌ૚െ૛࢚, and ࢟ሺ࢚ሻ ൌ૚൅࢚ ࢟ൌ െ ࢞ ࢟ሺ࢚ሻ ૚ ૚ ૛ ૛ ࢞ሺ࢚ሻ െ૚ ૚ Particle 2: ൬ ൰ൌቀ ቁ൅ቀ ቁ࢚ ࢞ሺ࢚ሻ ൌെ૚൅࢚, and ࢟ሺ࢚ሻ ൌ૚൅૛࢚ ࢟ൌ૜൅૛࢞ ࢟ሺ࢚ሻ ૚ ૛ െ૙Ǥ ૟ The particles will not collide. While they do cross the same point at ቀ ቁ, they do not cross this point at the same time. ૚Ǥ ૡ Particle 1 crosses this point at ࢚ൌ૙Ǥ૝, and Particle 2 does not cross this point until ࢚ൌ૙Ǥૡ.

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The two lines intersect at ሺെͲǤ͸ǡͳǤͺሻ. Why do the particles not collide at that point? à They do not collide because the two particles reach that point at two different times. ͵ ͳ .Particle 1 travels along the line ݕൌ െ ݔ, and Particle 2 travels along the line ݕൌ͵൅ʹݔ ʹ ʹ The graphs of these lines intersect at the point ሺെͲǤ͸ǡ ͳǤͺሻ as shown below. The blue portions of the graph show the .͵ parametric equations that represent the path of each particle on the interval െ͵ ൑ ݐ ൑

To show this result dynamically, enter the parametric equations for each particle on a graphing calculator or other graphing software, and set the ݐ-interval to be െ͵ ൑ ݐ ൑ ͵. Be sure the calculator is set to graph the two equations .simultaneously. Then, it can be seen that the particles do not arrive at the intersection point at the same ݐ-value The table below demonstrates numerically that two particles do not pass through the intersection point at the same time. The intersection point of the graphs of the lines occurs when ݕൌͳǤͺ, thus the two particles will never be at that .ݕ-value at the same time

ݐ ݔଵሺݐሻ ൌͳെʹݐ ݕଵሺݐሻ ൌͳ൅ݐ ݔଶሺݐሻ ൌെͳ൅ݐ ݕଶሺݐሻ ൌͳ൅ʹݐ െ͵ ͹ െʹ െͶ െͷ െʹ ͷ െͳ െ͵ െ͵ െͳ ͵ Ͳ െʹ െͳ Ͳ ͳ ͳ െͳ ͳ ͳ െͳ ʹ Ͳ ͵ ʹ െ͵ ͵ ͳ ͷ ͵ െͷ Ͷ ʹ ͹

ଷ ଵ .This system can be solved algebraically by solving the equation െ ݔൌʹݔ൅͵ for ݔ. The solution is െͲǤ͸ ଶ ଶ :Substituting െͲǤ͸ for ݔ into the parametric equations and solving for ݐ gives the following solutions െʹݐൌെͲǤ͸ʹ ͵ݐൌͳǤ and െͳ ൅ ݐ ൌ െͲǤ͸ ݐൌͲǤͶ

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You can see that these particles do not reach the ݔ-coordinate of the intersection point at the same time and therefore will not collide. Have students present their solution approaches to the class, highlighting different approaches.

Exercise 1 (5 minutes) Have students continue to work with a partner through Exercise 1. Then, debrief as a class, and hold the discussion that follows. This exercise represents lines using vectors as shown in previous lessons. If students struggle to make sense of part (a), encourage them to write the parametric equations for each line, and then eliminate the ݐ parameter to write .the equation of the line in terms of the variables ݔ and ݕ

Exercise 1

.ሽۄെ૚ǡ ૜ۃൌ࢚ ۄ࢞ǡ ࢟ۃሽ and ࢓ൌሼሺ࢞ǡ ࢟ሻȁۄ૚ǡ െ૛ۃൌ࢚ ۄ࢞ǡ ࢟ۃConsider lines रൌሼሺ࢞ǡ ࢟ሻȁ

a. To what graph does each line correspond?

Line र corresponds to the graph of ࢟ ൌ െ૛࢞. Line ࢓ corresponds to the graph of ࢟ ൌ െ૜࢞.

૜૚ b. Describe what happens to the vectors defining these lines under the transformation ࡭ൌቀ ቁ. ૛૚ .ۄ૙ǡ ૚ۃ ՜ ۄെ૚ǡ ૜ۃ and ,ۄ૚ǡ ૙ۃ ՜ ۄ૚ǡ െ૛ۃ .Each vector is mapped to a new vector

Students may need to be reminded that to apply the transformation ܣ, they must compute ܣήܞ, where ܞ is the vector that defines each line. ͵ͳ ͳ ͳ ቀ ቁήቀ ቁൌቀ ቁ Scaffolding: ʹͳ െʹ Ͳ Challenge advanced students and to come up with their own ͵ͳ െͳ Ͳ ቀ ቁήቀ ቁൌቀ ቁ example that illustrates the ʹͳ ͵ ͳ point that a linear transformation will preserve c. Show this transformation graphically. the solution set for lines that pass through the origin. and the vector ,ۄ૚ǡ െ૛ۃൌܝ The vector that defined र shown in the diagram is ,In the diagram, the image For struggling students .ۄെ૚ǡ ૜ۃൌܞ that defined ࢓ shown in the diagram is of र, when transformed by ࡭, results in the graph of the line ࢟ൌ૙,’ and the provide another example image of ࢓, when transformed by ࡭, results in the graph of the line ࢞ൌ૙. showing that if the solution set is the origin, it will still be the origin after a transformation. .ۄͳǡͶۃand ݉ൌ ۄǡͳʹۃLet κൌ Use a transformation of ͳͶ ܣൌቀ ቁ. Ͳെͳ ۄ͸ǡ െͳۃ ՜ ۄǡͳʹۃ ۄͳ͹ǡ െͶۃ ՜ ۄͳǡͶۃ and The solution is still ሺͲǡͲሻ.

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Discussion (5 minutes) What is the solution to the original system of equations given by κ and ݉? à The origin ሺͲǡͲሻ What is the solution to the system after the transformation given by ܣκ and ܣ݉ǫ à The solution is still the origin. Can we say that linear transformations preserve the solution set to a system of linear equations? à It seems this is true for lines that pass through the origin. What happens if we apply a linear transformation to two lines that do not intersect at the origin? Do you think the solution set of the system of equations is still preserved? à Answers will vary. The solution set may be preserved because the lines are simply rotated about the intersection point. Another conjecture would be that the solution set may not be preserved because the transformation rotates the lines about the origin, which moves the intersection point.

Exercise 2 (5 minutes) This exercise has students test their conjectures by having them consider a system of equations where the graphs of the two lines do not intersect at the origin. Have students continue to work with a partner through Exercise 2. Then, debrief as a class, and continue the discussion. Use the questions below to help students begin to think about the applied transformations. How does line κ in this exercise compare to line κ from Exercise 1? How does line ݉ in this exercise compare to line ݉ in Exercise 1? à The lines have the same slope but a different initial point. The new lines κ and ݉ have the same point .ሺͳǡͳሻ when ݐൌͲ Describe these lines as transformations of the lines from Exercise 1. ͳ à Both lines in Exercise 2 are a translation of the original lines by the vector ቀ ቁ. ͳ Knowing the solution to Exercise 1, how can you quickly find the solution to the system of equations represented by the lines in this exercise? à Since the solution to the original system in Exercise 1 is ሺͲǡͲሻ, when the two lines are both translated by ͳ the vector ቀ ቁ, all points on the graph of the lines shift ͳ unit right and ͳ unit up; therefore, the ͳ intersection point of the graphs of the lines that is the solution to the system becomes ሺͳǡͳሻ.

Exercise 2

.ሽۄെ૚ǡ ૜ۃ൅࢚ ۄ૚ǡ ૚ۃ ൌ ۄ࢞ǡ ࢟ۃሽ and ࢓ൌሼሺ࢞ǡ ࢟ሻȁۄ૚ǡ െ૛ۃ൅࢚ ۄ૚ǡ ૚ۃ ൌ ۄ࢞ǡ ࢟ۃConsider lines रൌሼሺ࢞ǡ ࢟ሻȁ

a. What is the solution to the system of equations given by lines र and ࢓?

The solution isሺ૚ǡ ૚ሻ.

૜૚ b. Describe what happens to the lines under the transformation ࡭ൌቀ ቁ. ૛૚ .ሽۄ૚ǡ ૙ۃ൅࢚ ۄ૝ǡ ૜ۃ ൌ ۄ࢞ǡ ࢟ۃLine र becomes ࡭र ൌ ሼሺ࢞ǡ ࢟ሻȁ

.ሽۄ૙ǡ ૚ۃ൅࢚ ۄ૝ǡ ૜ۃ ൌ ۄ࢞ǡ ࢟ۃLine ࢓ becomes ࡭࢓ ൌ ሼሺ࢞ǡ ࢟ሻȁ

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c. What is the solution to the system of equations after the transformation?

The solution is the point ሺ૝ǡ ૜ሻ.

Discussion (10 minutes) What happened to the solution to the system when we applied a linear transformation to two lines that did not intersect at the origin? à The solution to the system changed. Why did the solution change when we applied the transformation ܣ to this system? à The linear transformation ܣ not only transforms the velocity vectors for each line, but it also transforms ͳ the position vector ቀ ቁ. ͳ Can we say that linear transformations will preserve the solution set to a system of linear equations? à If the lines intersect at a point other than the origin, then the solution set is NOT preserved. In Exercise 2, the solution set changed after the linear transformation. ͳ In Exercise 2, we translated each linear equation by the vector ቀ ቁ to put the solution at ሺͳǡͳሻ. Could we ͳ translate any system of linear equations so that the intersection point of the graphs of the lines is ሺͲǡͲሻ? Explain your reasoning. à The coordinates of the intersection point are the terminal point of a position vector. If we translate the linear equations by the opposite of this vector, the intersection point is ሺͲǡͲሻ. What if I translated a system of equations to the origin before applying a linear transformation to the system? Explain your reasoning. à Then I would know that the transformation would not change the solution to the system. (The solution would be the origin.) The diagrams below can show the sequence of transformations that preserve the solution set to a system of equations. As the discussion continues, draw the pictures below on the board. Do not be concerned with using this as an algebraic means for solving a system of equations. What should be emphasized are the ideas behind performing these transformations, which have some important ramifications in future math courses. Some of these ideas are explored further in the Problem Set.

Translate the system to the origin ܶି௉.

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What happens to the solution of this new system when we apply a linear transformation? à If the intersection point of the graphs of the lines is ሺͲǡͲሻ, then the solution to the system does not change when we apply a linear transformation. Further, there exists a matrix that takes the vectors .that define each line to the ݔ- and ݕ-axes when we apply the linear transformation But this is not the solution to the original system of equations. What would we need to do next to find the solution? à We would need to reverse the translation of the transformed system. How can we transform this system into a system with the same solution set as the original one? à We can translate the system by the original vector that represented the solution to the system. The diagram below shows the result of applying the linear transformation ܣ to the new system of equations whose graphs intersect at the origin and then the subsequent translation ܶ௉ to return the intersection point back to the original intersection point.

Then, translate by ܶ௉ to return the system to the original intersection point.

Apply ܣ to ܶି௣ to take the lines to the .ݔ- and ݕ-axes

Exercise 3 (5 minutes) The next exercise lets students practice this approach with a fairly simple system of linear equations. The lines are at right angles to one another, so the linear transformation that maps the translated system to the ݔ- and ݕ-axes is not that difficult to determine. Have students continue working with their partners.

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Exercise 3

The system of equations is given below. A graph of the equations and their intersection point is also shown.

࢞൅࢟ൌ૟ ࢞െ࢟ൌ૛

is the position vector whose terminal point is the solution ܘ ࢚ whereܞ൅ܘሺ࢚ሻ ൌۺ a. Write each line in the form of the system and ܞ is the velocity vector that defines the path of a particle traveling along the line such that when ࢚ൌ૙, the solution to the system is ൫࢞ሺ૙ሻǡ࢟ሺ૙ሻ൯. ૝ െ૚ ࡸ ሺ࢚ሻ ൌ ቀ ቁ൅ቀ ቁ࢚ ૚ ૛ ૚ ૝ ૚ ࡸ ሺ࢚ሻ ൌቀ ቁ൅ቀ ቁ࢚ ૛ ૛ ૚

b. Describe a translation that takes the point ሺ࢞ሺ૙ሻǡ࢟ሺ૙ሻሻ to the origin. What is the new system?

.ۄെ૝ǡ െ૛ۃ Translate the system by the vector

࢟ൌെ࢞ ࢟ൌ࢞

c. Describe a transformation matrix ࡭ that rotates the lines to the ࢞- and ࢟-axes. What is the new system? ࢇ࢈ െ૚ ૚ ૚ ૙ Let ࡭ൌቀ ቁ. We want ࡭ήቀ ቁൌቀ ቁ, and ࡭ήቀ ቁൌቀ ቁ. Thus ࢉࢊ ૚ ૙ ૚ ૚ െࢇ ൅ ࢈ ൌ ૚ െࢉ ൅ ࢊ ൌ ૙ and

ࢇ൅࢈ൌ૙ ࢉ൅ࢊൌ૚ Solving for ࢇ, ࢈, ࢉ, and ࢊ gives ૚ ૚ െ ࡭ൌ൮ ૛ ૛൲ ૚ ૚ ૛ ૛ The new system is

࢞ൌ૙ ࢟ൌ૙

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d. Describe a translation that results in a system that has the same solution set as the original system. What is the new system of equations?

The new system is .ۄ૝ǡ ૛ۃ Translate by the vector

࢞ൌ૝ ࢟ൌ૛, which does intersect at the point ሺ૝ǡ ૛ሻ, so the solution is ሺ૝ǡ ૛ሻ.

The diagram below details the transformations graphically.

Closing (3 minutes) Ask students to summarize the key points of the lesson first in writing and then with a partner. Share key points as a class. A linear transformation preserves the solution set of a system of linear equations if they intersect at the origin. If a system of linear equations does not intersect at the origin, the solution set may not be preserved by a linear transformation. A sequence of transformation can be applied to a system of equations to preserve the solution set.

Exit Ticket (5 minutes)

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Name Date _

Lesson 24: Why Are Vectors Useful?

Exit Ticket

ݕൌͷݔ൅ʹ ݔ 1. Consider the system of equations ൜ , and perform the following operations on an arbitrary point ቀ ቁ: ݕൌ͵ݔ ݕ a. Rotate around the origin by ߠ.

b. Translate by the opposite of the solution to the system.

ଶ c. Apply a dilation of . ଷ

2. What effect does each of the transformations in Problem 1 have on the solution of the system and on the origin?

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Exit Ticket Sample Solutions

࢟ൌ૞࢞൅૛ ࢞ 1. Consider the system of equations ൜ , and perform the following operations on an arbitrary point ቀ ቁ: ࢟ൌ૜࢞ ࢟ a. Rotate around the origin by ࣂ. ܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ ࢞ ࢞᩺܋ܗܛሺࣂሻ െ࢟᩺ܛܑܖሺࣂሻ ൬ ൰ቀ ቁൌ൬ ൰ ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ࢟ ࢞᩺ܛܑܖሺࣂሻ ൅࢟᩺܋ܗܛሺࣂሻ

b. Translate by the opposite of the solution to the system. ࢞ െ૚ ࢞൅૚ ቀ ቁെቀ ቁൌ൬ ൰ ࢟ െ૜ ࢟൅૜

૛ c. Apply a dilation of . ૜ ૛ ૛ ૙ ࢞ ࢞ ൮૜ ൲ቀ ቁൌ൮૜ ൲ ૛ ࢟ ૛ ૙ ࢞ ૜ ૜

2. What effect does each of the transformations in Problem 1 have on the solution of the system and on the origin? ૛ െ૚ െ܋ܗܛሺࣂሻ ൅૜᩺ܛܑܖሺࣂሻ ૙ െ The solution of the system ቀ ቁ maps to ൬ ൰, ቀ ቁ, and ቆ ૜ቇ respectively. െ૜ െܛܑܖሺࣂሻ െ૜᩺܋ܗܛሺࣂሻ ૙ െ૛ The origin maps to itself for the first and third transformations, which were linear transformations, and maps to the ૚ opposite of the system’s solution for the second transformation: ቀ ቁ. ૜

Problem Set Sample Solutions

࢟ൌ૜࢞൅૛ 1. Consider the system of equations ൜ . ࢟ൌെ࢞൅૚૝ a. Solve the system of equations.

The point ሺ૜ǡ ૚૚ሻ is a simultaneous solution to the two equations.

b. Ilene wants to rotate the lines representing this system of equations about their solution and wishes to apply ܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ the matrix ൬ ൰ to any point ࡭ on either of the lines. If Ilene is correct, then applying a ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ rotation to the solution will map the solution to itself. Let ࣂൌૢ૙ι, and find where Ilene’s strategy maps the solution you found in part (a). What is wrong with Ilene’s strategy?

܋ܗܛሺૢ૙ιሻ െܛܑܖሺૢ૙ιሻ ૜ ૙െ૚ ૜ ൬ ൰ቀ ቁൌቀ ቁቀ ቁ ܛܑܖሺૢ૙ιሻ ܋ܗܛሺૢ૙ιሻ ૚૚ ૚૙ ૚૚ െ૚૚ ൌቀ ቁ ૜

Her transformation does not map the pivot point to itself, so the system is not rotating around the pivot point.

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c. Jasmine thinks that in order to apply a rotation to some point on either of these two lines, the entire system needs to be shifted so that the pivot point is translated to the origin. For an arbitrary point ࡭ on either of the two lines, what transformation needs to be applied so that the pivot point is mapped to the origin?

To map the pivot point to the origin, we can take a translation equal to the opposite of the pivot point. In this ࢞ ࢞ െ૜ case, that means for ࡭ൌቀ ቁ on either of the two lines, we do the following: ቀ ቁ൅ቀ ቁ. ࢟ ࢟ െ૚૚ ૜ െ૜ ૙ We see that ቀ ቁ൅ቀ ቁൌቀ ቁ. ૚૚ െ૚૚ ૙

d. After applying your transformation in part (c), apply Ilene’s rotation matrix for ࣂൌૢ૙ι. Show that the pivot point remains on the origin. What happens to the point ሺ૙ǡ ૛ሻ after both of these transformations? ૙െ૚ ૙ ૙ ቀ ቁቀ ቁൌቀ ቁ ૚૙ ૙ ૙ ૙െ૚ ૙ െ૜ ૙െ૚ െ૜ ቀ ቁ ቆቀ ቁ൅ቀ ቁቇ ൌቀ ቁቀ ቁ ૚૙ ૛ െ૚૚ ૚૙ െૢ ૢ ൌቀ ቁ െ૜

The point is translated and then rotated ૢ૙ι.

e. Although Jasmine and Ilene were able to work together to rotate the points around the pivot point, now their lines are nowhere near the original lines. What transformation brings the system of equations back so that the pivot point returns to where it started and all other points have been rotated? Find the final image of the point ሺ૙ǡ ૛ሻ.

Applying the inverse translation that we did in part (c) brings the points back to where they should be. That ࢞ ࢞ ૜ ૙ is, for ࡭Ԣ ൌ ቀ ቁ, a point on the transformed system, add ቀ ቁ൅ቀ ቁ. For ቀ ቁ, we get ࢟ ࢟ ૚૚ ૛ ૢ ૜ ૚૛ ቀ ቁ൅ቀ ቁൌቀ ቁ . െ૜ ૚૚ ૡ

f. Summarize your results in parts (a)–(e). ࢇ To rotate a system of equations around its solution ቀ ቁ, we apply the following transformations to any ࢈ ࢞ arbitrary point ቀ࢟ቁ:

܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ ࢞ ࢇ ࢞ ൬ ൰ ቆቀ ቁെቀ ቁቇ ൅ቀ ቁ . ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ࢟ ࢈ ࢟

We translate the system so that the solution is mapped to the origin, then rotate the system, and then apply the inverse translation.

Extension: ૚ ૙ 2. Let ࢈ ൌቀ ቁ and ࢈ ൌቀ ቁ. Then answer the following: ૚ ૙ ૛ ૚

.࢈૛ڄ࢈૚ ൅૙ڄa. Find ૚ ૚ ૙ ૚ ૙ ૚ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁൌ࢈ڄቀ ቁ൅૙ڄ૚ ૙ ૚ ૙ ૙ ૙ ૚

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.࢈૛ڄ࢈૚ ൅૚ڄb. Find ૙ ૚ ૙ ૙ ૙ ૙ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁൌ࢈ڄቀ ቁ൅૚ڄ૙ ૙ ૚ ૙ ૚ ૚ ૛

.࢈૛ڄ࢈૚ ൅૚ڄc. Find ૚ ૚ ૙ ૚ ૙ ૚ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁڄቀ ቁ൅૚ڄ૚ ૙ ૚ ૙ ૚ ૚

.࢈૛ڄ࢈૚ ൅૛ڄd. Find ૜ ૚ ૙ ૜ ૙ ૜ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁڄቀ ቁ൅૛ڄ૜ ૙ ૚ ૙ ૛ ૛

.࢈૛ڄ࢈૚ ൅૙ڄe. Find ૙ ૚ ૙ ૙ ૙ ૙ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁڄቀ ቁ൅૙ڄ૙ ૙ ૚ ૙ ૙ ૙

.࢈૛ for ࢞ǡ ࢟ real numbersڄ࢈૚ ൅࢟ڄf. Find ࢞ ૚ ૙ ࢞ ૙ ࢞ ቀ ቁൌቀ ቁ൅൬ ൰ൌቀ ቁڄቀ ቁ൅࢟ڄ࢞ ૙ ૚ ૙ ࢟ ࢟

૛ g. Summarize your results from parts (a)–(f). Can you use ࢈૚ and ࢈૛ to define any point in Թ ? In each case, the resultant vector was always equal to the scalar multiplied by the first vector for the ࢞-coordinate and the scalar multiplied by the second vector for the ࢟-coordinate.

૜ െ૛ 3. Let ࢈ ൌቀ ቁ and ࢈ ൌቀ ቁ. Then answer the following: ૚ ૛ ૛ ૜

.࢈૛ڄ࢈૚ ൅૚ڄa. Find ૚ ૜ െ૛ ૜ െ૛ ૚ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁڄቀ ቁ൅૚ڄ૚ ૛ ૜ ૛ ૜ ૞

.࢈૛ڄ࢈૚ ൅૚ڄb. Find ૙ ૜ െ૛ ૙ െ૛ െ૛ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁൌ࢈ڄቀ ቁ൅૚ڄ૙ ૛ ૜ ૙ ૜ ૜ ૛

.࢈૛ڄ࢈૚ ൅૙ڄc. Find ૚ ૜ െ૛ ૜ ૙ ૜ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁൌ࢈ڄቀ ቁ൅૙ڄ૚ ૛ ૜ ૛ ૙ ૛ ૚

.࢈૛ڄ࢈૚ ൅૛ ڄ d. Find െ૝ ૜ െ૛ െ૚૛ െ૝ െ૚૟ ቀ ቁൌቀ ቁ൅ቀ ቁൌቀ ቁڄቀ ቁ൅૛ ڄ െ૝ ૛ ૜ െૡ ૟ െ૛

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.࢈૛ ൌ૙ڄ࢈૚ ൅࢙ڄe. Solve ࢘ ૜ െ૛ ૙ ቀ ቁൌቀ ቁڄቀ ቁ൅࢙ڄ࢘ ૛ ૜ ૙ ૜࢘ െ૛࢙ ૙ ቀ ቁ൅ቀ ቁൌቀ ቁ ૛࢘ ૜࢙ ૙ ૜࢘ െ ૛࢙ ૙ ቀ ቁൌቀ ቁ ૛࢘ ൅ ૜࢙ ૙ ૛ ૜࢘ ൌ ૛࢙ ֜ ࢘ ൌ ࢙ ૜ ૛ ૛൬ ࢙൰ ൅ ૜࢙ ൌ ૙ ૜ ૝ ࢙൅૜࢙ൌ૙ ૜ ૝࢙ ൅ ૢ࢙ ൌ ૙ ૚૜࢙ ൌ ૙ ࢙ൌ૙ ૛ ૙ൌ૙ڄ ࢘ൌ ૜

So both ࢘ and ࢙ are zero.

૛૛ .࢈ ൌቀ ቁڄ࢈ ൅࢙ڄf. Solve ࢘ ૚ ૛ െૠ ૜ െ૛ ૛૛ ቀ ቁൌቀ ቁڄቀ ቁ൅࢙ڄ࢘ ૛ ૜ െૠ ૜࢘ െ૛࢙ ૛૛ ቀ ቁ൅ቀ ቁൌቀ ቁ ૛࢘ ૜࢙ െૠ ૜࢘ െ ૛࢙ ૛૛ ቀ ቁൌቀ ቁ ૛࢘ ൅ ૜࢙ െૠ

૛૛ ૛ ૛૛ ૛ ࢘ൌ ൅ ࢙ ૛ቀ ൅ ࢙ቁ ൅ ૜࢙ ൌ െૠ Thus, ૜ ૜ , and we get ૜ ૜ . ૝૝ ૝ ൅ ࢙൅૜࢙ൌെૠ ૜ ૜ ૝૝ ൅ ૝࢙ ൅ ૢ࢙ ൌ െ૛૚ ૚૜࢙ ൌ െ૟૞ ࢙ൌെ૞ ૛૛ ૛ ૛૛ ૚૙ ૚૛ ࢘ൌ ൅ ሺെ૞ሻ ൌ െ ൌ ൌ૝ ૜ ૜ ૜ ૜ ૜ So, ࢘ൌ૝ and ࢙ൌെ૞.

࢞ g. Is there any point ቀ࢟ቁ that cannot be expressed as a linear combination of ࢈૚ and ࢈૛ ࢞ ?(࢈૛ ൌቀ࢟ቁ has real solutions, for ࢞ǡ ࢟ real numbersڄ࢈૚ ൅࢙ڄi.e., where ࢘) ૛૛ No. There was nothing special about the point ቀ ቁ. We could have substituted in any other two values, െૠ performed the same steps, and arrived at a valid solution.

h. Explain your response to part (g) geometrically.

Answers may vary. ࢈૚ and ࢈૛ are two vectors that intersect at the origin. They are not parallel, so through ࢞ linear transformations, we can arrive at any point ቀ࢟ቁ. Alternatively, you can think of the equations ૜࢘ െ ૛࢙ ൌ ࢞ and ૛࢘ െ ૜࢙ ൌ ࢟ as lines in terms of ࢘ and ࢙; then these lines are not parallel and thus are always consistent when used in a system of linear equations.

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PRECALCULUS AND ADVANCED TOPICS ͻDKh>2

Topic E &ŝƌƐƚ-PĞƌƐŽŶsŝĚĞŽ GĂŵĞƐ—WƌŽũĞĐƚŝŽŶ DĂƚƌŝĐĞƐ

&ŽĐƵƐ^ƚĂŶĚĂƌĚƐ: (+) Add, subtract, and multiply matrices of appropriate dimensions. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of Ͳ and ͳ in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable- dimensions to produce another vector. Work with matrices as transformations of vectors. /ŶƐƚƌƵĐƚŝŽŶĂůĂǇƐ͗ 3 >ĞƐƐŽŶϮ5: First-Person Computer Games (E)1 >ĞƐƐŽŶϮ6: Projecting a 3-D Object onto a 2-D Plane (P) >ĞƐƐŽŶ27: Designing Your Own Game (P) The module ends with Topic E, which asks students to apply the knowledge they developed in this module to understand how first-person video games use matrix operations to project three-dimensional objects onto two-dimensional screens and to animate those images to give the illusion of motion.

Throughout this topic, students explore the projection of three-dimensional objects onto two-dimensional space. In Lesson 25, students create and manipulate objects in three dimensions and attempt a projection of a cube onto a flat screen. Through a discussion of the historical development of perspective drawing, students learn how to use vanishing points to create realistic two-dimensional representations of three- dimensional objects. In Lesson 26, students engage the mathematics that underlies the kind of animations seen in video games.

1Lesson Structure Key: P-Problem Set Lesson, D-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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In particular, students learn to take points in three-dimensional space and map them onto a two-dimensional plane in the same way that a programmer would seek to model the 3-D world on a 2-D screen. Students analyze the 2-D projection of a path followed by an object being rotated through three-dimensional space. Topic E and Module 2 end with Lesson 27, where students create a brief scene in a video game including one- step turn and roll procedures. Students describe the motions of the characters with respect to the scene and as rotations about specific axes. Students apply the mathematics from the previous lessons to use matrices to represent the motions of the characters in their scenes. Students make characters move in a straight line, turn, and flip in all directions by exploring rotations about each axis, consistently relating these motions back to their matrix representations. The materials support the use of geometry and game-creating software, such as GeoGebra and the freely available ALICE 3.1. In Topic E, students are making sense of vectors, understanding them as abstract representations of real- world situations such as programming video games and modeling video games with vectors and matrices. Through the use of computer games (ALICE 3.1) and software (GeoGebra), students are also using appropriate tools strategically to apply and understand mathematical concepts; further, they come to see matrices as tools to create transformations.

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Lesson 25: First-Person Computer Games

Student Outcomes Through discovery, students understand how challenging it is to project three-dimensional objects onto a two- dimensional space. Students understand that vanishing points can be used to project three-dimensional objects onto two- dimensional space. They demonstrate this process by showing how a cube is projected onto the two- dimensional surface of a screen. Students recognize that projecting three-dimensional images onto a two-dimensional surface can be accomplished by applying linear transformation matrices to the pre-image points.

Lesson Notes Have students download the ALICE program before starting the lesson. Homework assigned prior to this lesson should familiarize students with the ALICE 3.1 program. After sharing their experiences creating and manipulating objects in ALICE, students discuss the projection of three-dimensional objects onto two-dimensional space and attempt a projection of a cube onto a flat screen, which should help them appreciate the difficulty of sketching a two-dimensional projection of a three-dimensional object by hand. They also use right-triangle trigonometry to explore how projections onto different-sized screens affect the viewer’s field of vision. Through a discussion of the historical development of perspective drawing, students learn how vanishing points can be used to create realistic two-dimensional representations of three-dimensional objects. They apply this understanding by revisiting the task of projecting a cube onto a two-dimensional screen and form conjectures about how vectors, representing points on the three-dimensional objects, can be multiplied by matrices to generate two-dimensional projections. This prepares them to apply matrix multiplication to generate two-dimensional projections in the next lesson.

Classwork Opening (3 minutes) Open the ALICE 3.1 program, and encourage students to share briefly the types of items they created or manipulated while exploring the program. Allow a few volunteers to demonstrate their new skills.

Discussion (2 minutes) Are the objects created in ALICE actually three-dimensional? If not, how can we describe them? à They are representations or projections of three-dimensional objects onto a two-dimensional screen. How do people create realistic looking three-dimensional objects on a flat surface? à Answers may vary and could include the concept of depicting objects from a certain vantage point.

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Exploratory Challenge (10 minutes) Display the ALICE program Projection_No_Wires. Note: If the program is not available, the example could be completed using a clear plastic sheet (such as a sheet protector or transparency sheet), a small cube, and a larger ball. Student volunteers could change the position of the objects to represent the different vantage points shown in the scene from the ALICE program. When the program opens, the instructor should click run. Typing ͷ should display a blue cube to the right of a gray rectangle.

Explain to students that the cube is a three-dimensional object that needs to be projected onto a screen represented by the gray rectangle. By typing ʹ when the program is running, students should see from an observer’s perspective, looking forward through the screen to the cube and a red sphere behind it. Note: If real objects are used, the sheet would be directly in front of the viewer’s eye, with the cube in the foreground and the larger ball directly behind it.

Typing Ͷ provides a wide view of the scene.

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After discussing the different viewpoints, provide each student with a copy of the image shown in the example. Ask students to attempt the construction and, after a few minutes, have them share their ideas with a partner. Then, facilitate a discussion that emphasizes the difficulty of projecting the cube onto the screen by hand. The program projection should help students visualize what the projection of the cube onto the screen represents (i.e., the points where the line segments from the viewer’s eye to the cube intersect the surface of the screen). This is viewpoint ͷ, where the “strings” represent the line of sight from the viewer to the cube.

The first screen we saw is viewpoint ͷ. The blue cube represents the three-dimensional object to be projected onto a two-dimensional screen, which is represented by the gray rectangle. Now when we look at viewpoint ʹ, how did the appearance of the scene change? à The screen is in the foreground with the blue cube and a red sphere behind it. So viewpoint ʹ could represent the vantage point of a viewer looking directly at a TV screen. You could think of the screen as a window through which the viewer looks at the real-world objects on the other side. Now let’s look at viewpoint Ͷ. How does this viewpoint change the appearance of the scene? à Answers may vary but will probably indicate that this seems to be a wide view from the side, and the sphere looks much larger than the cube.

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How is it possible that viewpoints ʹ and Ͷ could represent the same scene? à They show the objects from different positions. Viewpoint ʹ is from the perspective of a viewer looking at the two objects directly in front of him, and viewpoint Ͷ is shown from a perspective that is to the side of the objects. Why does the relative size of the sphere look so different between the viewpoints? à Answers may vary, but students might recognize that when two objects are aligned in a viewer’s line of sight, a much smaller object in the foreground can appear to be similar in size to a much larger object in the background (e.g., one’s index finger placed a few inches directly in front of one’s eyes might look similar in height to a building in front of the viewer a mile away). Now look at the example. It is drawn from a vantage point similar to viewpoint ͷ in the ALICE program. By drawing line segments from the point in the lower-left corner of the diagram (which represents the viewer’s eye) to the vertices of the cube, we can represent the viewer’s line of sight. Once we draw the line segments in, how can we draw an image of the cube on the screen? à Answers will vary but might include that the vertices of the projection are located where the line segments intersect the rectangle. Now that you have tried to draw the image of the cube projected on the screen, how would you characterize the activity? What challenges did you face? à Students may struggle with completing the construction. They might suggest that it is hard to see the points where the line segments intersect the screen. (Note: Students complete this activity later in the lesson through step-by-step instructions. The diagram later in the lesson shows the correctly drawn projection.) Let’s look at another program in ALICE. In this program, the cube is “attached” to the viewer’s eye by strings. In viewpoint ͷ, you can see where the strings intersect the screen. The intersection of the viewer’s line of sight and the screen represents the projection of the cube onto the screen. This is difficult to represent on paper.

Exploratory Challenge

In this drawing task, the “eye” or the “camera” is the point, and the shaded figure is the “TV screen.” The cube is in the 3-D universe of the computer game.

By using lines drawn from each vertex of the cube to the point, draw the image of the 3-D cube on the screen.

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Scaffolding: Consider asking advanced students to project the cube onto the screen without further prompting. Model how the first vertex is drawn to the “eye,” and have students sketch the remaining line segments independently. If the ALICE software is unavailable, a transparent sheet could be used to represent the screen. A volunteer could hold a wooden block on one side of the screen, and pieces of twine could be used to help students “see” the line of sight from the viewer’s eye to a point on the block. The point of intersection between the twine and

the transparency sheet would represent the two-dimensional projection of the point on the block Example (5 minutes) onto the screen.

This example allows students to use right-triangle trigonometry to explore an additional challenge programmers have in projecting images onto a screen: limitations on the field of view. Students apply trigonometry to determine the effect of screen size on a horizontal field of view. This example should be completed as part of a teacher-led discussion. Alternatively, students could complete the problem in pairs and share their responses after a few minutes. How does a person’s peripheral vision compare with the field of view of a screen? Explain your answer. à The field of view of the screen is smaller. When looking at a screen, you can generally see objects in your peripheral vision that are to the right and left of the screen. We have experienced the limiting effect of a screen on our field of view. If we model the field of view of a television screen with a diagram, what components do we need to include? à The screen; the field of view, which is the angle formed by the rays representing the line of sight of the viewer to the right and left ends of the screen; the width of the screen; and the distance from the middle of the screen to the viewer’s eyes What does our model look like when all these components are included? ௪ ఏ à Two right triangles are formed, each with leg lengths of and ݀, acute angle formed by ݀, and the ଶ ଶ line of sight to the right or left side of the screen. How can we use trigonometry to find the line of sight of the person sitting in front of the television screen? ௪ ఏ à We can apply the inverse tangent function to the ratio of and ݀ to find . ଶ ଶ

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How does your own experience with screen viewing compare with our findings regarding the relationship between screen size, distance from a screen, and field of view? à Answers will vary but should address that larger screens, such as movie theater screens, require a greater viewing distance for a comfortable field of view than smaller screens (similarly a comfortable viewing distance from a television screen is usually greater than that for a computer screen).

Example

When three-dimensional objects are projected onto screens with finite dimensions, it often limits the field of view (FOV), or the angle the scene represents. This limiting effect can vary based on the size of the screen and position of the observer.

a. Sketch a diagram that could be used to calculate a viewer’s field of view ࣂ in relation to the horizontal width of the screen ࢝ and the distance the viewer is from the screen ࢊ.

b. Assume that a person is sitting directly in front of a television screen whose width is ૝ૡinches at a distance of ૡfeet from the screen. Use your diagram and right-triangle trigonometry to find the viewer’s horizontal field of view ࣂ. ࢝ ࣂ ܜ܉ܖ ൬ ൰ൌ ૛ ૛ ࢊ ૝ ࣂ ܜ܉ܖ ൬ ൰ൌ૛ ൌ૙Ǥ૛૞ ૛ ૡ ࣂ ܜ܉ܖି૚ሺ૙Ǥ ૛૞ሻ ൌ ૛ ࣂ ૚૝ι ൎ ૛ ࣂ ൎ ૛ૡι

c. How far would a viewer need to be from the middle of a computer screen with a width of ૚૞inches to produce the same field of view as the person in front of the television? ࢝ ࣂ ܜ܉ܖ ൬ ൰ൌ ૛ ૛ ࢊ ૚૞ ૛ૡι ܜ܉ܖ ൬ ൰ൌ ૛ ૛ ࢊ ૚૞ ࢊൌ ૛ ൎ૜૙ inches ܜ܉ܖሺ૚૝ιሻ

d. Write a general statement about the relationship between screen size and field of view.

The smaller the screen, the closer a viewer must be to the screen to result in a given horizontal FOV.

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Discussion (8 minutes): Development of Perspective in Renaissance Art This discussion exposes students to the difficulties artists have faced historically in representing three-dimensional objects on a two-dimensional surface. We have now experienced the challenges of trying to represent a three-dimensional object on a two- dimensional surface and explored some of the visual challenges in representing scenes on a surface with finite dimensions. Let us see how artists addressed the challenge of projecting images during the medieval and Renaissance periods. Display an image similar to the one found at the following site as well as the two images shown. http://c300221.r21.cf1.rackcdn.com/medieval-painting-1384196679 org.jpg

Giotto di Bondone. Scala. Leonardo da Vinci. Alinari. Licensed from Art Resource, NY Licensed from Art Resource, NY

Have students, in small groups, compare and contrast the paintings, focusing on the depiction of three-dimensional images. Facilitate a discussion based on students’ findings, and provide a brief explanation of the history of the development of linear perspective in art: How do the images differ in presenting realistic images of three-dimensional objects? à Answers will vary but should address that the image of the meal does not realistically represent the table (e.g., it looks like the food is falling off the table); the second image provides a sense of a top and bottom of the hill but some figures look unrealistic (e.g., the black sheep looks like it is floating on the hill); the painting with the eagle realistically portrays the objects. This first painting you looked at is an example of a painting from the Middle Ages. In the early 1300s, artists generally created works that appeared flat, without distinguishing between the foreground and background or accounting for differences in the size of objects leading to scenes that looked unrealistic (e.g., floors and tables that tilted up).

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In the early 1300s, an artist named Giotto di Bondone was the first Italian painter to attempt to paint using perspective, trying to represent objects with a realistic impression of their size and depth. He is credited with applying algebra to help place distant lines in his paintings. An example of his work is the second painting you looked at. In what ways does this painting show an improvement in representing realistic three-dimensional objects in two-dimensional space when you compare it to the first painting? à Answers will vary but might address the illusion of depth between the left (upper) and right (lower) parts of the painting. In the early 1400s, the artist Filipo Brunelleschi was attributed with advancing the use of linear perspective in art. In particular, he made use of a vanishing point, a point to which all lines in a painting would converge. The use of these lines helped artists to create the illusion of depth. Here is a simple example of using a vanishing point to create perspective:

A rectangle with a vanishing point at its center

A room with floor, back wall, and side door constructed using the vanishing point

A room with extraneous parts of the line segments erased

How was the vanishing point used to create the elements in the room? Share your ideas with a partner. à Answers will vary but should include that each of the line segments for the floor, ceiling, and door were created using line segments that intersect the vanishing point when extended. For the back wall, the vertices of the rectangle lie along line segments that intersect with the vanishing point.

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By the late 1400s, artists like Leonardo da Vinci were using multiple vanishing points in their artwork to create images that were very realistic, such as the following. How can you see the use of a vanishing point in this picture? à The lines in the drawing all converge at the vanishing point, which is at the eagle. This draws the eye to the eagle as the focal point of the drawing and creates a realistic sense of depth and size for the figures in the drawing. We can see the use of vanishing points in many realistic images.

One Vanishing Point Two Vanishing Points

“Inside Greenwich Foot Tunnel” by C.G.P Grey, “Perspective 1” created by Ejahng, is licensed under CC BY 2.0 is licensed under CC BY-SA 3.0 http://creativecommons.org/licenses/by/2.0 http://creativecommons.org/licenses/by-sa/3.0/deed.en

Three Vanishing Points Multiple Vanishing Points

“UT band during Rice game” by Mary Estrada, is licensed under CC BY 2.0 http://creativecommons.org/licenses/by/2.0

Berenice Abbott, 1934

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How might the use of vanishing points to create linear perspective relate to transformations of objects in three-dimensional space? à Answers will vary but might address that the rescaling of objects is similar to performing linear transformations. And how have we transformed objects in three-dimensional space? à We have multiplied points by transformation matrices. One way to use multiple vanishing points to project a three-dimensional object into two-dimensional space by hand is to create a horizon line through the two-dimensional surface and plot one or more vanishing points along that line. We are going to use this idea to try again to project the cube onto the screen.

Exercise 1 (6 minutes) Students should try to create the two-dimensional projection of the cube onto the screen, this time by using the horizon line and vanishing points provided. Point out that the first projected point is selected arbitrarily, and the rest of the points can be found using the vanishing points. Note: It is important to work through the example prior to modeling it with students in class. This is likely to take more than one attempt. Once students have completed the projection, have them paraphrase the procedure in writing, including an interpretation of the object they created. Have students share their projections and written responses with a partner.

Exercise 1

In this drawing task, the “eye” or the “camera” is the point, and the shaded figure is the “TV screen.” The cube is in the 3-D universe of the computer game.

By using lines drawn from each vertex of the cube to the point, draw the image of the 3-D cube on the screen. A horizon line and two additional vanishing points have been included to help you. The image point of the first vertex is shown.

Scaffolding: Try prompting advanced students to create a ܸଶ ܲଵ horizon line with two endpoints as vanishing ܲଵԢ points outside the screen. Allow them to then choose the location of the line. Lead struggling students step-by-step through the construction as shown.

ܸଷ

ܸଵ

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Draw line segments connecting the “eye” (ܸଵ) with the vertices of the cube.

Draw a vertical line segment on the screen from the given image point (ܲଵԢ) to the point where it intersects the line segment from ܸଵ to the vertex beneath it (ܲଶԢ).

ܲଵ ܸଶ

ܲଵԢ

ܲଶ ܲଶԢ

ܸଷ

ܸଵ

Draw dashed line segments connecting the two image points to vanishing points ʹ and ͵.

ܲ ܸଶ ଵ

ܲଵԢ

ܲଶ

ܲଶԢ

ܸଷ

ܸଵ

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Plot a point where each of the line segments from ܸଵ to the vertices intersect the line segments connecting the dashed line segments. These are the remaining image points for the vertices of the cube.

ܲହ ܲ଺ ܲ ܸଶ ଵ ܲଷ ܲ଺Ԣ ܲଵ ܲହԢ ܲ ܲ Ԣ ଷ ܲସ ܲ Ԣ ܲଶ ଻ ܲଶԢ ܲସԢ

ܸଷ

ܸଵ

Connect the image points to form the projected image of the cube.

ܸଶ

ܸଷ

ܸଵ

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Exercise 2 (3 minutes) This exercise should be completed as part of a teacher-led discussion. It provides the opportunity for students to explore using matrix operations to find the coordinates for projected points of three-dimensional objects.

Exercise 2

Let’s assume that the point ࢂ૚ in our projection diagram is at the origin and the upper-right vertex of the cube is located ૞ at ൭ૡ൱. If our screen represents the plane ࢟ൌ૛, use matrix multiplication to determine the vector that represents the ૝ line of sight from the observer to the projected point on the screen. Explain your thinking. ૞ The vector from the viewer to the vertex on the actual object is൥ૡ൩. Since the projected point and the actual vertex are ૝ along the same line of sight, the vector from the line of sight of the viewer to the projected point can be found by resizing the vector to the vertex so that the ࢟-coordinate of the projected point is ૛. This can be accomplished by rescaling with a ૙Ǥ ૛૞ ૙ ૙ ૞ ૚Ǥ ૛૞ factor of ૙Ǥ ૛૞ using the matrix multiplication ൥ ૙૙Ǥ૛૞૙൩൥ૡ൩ ൌ ൥ ૛ ൩. ૙૙૙Ǥ૛૞૝ ૚

Closing (3 minutes) Students should respond to the question in writing. After a minute, they could share their predictions with a partner. What do you predict that computer programmers might do to represent and manipulate three-dimensional objects in two-dimensional space? à Answers will vary but might include determining image points by multiplying vectors by transformation matrices and applying matrix operations to represent linear transformations of the points.

Exit Ticket (5 minutes)

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Name Date

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Exit Ticket

ʹ In a computer game, the camera eye is at the origin, and the tip of a dog’s nose has coordinates ൭ͳͲ൱. If the computer ͵ screen represents the plane ݕൌͳ, determine the coordinates of the projected point that represents the tip of the dog’s nose.

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Exit Ticket Sample Solutions

૛ In a computer game, the camera eye is at the origin, and the tip of a dog’s nose has coordinates ൭૚૙൱. If the computer ૜ screen represents the plane ࢟ൌ૚, determine the coordinates of the projected point that represents the tip of the dog’s nose. ૛ The vector representing the line of sight from the viewer to the tip of the dog’s nose is ൥૚૙൩ . To resize the vector so that ૜ the ࢟-coordinate of the projected point is ૚, we need to rescale by a factor of ૙Ǥ ૚. ૙Ǥ ૚ ૙ ૙ ૛ ૙Ǥ ૛ ૙Ǥ ૛ ൥ ૙૙Ǥ૚૙൩൥૚૙൩ ൌ ൥ ૚ ൩, so the projected point is ൭ ૚ ൱. ૙૙૙Ǥ૚૜ ૙Ǥ ૜ ૙Ǥ ૜

Problem Set Sample Solutions

1. Projecting the image of a three-dimensional scene onto a computer screen has the added constraint of the screen size limiting our field of view, or FOV. When we speak of FOV, we wish to know what angle of view the scene represents. Humans have remarkably good peripheral vision. In New York State, the requirement for a driver’s license is a horizontal FOV of no less than ૚૝૙ι. There is no restriction placed on the vertical field of vision, but humans normally have a vertical FOV greater than ૚૛૙ι. a. Consider the (simulated) distance the camera is from the screen as ࢊ, the horizontal distance of the screen as ࢝, and the horizontal FOV as ࣂ. Then use the diagram below and right-triangle trigonometry to help you find ࣂ in terms of ࢝ and ࢊ.

࢝ ࣂ ܜ܉ܖ ൬ ൰ൌ ૛ ૛ ࢊ ࢝ ࣂ ൌܜ܉ܖି૚ ቌ ૛ ቍ ࢝ ૛ ࢊ

࢝ ࢊ ࣂൌ૛ܜ܉ܖି૚ ቌ ૛ ቍ ࢊ ࣂ ૛

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b. Repeat procedures from part (a), but this time let ࢎ represent the height of the screen and ࣒ represent the vertical FOV.

ࢎ ࣒ ܜ܉ܖ ൬ ൰ൌ ૛ ૛ ࢊ ࢎ ࣒ ࢎ ൌܜ܉ܖି૚ ቌ૛ቍ ૛ ࢊ ࢊ ࢎ ି૚ ૛ ࣒ ࣒ൌ૛ܜ܉ܖ ቌ ቍ ࢊ ૛

c. If a particular game uses an aspect ratio of ૚૟ǣ ૢ as its standard view and treats the camera as though it were ૡ units away, find the horizontal and vertical FOVs for this game. Round your answers to the nearest degree. ૚૟ ૚ ቌ ૛ ቍିܖ܉ܜڄࣂൌ૛ ૡ

૚ሺ૚ሻିܖ܉ܜڄൌ૛ ૝૞ڄൌ૛ ൌૢ૙

The horizontal field of view is ૢ૙ι. ૢ ૚ ቌ૛ቍିܖ܉ܜڄ࣒ൌ૛ ૡ

૝Ǥ ૞ ૚ ൬ ൰ିܖ܉ܜڄൌ૛ ૡ ૛ૢǤ૜૟ڄൎ૛ ൌ૞ૡǤૠ૛ ൎ૞ૢ

So, the vertical field of view is about ૞ૢι.

d. When humans sit too close to monitors with FOVs less than what they are used to in real life or in other games, they may grow dizzy and feel sick. Does the game in part (c) run the risk of that? Would you recommend this game be played on a computer or on a television with these FOVs?

The game in part (c) has a smaller field of vision than most humans are probably used to. Since computer games are played much closer and that increases the chance of dizziness and illness, this game would probably be better on a television than a computer.

2. Computers regularly use polygon meshes to model three-dimensional objects. Most polygon meshes are a collection of triangles that approximate the shape of a three-dimensional object. If we define a face of a polygon mesh to be a triangle connecting three vertices of the shape, how many faces at minimum do the following shapes require? a. A cube

A cube normally has ૟ square faces, but it requires ૛ triangles to make a square, so a polygon mesh of a cube would have ૚૛ faces.

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b. A pyramid with a square base

The pyramid has ૝ triangular faces already, and would require ૛ more to make up the base, so ૟ faces.

c. A tetrahedron

A tetrahedron is composed of ૝ triangles, so ૝ faces for its polygon mesh.

d. A rectangular prism

A rectangular prism is the same as a cube: It has ૚૛ faces.

e. A triangular prism

.૜, or ૡ facesڄA triangular prism has ૛ faces make up the ends, and each side requires 2 more faces, so ૛൅૛

f. An octahedron

An octahedron is ૡ triangular pieces, so ૡ faces.

g. A dodecahedron

A dodecahedron normally has ૚૛ pentagonal faces. Each pentagon requires a minimum of ૜ triangles to construct it, so a dodecahedron would have ૜૟ faces in its polygon mesh.

h. An icosahedron

An icosahedron has ૛૙ triangular faces, so its polygon mesh does as well.

i. How many faces should a sphere have?

Answers will vary. An icosahedron is a rough approximation of a sphere, so ૛૙ is a good estimate, especially for graphics in the early years of computers and video games. Some students might suggest ૜૟૙, which is a good estimate of the number of lines to approximate a circle. The best polygon mesh for a sphere would have as many faces as the computer can handle without slowing down..

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3. In the beginning of 3-D graphics, objects were created only using the wireframes from a polygon mesh without shading or textures. As processing capabilities increased, 3-D models became more advanced, and shading and textures were incorporated into 3-D models. One technique that helps viewers visualize how shading works on a 3-D figure is to include both an “eye” and a “light source.” Vectors are drawn from the eye to the figure, and then reflected to the light (this technique is called ray tracing). See the diagram below.

“light source”

ܞ୪

ܞୱ

“eye” or “camera”

a. Using this technique, the hue of the object depends on the sum of the magnitudes of the vectors. Assume the ૝ “eye” in the picture above is located at the origin, ܞܛ is the vector from the eye to the location ൭૞൱, and the ૜ ૞ “light source” is located at ൭૟൱. Then find ܞܔ, the vector from ܞܛ to the light source, and the sum of the ૡ magnitudes of the vectors. ૞ ૝ ૚ ܞܔ is the difference between the light source and the object, so ܞܔ ൌ൭૟൱െ൭૞൱ൌ൭૚൱. ૡ ૜ ૞

૛ ૛ ૛ ԡܞܔԡ ൌ ඥ૚ ൅૚ ൅૞ ൌ ξ૛ૠ ൌ૜ξ૜

૛ ૛ ૛ ԡܞܛԡ ൌ ඥ૝ ൅૞ ൅૜ ൌξ૚૟൅૛૞൅ૢൌξ૞૙ൌ૞ξ૛

ԡܞܔԡ ൅ ԡ࢙ԡ ൌ૜ξ૜ ൅૞ξ૛

b. What direction does light travel in real life, and how does this compare to the computer model portrayed above? Can you think of any reason why the computer only traces the path of vectors that start at the “eye?”

In reality, light travels from the light source, bouncing off objects until it reaches our eyes. If the computer starts drawing vectors at the light source, then it will draw many vectors that will never reach our “eye,” wasting resources.

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Lesson 26: Projecting a 3-D Object onto a 2-D Plane

Student Outcomes Students learn to project points in 3-dimensional space onto a 2-dimensional plane. Students learn how to simulate 3-dimensional turning on a 2-dimensional screen.

Lesson Notes Students engage the mathematics that provides a foundation for animations like the kind seen in video games. In particular, students learn to take points in 3-dimensional space and map them onto a 2-dimensional plane in the same way that a programmer would seek to model the 3-D world on a 2-D screen. Students also analyze the 2-D projection of a path followed by an object being rotated through 3-dimensional space.

Classwork In the context of this lesson, the statement ݕൌͳ is used to describe a plane in 3-dimensional space that is parallel to the ݔݖ-plane. However, the same statement could be used to describe a line in the coordinate plane that is parallel to the ݔ-axis, or to describe a point on the number line. This series of exercises gives students an opportunity to look for and make use of structure as they learn to differentiate between these situations based on context. Students may represent their responses to the questions using words, pictures, or graphs.

Discussion (4 minutes): Visualizing ࢟ൌ૚ :On the real number line, there is exactly one point ݕ whose coordinate is ͳ

.Describe the set of points ሺݔǡ ݕሻ in Թଶ whose ݕ-coordinate is ͳ. Give several specific examples of such points à The set of points whose ݕ-coordinate is ͳ consists of points that are ͳ unit above the ݔ-axis. This set of ,points forms a line that is parallel to the ݔ-axis. For example, this set includes the points ሺെͳǡ ͳሻ, ሺʹǡ ͳሻ and ሺͷǡ ͳሻ.

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Describe the set of points ሺݔǡݕǡݖሻ in Թଷ whose ݕ-coordinate is ͳ. Give several specific examples of such points. .à The set of points whose ݕ-coordinate is ͳ consists of points that are ͳ unit to the right of the ݔݖ-plane This set of points forms a plane that is parallel to the ݔݖ-plane. For example, this set includes the points ሺെͳǡ ͳǡ ͸ሻ, ሺͲǡͳǡͲሻ, and ሺͷǡͳǡ͵ሻ.

Discussion (11 minutes): Projections Now imagine that the plane ݕൌͳ represents a TV screen, with the origin located where your eye is. Suppose we have a point ܣ in 3-dimensional space that we wish to project on to the screen. Let’s call the image ܣԢ. How should we go about locating ܣԢ? à ܣԢ must be located at a point on the screen that lies on the line that connects ܱ to ܣ.

Good! We would like to be able to describe the line through ܱ and ܣ, and then find out where it hits our screen. Before we attempt to do this, let’s examine the 2-dimensional case to see if we can get any ideas.

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Suppose you have a screen located along the line ݕൌͳ, and the origin is located where your eye is. Let’s take the point ܣሺ͹ǡ ͷሻ and project it onto this screen. Where is the image of ܣ? Take a few moments to think about this, then discuss your thoughts with the students around you. In a few minutes, please be ready to present your argument to the class.

Any point ܲ on the line joining ܱ and ܣ makes a triangle that is similar to ᇞܣܱܤ in the figure below. Thus, the ௬ ହ coordinates of ܲ must satisfy the proportion ൌ . In the case of ܣԢ, which lies on our screen, the ௫ ଻ ௬ ହ ͷ ݔ-coordinate must be ͳ. Thus, its ݕ-coordinate can be found by solving ൌ , so ݕൌ . In summary, the ଵ ଻ ͹ ͷ image of point ܣ is the point ܣᇱ ൌቀͳǡ ቁ. ͹ Scaffolding: Show students the diagram to the left, and assist them with cues such as the following: What can we say about the triangles in the figure? Can you write an equation that shows the relationship between the lengths in the figure?

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using a parameter ݐ. In other ܣ Next we attempt to describe the set of points on the line through ܱ and .words, describe the points on this line as ൫ݔሺݐሻǡݕሺݐሻ൯. Justify your answer à We have ൫ݔሺݐሻǡݕሺݐሻ൯ൌሺ͹ݐǡͷݐሻ. We know this is correct because the coordinates satisfy the ହ௧ ହ proportion ൌ . ଻௧ ଻ To round out our discussion, we look at this from one more point of view. Give a geometric description of the .transformation ሺ͹ǡ ͷሻ հ ሺ͹ݐǡ ͷݐሻ à This is a dilation of the point ሺ͹ǡ ͷሻ, where the origin is the center of the dilation, and the parameter ݐ represents the scale factor. We know that a point and its image lie on the same line through the origin when dilations are involved, so our work with the projection makes sense from that perspective as well. Do you see how we can find the image of ?ܣ ሺ͹ǡ ͷሻ using the parametric description ሺ͹ݐǡ ͷݐሻ of points on the line through ܱ and ͳ à Yes, we can find the value of ݐ that makes the ݔ-coordinate ͳ. With ͹ݐ ൌ ͳ, we have ݐൌ . Thus, the ͹ ͳ ͷ . ݕ-coordinate must be ͷቀ ቁൌ ͹ ͹

Exercises 1–3 (2 minutes) Give students time to work on the problems below, and then ask them to share their work with a partner. When they are ready, select students to present their solutions to the class.

Exercises

1. Describe the set of points ሺૡ࢚ǡ ૜࢚ሻ, where ࢚ represents a real number.

The point ሺૡ࢚ǡ ૜࢚ሻ is a dilation of the point ሺૡǡ ૜ሻ with a scale factor of ࢚. Thus, each such point lies on the line that joins ሺૡǡ ૜ሻ to the origin.

2. Project the point ሺૡǡ ૜ሻ onto the line ࢞ൌ૚. ૚ ૚ ૜ ૜ ࢚ ૡ࢚ ൌ ૚ ࢚ൌ ૜࢚ ൌ ૜ ቀ ቁൌ ቀ૚ǡ ቁ We need the value of that makes , so ૡ. Thus, ૡ ૡ, and so the image is ૡ .

3. Project the point ሺૡǡ ૜ሻ onto the line ࢞ൌ૞. ૞ ૜ ૚૞ ૚૞ ࢚ ૡ࢚ ൌ ૞ ࢚ൌ ૜࢚ ൌ ૜ ቀ ቁൌ ቀ૞ǡ ቁ We need the value of that makes , so ૡ. Thus, ૡ ૡ , and so the image is ૡ .

Discussion (8 minutes): Projecting a Point onto a Plane We have just examined the problem of projecting a point in 2-dimensional space onto a 1-dimensional screen. Now let’s return to the 3-dimensional problem from the start of this discussion. is located at ሺͶǡ ͳͲǡ ͷሻ. We want to project this point onto the plane ݕൌͳ. In order ܣ Suppose that the point to do this, let’s describe the line through ܱ and ܣ parametrically. We practiced these descriptions in a 2-dimensional context. Can you see how to extend this to the 3-dimensional case? à We should have ൫ݔሺݐሻǡݕሺݐሻǡݖሺݐሻ൯ ൌ ሺͶݐǡ ͳͲݐǡ ͷݐሻ, which is a dilation of the point ሺͶǡ ͳͲǡ ͷሻ with a scale .factor of ݐ

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We expect points of this form to lie on the line through ܱ and ܣ. Let’s use similar triangles to confirm this just as we did in the 2-dimensional case. Can you make an argument that ሺͶǡ ͳͲǡ ͷሻ and ሺͶݐǡ ͳͲݐǡ ͷݐሻ are in fact on the same line through the origin? The diagram below may help with this.

Scaffolding: Show students the diagram to the left, and give them the following cues as needed: Find the height of each triangle. The base of each triangle is a segment in the ݔݕ-plane. How could we find the length of each base? à The large triangle contains the points ሺͲǡͲǡͲሻ, ሺͶǡ ͳͲǡ Ͳሻ, and ሺͶǡ ͳͲǡ ͷሻ. .à The small triangle contains the points ሺͲǡ Ͳǡ Ͳሻ, ሺͶݐǡ ͳͲݐǡ Ͳሻ, and ሺͶݐǡ ͳͲݐǡ ͷݐሻ ହ௧ à The vertical segments are in the ratio ൌݐ. The ratio of the segments in the ݔݕ-plane is ହ .also ݐ, proving that the triangles are similar .à The long segment in the ݔݕ-plane has length ξͶଶ ൅ͳͲଶ .à The short segment in the ݔݕ-plane has length ඥሺͶݐሻଶ ൅ሺͳͲݐሻଶ ൌ ξݐଶ ήͶଶ ൅ݐଶ ήͳͲଶ

Thus, we have ඥݐଶሺͶଶ ൅ͳͲଶሻ ൌݐξͶଶ ൅ͳͲଶ. This confirms that the ratio of the corresponding sides is ݐ, proving that the triangles are similar. From this we conclude that the original point and its dilated image both lie on the same line through the origin. Now return to the problem at hand: What is the image of ሺͶǡ ͳͲǡ ͷሻ when it is projected onto the plane? ?ݕൌͳ ͳ . à We want the value of ݐ that takes the ݕ-coordinate to ͳ, so we need ͳͲݐ ൌ ͳ, which gives ݐൌ ͳͲ So the image must be ሺͲǤͶǡ ͳǡ ͲǤͷሻ.

Let’s do some more practice with 3-D projections onto a 2-D screen.

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A STORY OF FUNCTIONS Lesson 26 M2 PRECALCULUS AND ADVANCED TOPICS

Exercises 4–5 (3 minutes) Give students time to work on the problems below. Ask them to share their work with a partner when they are ready. Select students to present their solutions to the class at an appropriate time.

4. Project the point ሺെ૚ǡ ૝ǡ ૞ሻ onto the plane ࢟ൌ૚. ૚ ૚ ૞ ࢚ ૝࢚ ൌ ૚ ࢚ൌ ቀെ ǡ૚ǡ ቁ We need the value of that makes , so ૝. Thus, the image is ૝ ૝ .

5. Project the point ሺૢǡ ૞ǡ െૡሻ onto the plane ࢠൌ૜. ૜ ૛ૠ ૚૞ ࢚ െૡ࢚ ൌ ૜ ࢚ൌെ ቀെ ǡെ ǡ૜ቁ We need the value of that makes , so ૡ. Thus, the image is ૡ ૡ .

Example (10 minutes): Rotations in 3D

In this example, students explore rotations around one of the coordinate axes. ሺߠሻ െ ሺߠሻ Ͳ .ሺߠሻ Ͳ൱ represents a rotation in 3-dimensional space about the ݖ-axis Recall that the matrix ൭ሺߠሻ ͲͲͳ As ߠ varies, what path does ሺͳͲǡ ͳͲǡ ͳͲሻ trace out in 3-dimensional space? à The path is a circle whose center is ሺͲǡ Ͳǡ ͳͲሻ. Now turn your attention to the screen, that is, the plane where ݕൌͳ. What is the path traced out by the projected image of ሺͳͲǡ ͳͲǡ ͳͲሻ as ߠ varies? ሺߠሻ െ ሺߠሻ Ͳ ͳͲ ͳͲ ሺߠሻ െ ͳͲ ሺߠሻ à The rotated image is the point ൭ሺߠሻ ሺߠሻ Ͳ൱൭ͳͲ൱ൌ൭ͳͲ ሺߠሻ ൅ ͳͲ ሺߠሻ൱. ͲͲͳͳͲ ͳͲ à To calculate the image of the point when it is projected onto the screen, we need the value of ݐ that ͳ . ሺߠሻሻ ൌ ͳ, which is ݐൌ causes ݐሺͳͲ ሺߠሻ ൅ ͳͲ ͳͲሺሺߠሻ൅ ሺߠሻሻ ሺߠሻെሺߠሻ ۊ ሺߠሻ ሺߠሻ൅ ۇ ,Using graphing software we get the following path .ۋ ͳ ۈ à Thus, the projected image is ͳ یሺߠሻሻ ͳͲሺሺߠሻ൅ۉ which appears to be a hyperbola:

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?If instead we wanted to model a rotation of ሺͳͲǡ ͳͲǡ ͳͲሻ around the ݕ-axis, what matrix should be used ሺߠሻ Ͳ െ ሺߠሻ à We should use ൭ Ͳͳ Ͳ൱. This leaves the ݕ-coordinate fixed and has the structure of a ሺߠሻ Ͳ ሺߠሻ rotation matrix. .As ߠ varies, let’s project the points onto the plane ݕൌͳ. Describe the path traced out ሺߠሻ Ͳ െ ሺߠሻ ͳͲ ͳͲ ሺߠሻ െ ͳͲ ሺߠሻ à The rotated image is the point ൭ Ͳͳ Ͳ൱൭ͳͲ൱ൌ൭ ͳͲ ൱. ሺߠሻ Ͳ ሺߠሻ ͳͲ ͳͲ ሺߠሻ ൅ ͳͲ ሺߠሻ à To calculate the image of the point when it is projected onto the screen, we need the value of ݐ that ͳ . causes ͳͲݐ ൌ ͳ, which is ݐൌ ͳͲ ሺߠሻ െ ሺߠሻ à Thus, the projected image is ൭ ͳ ൱. Using graphing software we get the following path, ሺߠሻ ൅ ሺߠሻ which is a circle. This circle is simply a smaller version of the circle traced out during the rotation:

These examples demonstrate how programmers give viewers the impression that objects are turning on the screen.

Closing (2 minutes) Use your journal to write a brief summary of what you learned in today’s lesson. à We learned how to project points in 3-D space onto a plane, which represents a 2-D screen. à We learned how to model a 3-D rotation using a 2-D screen.

Exit Ticket (5 minutes)

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Name Date

Lesson 26: Projecting a 3-D Object onto a 2-D Plane

Exit Ticket

͵ ʹ .Consider the plane defined by ݖൌʹ and the points, ݔൌ൭͸൱ and ݕൌ൭െ͵൱ .1 ͺ ͷ .a. Find the projections of ݔ and ݕ onto the plane ݖൌʹ if the eye is placed at the origin

ͳ .b. Consider ݓൌ൭ʹ ൱; does it make sense to find the projection of ݓ onto ݖൌʹ? Explain െͳ

͵ Consider an object located at ൭Ͷ൱ and rotating around the ݖ-axis. At what ߠ value is the object out of sight of the .2 Ͳ ?plane ݕൌͳ

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Exit Ticket Sample Solutions

૜ ૛ 1. Consider the plane defined by ࢠൌ૛ and the points, ࢞ൌ൭૟൱ and ࢟ൌ൭െ૜൱. ૡ ૞ a. Find the projections of ࢞ and ࢟ onto the plane ࢠൌ૛ if the eye is placed at the origin. ૜ ૝ ۊ ૞ ۇ ۊ૝ۇ ࢞ൌ ૜ ,࢟ ൌ ૟ ۋ െۈ ۋ ۈ ૛ ૞ ی ૛ ۉ ی૛ۉ

૚ b. Consider ࢝ൌ൭૛ ൱; does it make sense to find the projection of ࢝ onto ࢠൌ૛? Explain. െ૚ ૚ ࢚ൌെ No, ૛.

૜ 2. Consider an object located at ൭૝൱ and rotating around the ࢠ-axis. At what ࣂ value is the object out of sight of the ૙ plane ࢟ൌ૚? ܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ ૙ ૜ ૜᩺܋ܗܛሺࣂሻ െ૝᩺ܛܑܖሺࣂሻ To rotate around the ࢠ-axis and project onto ࢟ൌ૚, ൭ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ૙൱൭૝൱ൌ൭૜᩺ܛܑܖሺࣂሻ ൅૝᩺܋ܗܛሺࣂሻ൱. To ૙૙૚૙ ૙ ૜᩺܋ܗܛ൫ࣂ൯െ૝᩺ܛܑܖሺࣂሻ ૚ ૜᩺ܛܑܖ൫ࣂ൯൅૝᩺܋ܗܛሺࣂሻ project onto ࢟ൌ૚, ࢚ൌ . The projection would be ൮ ൲. The value of ࣂ that ૜᩺ܛܑܖ൫ࣂ൯൅૝᩺܋ܗܛሺࣂሻ ૚ ૙ ૜᩺܋ܗܛሺࣂሻି૝᩺ܛܑܖሺࣂሻ would make the object out of site is when ൌ૙. ૜᩺ܛܑܖሺࣂሻା૝᩺܋ܗܛሺࣂሻ ૜ ૜܋ܗܛሺࣂሻ െ૝ܛܑܖሺࣂሻ ൌ૙ ࣂൌܜ܉ܖି૚ ቀ ቁ This occurs when or when ૝ .

Problem Set Sample Solutions

૚૙ ૚૜ ૚૙ ૚૙ ૚૜ ૚૜ ૚૙ ૚૜ 1. A cube in ૜-D space has vertices ൭૚૙൱ǡ൭૚૙൱ǡ൭૚૜൱ǡ൭૚૙൱ǡ൭૚૜൱ǡ൭૚૙൱ǡ൭૚૜൱ǡ൭૚૜൱. ૚૙ ૚૙ ૚૙ ૚૜ ૚૙ ૚૜ ૚૜ ૚૜ a. How do we know that these vertices trace a cube?

The edges between the vertices form right angles, and all have a length of ૜units.

b. What is the volume of the cube?

૜૜ ൌ૛ૠ cubic units

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c. Let ࢠൌ૚. Find the eight points on the screen that represent the vertices of this cube (some may be obscured). ૚ ૚ ૚ ૚Ǥ ૜ ૚ ૚ ۊ૚Ǥ ૜ۇ ૚ ૚Ǥ ૜ ૚ ൭૚൱ǡ൭ ૚ ൱ǡ൭૚Ǥ ૜൱ǡ ૚ ǡ൭૚Ǥ ૜൱ǡቌ ቍ ǡ൮૚Ǥ ૜൲ǡ൭૚൱ ૚ ۋ ۈ ૚ ૚ ૚ ૚ ૚Ǥ ૜ ૚ ૚Ǥ ૜ ૚ ૚ ی ૚ ۉ

d. What do you notice about your result in part (c)?

Some of the points went to the same position. These were points that lay along the same path from the ૚ ૚ camera. Some points were dilated by a factor of ૚૙ and some by a factor of ૚૜ depending on how far they were from the plane in the ࢠ-direction.

૚ ૙ ૙ െ૚ ૙ 2. An object in 3-D space has vertices ൭૞൱ǡ൭૟൱ǡ൭૞൱ǡ൭ ૞ ൱ǡ൭૝൱. ૙ ૙ ૚ ૙ ૙ a. What kind of shape is formed by these vertices?

It appears this is a pyramid with a square base.

b. Let ࢟ൌ૚. Find the five points on the screen that represent the vertices of this shape. ૙Ǥ ૛ ૙ ૙ െ૙Ǥ ૛ ૙ ൭ ૚ ൱ǡ൭૚൱ǡ൭ ૚ ൱ǡ൭ ૚ ൱ǡ൭૚൱ ૙ ૙ ૙Ǥ ૛ ૙ ૙

3. Consider the shape formed by the vertices given in Problem 2. a. Write a transformation matrix that rotates each point around the ࢟-axis ࣂ degrees. ܋ܗܛሺࣂሻ ૙െܛܑܖሺࣂሻ ൭ ૙૚ ૙൱ ܛܑܖሺࣂሻ ૙܋ܗܛሺࣂሻ

b. Project each rotated point onto the plane ࢟ൌ૚ if ࣂൌ૝૞ι.

ඥ૛ ඥ૛ ඥ૛ െ െ ૙ ۊ ૛ ۇ ۊ ૛ ۇ ૙ ۊ ૛ ۇ .ǡ൭૝൱ ۋ ૞ ۈ ǡ ۋ ૞ ۈǡ൭૟൱ǡ ۋ ૞ ۈ :First we rotate each point ඥ૛ ૙ ඥ૛ ඥ૛ ૙ െ ی ૛ ۉ ی ૛ ۉ ی ૛ ۉ

c. Is this the same as rotating the values you obtained in Problem 2 by ૝૞ι?

Yes. Since the projection is interpreted as a dilation, order does not matter.

4. In technical drawings, it is frequently important to preserve the scale of the objects being represented. To accomplish this, we use an orthographic projection instead of a perspective projection. The idea behind the orthographic projection is that the points are translated at right angles to the screen (the word stem ortho- means ૚૙૙ straight or right). To project onto the ࢞࢟-plane for instance, we can use the matrix ቀ ቁ. ૙૚૙ a. Project the cube in Problem 1 onto the ࢞࢟-plane by finding the 8 points that correspond to the vertices. ૚૙ ૚૜ ૚૙ ૚૙ ૚૜ ૚૜ ૚૙ ૚૜ ቀ ቁǡቀ ቁǡቀ ቁǡቀ ቁǡቀ ቁǡቀ ቁǡቀ ቁǡቀ ቁ ૚૙ ૚૙ ૚૜ ૚૙ ૚૜ ૚૙ ૚૜ ૚૜

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b. What do you notice about the vertices of the cube after projecting?

They form a square with all ૡ points being mapped to ૝ points.

c. What shape is visible on the screen?

It is a square with sides of length ૜.

d. Is the area of the shape that is visible on the screen what you expected from the original cube? Explain.

The area of the visible shape is ૢ, which is the same as the area of any of the faces of the original cube.

e. Summarize your findings from parts (a)–(d).

Orthographic projections preserve the scale of objects but much of the original information is lost.

f. State the orthographic projection matrices for the ࢞ࢠ-plane and the ࢟ࢠ-plane. ૚૙૙ For ࢞ࢠ-plane: ቀ ቁ ૙૙૚ ૙૚૙ For ࢟ࢠ-plane: ቀ ቁ ૙૙૚

g. In regard to the dimensions of the orthographic projection matrices, what causes the outputs to be two-dimensional?

The transformation matrices are ૛ൈ૜, meaning they can operate on 3-dimensional points but generate 2-dimensional outputs.

ࢇ࢞ 5. Consider the point ࡭ൌ൭ࢇ࢟൱ in the field of view from the origin through the plane ࢠൌ૚. ࢇࢠ a. Find the projection of ࡭ onto the plane ࢠൌ૚.

ࢇ࢞ ࢇࢠ ࢇ .ۊ࢟ ۇ :Since the point is in the field of view, we know that ࢇࢠ ്૙, and we get ࢇࢠ ی ૚ ۉ

b. Find a ૜ൈ૜ matrix ࡼ such that ࡼ࡭ finds the projection of ࡭ onto the plane ࢠൌ૚. ૚ ૙૙ ۊ ࢇࢠۇ ۋ ૚ ۈ ۋ ૙ ૙ ۈ ۋ ࢇࢠ ۈ ૚ ૙૙ یࢇࢠ ۉ

c. How does the matrix change if instead of projecting onto ࢠൌ૚, we project onto ࢠൌࢉ, for some real number ࢉ്૙?

The numerator of each fraction is ࢉ instead of ૚.

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d. Find the scalars that generate the image of ࡭ onto the planes ࢞ൌࢉ and ࢟ൌࢉ, assuming the image exists. Describe the scalars in words. ࢉ ࢉ and . Whenever we are projecting onto a plane parallel to the ࢞࢟-, ࢞ࢠ-, or ࢟ࢠ-planes, the scalar by ࢇ࢞ ࢇ࢟ which we multiply the point ࡭ is always going to be the reciprocal of ࢠ-, ࢟-, or ࢞-coordinate times the ࢠ-, ࢟-, or ࢞-location of the plane.

Extension:

6. Instead of considering the rotation of a point about an axis, consider the rotation of the camera. Rotations of the camera cause the screen to rotate along with it, so that to the viewer, the screen appears immobile.

a. If the camera rotates ࣂ࢞ around the ࢞-axis, how does the computer world appear to move?

A rotation of െࣂ࢞ about the ࢞-axis

b. State the rotation matrix we could use on a point ࡭ to simulate rotating the camera and computer screen by

ࣂ࢞ about the ࢞-axis but in fact keeping the camera and screen fixed. ૚૙ ૙ ൭૙܋ܗܛሺെࣂ࢞ሻ െܛܑܖሺെࣂ࢞ሻ൱ ૙ܛܑܖሺെࣂ࢞ሻ܋ܗܛሺെࣂ࢞ሻ

c. If the camera rotates ࣂ࢟ around the ࢟-axis, how does the computer world appear to move?

A rotation of െࣂ࢟ about the ࢟-axis

d. State the rotation matrix we could use on a point ࡭ to simulate rotating the camera and computer screen by ࣂ࢟ about the ࢟-axis but in fact keeping the camera and screen fixed.

܋ܗܛሺെࣂ࢟ሻ ૙ െܛܑܖ൫െࣂ࢟൯ ቌ ૙૚ ૙ቍ ܛܑܖ൫െࣂ࢟൯ ૙܋ܗܛ൫െࣂ࢟൯

e. If the camera rotates ࣂࢠ around the ࢠ-axis, how does the computer world appear to move?

A rotation of െࣂࢠ about the ࢠ-axis

f. State the rotation matrix we could use on a point ࡭ to simulate rotating the camera and computer screen by

ࣂࢠ about the ࢠ-axis but in fact keeping the camera and screen fixed.

܋ܗܛሺെࣂࢠሻ െܛܑܖሺെࣂࢠሻ૙ ൭ܛܑܖሺെࣂࢠሻ ܋ܗܛሺെࣂࢠሻ ૙൱ ૙૙૚

g. What matrix multiplication could represent the camera starting at a relative angle ൫ࣂ࢞ǡࣂ࢟ǡࣂࢠ൯? Apply the transformations in the order ࢠ-࢟-࢞. Do not find the product.

૚૙ ૙ ܋ܗܛሺെࣂ࢟ሻ૙െܛܑܖ൫െࣂ࢟൯ ܋ܗܛሺെࣂࢠሻ െܛܑܖሺെࣂࢠሻ૙ ૙܋ܗܛሺെࣂ ሻ െܛܑܖሺെࣂ ሻ ൭ ࢞ ࢞ ൱ ቌ ૙૚ ૙ቍ ൭ܛܑܖሺെࣂࢠሻ ܋ܗܛሺെࣂࢠሻ ૙൱ ૙ܛܑܖሺെࣂ࢞ሻ܋ܗܛሺെࣂ࢞ሻ ܛܑܖ൫െࣂ࢟൯ ૙܋ܗܛ൫െࣂ࢟൯ ૙૙૚

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Lesson 27: Designing Your Own Game

Student Outcomes Students create a short animation in ALICE 3.1 using one-step procedures. Students apply their understanding of the mathematics of projecting three-dimensional images onto two- dimensional surfaces by writing matrices to represent the motions in the animations they create in ALICE.

Lesson Notes Lesson 27 is the culminating lesson in this topic and module where students see how their work with functions, matrices, vectors, and linear transformations comes together. Prior to this lesson, students should have become familiar with the ALICE program, having opportunities to explore it in previous homework assignments. The ALICE 3.1 program should be available to students during the lesson. Students use it to create a brief scene including one-step turn and roll procedures. They describe the motions of the ALICE characters with respect to the scene and as rotations about specific axes. Then students apply the mathematics from the previous lesson to use matrices to represent the motions of the characters in their scenes. If there is only one copy of ALICE available, this lesson could be completed by having the class create a collective animation with selected students choosing the setting and characters. Then, volunteers could each apply a one-step roll or turn procedure, while the rest of the students determine a matrix operation that would appropriately represent the movement. Prompt students to create distinct procedures, so the movements represent rotations of various degrees and around various axes. The activities and procedures addressed in this lesson provide opportunities for students to reason abstractly about the movements of the animated objects and to model the movements of objects using matrix operations.

Classwork Discussion (10 minutes): Reviewing Projections of 3D Objects into 2D Space Through a teacher-led series of questions, students review what they have learned about projecting three-dimensional objects into two-dimensional space, including using matrices to represent projected images and rotations of points. If technology is not available, the question addressing the ALICE program could be skipped. In a sentence, explain the function of the ALICE program to a person new to the program. à It creates 3D animations using procedures to manipulate objects within the ALICE world. What is actually happening to create the illusion of three-dimensional objects in a two-dimensional space? à The points composing the three-dimensional objects are projected onto a two-dimensional screen. And how is this projection formed? à It is the intersection of the viewer’s line of sight to the real object with the flat surface onto which it is projected.

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How can this type of projection be drawn by hand? à Answers may vary but should include using a horizon line and vanishing points. And do all projections of a point look exactly the same? à No, they vary based on the placement of the horizon line and the vanishing points. We attempted this process in Lesson 25 by projecting a cube onto a flat screen. What challenges did we encounter? à Answers may vary but could include difficulties in seeing the points of intersection between the line of sight and the screen prior to the addition of the horizon line and vanishing points. What might be some additional challenges we would face in trying to project more complex three-dimensional images onto a screen by hand? à Answers may vary and could include the need for projecting numerous points to get an accurate projection, which could be time consuming and cumbersome when done by hand. How can mathematics be applied to expedite the process of projecting three-dimensional objects into two- dimensional space? à We can set the viewer’s eye at the origin and write an equation from the eye to each point to be projected. The image point on the screen will be located on that line at the intersection with the screen. ܽଵ So if we have the point ܣൌ൥ܽଶ൩, how can we represent the line through the viewer’s eye and point ܣ ܽଷ parametrically?

ݐܽଵ .à The equation of the line can be represented as ൥ݐܽଶ൩, where ݐ is a real number ݐܽଷ And what does this line represent? ܽଵ à It is a dilation of the point ൥ܽଶ൩ about the origin. ܽଷ onto a screen defined by the plane ݕൌͳ. How could we find the ܣ Let’s say we are projecting our point coordinates of the point? ͳ à Since ݕൌͳ, we need to find a value ݐ such that ݐܽଶ ൌͳ. Therefore, ݐൌ , and the image point is ܽʹ ܽͳ ېʹܽۍ ݐܽଵ .ۑ ͳ ێ൥ݐܽଶ൩ൌ ۑ ܽێ ͵ ݐܽଷ ےʹܽۏ ?ʹHow would our projected point change if we chose the screen to be at ݕൌ

.à We would need to find the value ݐ so that ݐܽଶ ൌʹ, but otherwise, the procedure would be the same ?How would our projected point change if we chose the screen to be at ݖൌͳ

.à We would need to find the value t so that ݐܽଷ ൌͳ, but otherwise, the procedure would be the same Let’s say we have created a seafloor scene in ALICE. In the scene, we placed a shark whose pivot point is represented by the coordinates ሺ͵ǡͷǡͲǤʹሻ. What would be the coordinates of this point projected onto a ?ʹscreen at ݕൌͳ? At ݖൌ .à For ݕൌͳ, ͷݐ ൌ ͳ, so ݐൌͲǤʹ, and the projected image point is ሺͲǤ͸ǡ ͳǤͲǡ ͲǤͲͶሻ .à For ݖൌʹ, ͲǤʹݐ ൌ ʹ, so ݐൌͳͲ, and the projected image point is ሺ͵Ͳǡ ͷͲǡ ʹሻ

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Now let’s discuss rotating points in three-dimensional space. How can we use matrices to represent the ?rotation of a point about the ݖ-axis through an angle ߠ ሺߠሻ െሺߠሻ Ͳ ܽଵ à ൥ሺߠሻ ሺߠሻ Ͳ൩൥ܽଶ൩ ͲͲͳܽଷ ?How can we use matrices to represent the rotation of a point about the ݕ-axis through an angle ߠ

ሺߠሻ Ͳെሺߠሻ ܽଵ à ൥ Ͳͳ Ͳ൩൥ܽଶ൩ ሺߠሻ Ͳ ሺߠሻ ܽଷ ?And a rotation about the ݔ-axis through an angle ߠ

ͳͲ Ͳ ܽଵ à ൥Ͳ ሺߠሻ െሺߠሻ൩൥ܽଶ൩ Ͳሺߠሻ ሺߠሻ ܽଷ ?rotated about the ݔ-axis ܣ And what are the coordinates of point ܽଵ à ൥ܽଶ ሺߠሻ െܽଷሺߠሻ൩ ܽଶሺߠሻ ൅ܽଷ ሺߠሻ ?How would we project these points onto the plane ݕൌͳ

.ሺߠሻ െܽଷԝሺߠሻ൯ൌͳ à Multiply the rotation matrix by ݐ so that ݐ൫ܽଶԝ And what would that matrix look like? ͳ ሺߠሻ െܽଷ ሺߠሻሻ ൌͳ, then ݐൌ , and the projected points are represented à If ݐሺܽଶ ܽʹ ൫ߠ൯െܽ͵ ൫ߠ൯ ܽͳ ې൫ߠ൯െܽ ൫ߠ൯ ܽۍ ۑ ͵ ʹ ێ .ۑ ͳ ێ by ۑ൫ߠ൯ ൫ߠ൯൅ܽ͵ ʹܽێ ے൫ߠ൯െܽ͵ ൫ߠ൯ ʹܽۏ

Example (10 minutes)

Open the ALICE program, and demonstrate how it works by completing the Example. As Scaffolding: students observe the setup of a simple scene with one character and the manipulation of Advanced students could the character within the ALICE world, they are reminded of how to create and manipulate represent multiple movements objects in ALICE, which should help them as they create their own scene with a partner in using matrix operations, such the exercises that follow. While presenting the example, use matrix operations to as multiple rotations or a represent the motions of characters. If technology is not available, students could work in rotation followed by a pairs or small groups to create an idea for a 3D animation. They could create a scene with translation. multiple characters and determine a few motions that each character would make in the scene. These motions could be described using captioned sketches, or the students could create paper characters, which could be manipulated to demonstrate their motions. Parameters for the motions of the characters could be set (e.g., between 2–4 characters per scene; 1–2 movements per character; at least two rotations around different axes must be represented). The scenes could be presented, and students who are not presenting their scene could use matrix operations to represent the movements of the characters (presentations discussed in the next lesson section).

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Once we have selected the dolphin, I am going to move it left and right. Look at the coordinates of the dolphin on the right side of the screen. What coordinate is changing, and how is it changing? à The ݔ-coordinate changes when the dolphin moves left and right. Moving to the left increases the .value of ݔ, and moving to the right decreases it So our positive ݔ-axis is to the left from the viewer’s vantage point when the dolphin is facing with its nose toward the viewer. Now, if I click on the arrow pointing up, I can move the dolphin up and down. What coordinate is changing now, and how is it changing? à The ݕ-coordinate changes when the dolphin moves up and down, with the up direction increasing the .value of ݕ So our ݕ-axis is vertical when the dolphin is facing forward with its back to the sky. How about when we move the dolphin forward and backward? .à The ݖ-coordinate changes. It increases as the dolphin moves back Now we have moved the dolphin so its pivot point is at the origin. How can we represent translations of the dolphin using matrix operations? ܽ Ͳ à Answers should address adding the matrix ቈܾ቉ to ൥Ͳ൩, where ܽ represents the number of units the ܿ Ͳ dolphin is translated to the left, ܾ represents the number of units the dolphin is translated up, and ܿ represents the number of units the dolphin is translated backward. Now we have selected the rotation button and the dolphin. What does the dolphin appear to do when we ?rotate it about the ݔ-axis à A forward or backward somersault ?And the ݕ-axis à The dolphin turns in the water. ?And the ݖ-axis à The dolphin rolls on its side. And how can we generalize the rotations of the dolphin about the axes using matrices? :à Rotation about ݔ-axis ͳͲ Ͳ ൥Ͳ ሺܿሻെሺܿሻ൩ where ܿ is the degrees of rotation counterclockwise about the axis Ͳ ሺܿሻ ሺܿሻ :à Rotation about ݕ-axis ሺܿሻ Ͳ െሺܿሻ ൥ Ͳͳ Ͳ൩ where ܿ is the degrees of rotation counterclockwise about the axis ሺܿሻ Ͳ ሺܿሻ :à Rotation about ݖ-axis ሺܿሻ െሺܿሻ Ͳ ൥ሺܿሻ ሺܿሻ Ͳ൩ where ܿ is the degrees of rotation counterclockwise about the axis ͲͲͳ Now we have selected the procedure: this.dolphin, roll: RIGHT 0.25, asSeenBy: this, duration: 2.0, animationStyle: BEGIN_AND_END_ABRUPTLY. What happened when we ran the program? à The dolphin does a quarter backward somersault out of the water in a counterclockwise rotation about .the negative ݖ-axis

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How does this make sense based on the procedure we created? à Answers will vary but should address that in the default orientation, a roll to its right would appear to the viewer as a roll onto its left side, which is a counterclockwise motion. Now you get to use what we have reviewed together to create your own animation and to describe the motions using matrices. Scaffolding:

Example Students who feel a. Select the Sea Surface as your background scene, and select the setup scene mode. comfortable in ALICE can From the Gallery By Class Hierarchy, select Swimmer Classes, then Marine Mammals, proceed to creating their and a Dolphin. Name your dolphin. Describe what you see. scene without the tutorial. We have a scene with an ocean surface and a dolphin facing forward in the center. Struggling students may benefit from a written set b. Click on the dolphin, and from the handle style buttons in the top-right corner of the of directions. screen, select translation. By selecting the arrows on the dolphin, we can move it to different locations in the screen. Move the dolphin left and right, up and down, forward and backward. Then, move it so that its coordinates are ሺ૙ǡ ૙ǡ ૙ሻ.

c. Use a matrix to describe each of the movements of the dolphin from its location at the origin. i. Move ૛ units right. ૙ െ૛ ൥૙൩ ൅ ൥ ૙ ൩ ૙ ૙

ii. Move ૝ units down. ૙ ૙ ൥૙൩ ൅ ൥െ૝൩ ૙ ૙

iii. Move ૜ units forward. ૙ ૙ ൥૙൩ ൅ ൥ ૙ ൩ ૙ െ૜

d. Click on the rotation button from the handle style buttons, and practice rotating the dolphin about the three axes through its center. Use a matrix to represent the motion of the dolphin described. Assume the center of rotation of the dolphin is at the origin. i. Rotation counterclockwise one full turn about the ࢠ-axis ܋ܗܛሺ૜૟૙ιሻ െܛܑܖሺ૜૟૙ιሻ ૙ ૚૙૙ ൥ܛܑܖሺ૜૟૙ιሻ ܋ܗܛሺ૜૟૙ιሻ ૙൩ ൌ ൥૙૚૙൩ ૙૙૚૙૙૚

ii. Rotation counterclockwise one half turn about the ࢞-axis

૚૙ ૙ ૚૙ ૙ ൥૙ ܋ܗܛሺ૚ૡ૙ιሻ െܛܑܖሺ૚ૡ૙ιሻ൩ ൌ ൥૙െ૚૙൩ ૙ ܛܑܖሺ૚ૡ૙ιሻ ܋ܗܛሺ૚ૡ૙ιሻ ૙૙െ૚

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e. Select Edit Code from the screen. Drag and drop “this.dolphin turn” from the “Procedures” menu and drop it into the “declare procedure” region on the right. Select from the drop-down menus: this.dolphin, turn: LEFT 0.25, asSeenBy: this, duration: 2.0, animationStyle: BEGIN_AND_END_ABRUPTLY. Then run the program. Describe what you see. Represent the motion using a matrix.

The dolphin undergoes a quarter counterclockwise rotation in a horizontal circle about the ࢟-axis. ܋ܗܛሺૢ૙ιሻ ૙ െܛܑܖሺૢ૙ιሻ ૙૙െ૚ ൥ ૙૚ ૙൩ ൌ ൥૙૚ ૙൩ ܛܑܖሺૢ૙ιሻ ૙ ܋ܗܛሺૢ૙ιሻ ૚૙ ૙

f. Drag and drop “this.dolphin roll” from the “Procedures” menu, and drop it into the “declare procedure” region on the right beneath the turn procedure. Select from the drop-down menus: this.dolphin, roll: RIGHT 3.0, asSeenBy: this, duration: 2.0, animationStyle: BEGIN_AND_END_ABRUPTLY. Then run the program. Describe what you see. Represent the motion using a matrix.

The dolphin performs three backward counterclockwise somersaults in the vertical plane about the negative ࢠ-axis. ܋ܗܛሺ૚૙ૡ૙ιሻ െܛܑܖሺ૚૙ૡ૙ιሻ ૙ ૚૙૙ ൥ܛܑܖሺ૚૙ૡ૙ιሻ ܋ܗܛሺ૚૙ૡ૙ιሻ ૙൩ ൌ ൥૙૚૙൩ ૙૙૚૙૙૚

Exercises (15 minutes) Give students time to work on the problems below in ALICE. If possible, have students complete the tasks in pairs. Once they have completed the exercises, they may consult with another pair to verify their solutions. Select students to present their scenes and solutions to the class as time permits; for example, perhaps each pair could demonstrate one procedure while the pair with whom they partnered displays the matrix that represents the motion in the 3D ALICE world, as well as on a projected screen. Some pairs of students may need some one-on-one assistance either setting up the scene or representing the projecting points using matrices. When students do pivots, have them assume the pivot point is at the origin. If technology is not available, continue the activity begun earlier in the lesson, where students present their ideas for animations, including descriptions or demonstrations of character movements. Have students who are not presenting represent the movements of each character using matrix operations. These students could share their matrices and justify their responses, while other students provide supporting or refuting evidence.

Exercises

1. Open ALICE 3.1. Select a background and three characters to create a scene. Describe the scene, including the coordinates of the pivot point for each character and the direction each character is facing.

Scene description: Answers will vary. I created a sea surface scene that contains an adult walrus facing forward, a baby walrus to the left of the adult and facing it (looking to the right), and a dolphin breaching on its back, with its nose pointed up and to the right. The dolphin is behind and to the left of the walruses.

Coordinates of the pivot point for each character:

Answers will vary. For example, dolphin ሺ૝Ǥ ૚૟ǡ ૙Ǥ ૞ǡ ૝Ǥ ૚ሻ; baby walrus ሺ૙Ǥ ૟ǡ ૙Ǥ ૚૚ǡ ૙Ǥ ૞૞ሻ; adult walrus ሺെ૙Ǥ ૝૟ǡ െ૙Ǥ ૙૜ǡ ૚Ǥ ૚૚ሻ.

2. a. Describe the location of a plane ࢞ൌ૞ in your scene from the perspective of the viewer.

The plane ࢞ൌ૞ is a vertical plane ૞ units to the left of the center of the screen.

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b. Determine the coordinates of the pivot points for each character if they were projected onto the plane ࢞ൌ૞.

Answers will vary. An example of an acceptable response is shown. ૞ For the dolphin, ૝Ǥ ૚૟࢚ ൌ ૞, so ࢚ൎ૚Ǥ૛૙ and image ൌ ൥ ૟ ൩. ૝Ǥ ૢ૛ ૞ ૛૞ ૙Ǥ ૟࢚ ൌ ૞ ࢚ൌ ൌ ൥ ൩ For the baby walrus, , so ૜ and image ૙Ǥ ૢ૛ . ૝Ǥ ૞ૡ ૞ For the adult walrus, െ૙Ǥ ૝૟࢚ ൌ ૞, so ࢚ ൎ െ૚૙Ǥ ૡૠ and image ൌ ൥ ૙Ǥ ૜૜ ൩. െ૚૛Ǥ ૙ૠ

3. Create a short scene that includes a one-step turn or roll procedure for each of the characters. The procedures should be unique. Write down the procedures in the space provided. After each procedure, describe what the character did in the context of the scene. Then describe the character’s motion using transformational language. Finally, represent each procedure using matrix operations.

Answers will vary. An example of an acceptable response is shown.

Procedure 1, adult walrus: this.adultWalrus turn: FORWARD 4.0, asSeenBy: this, duration: 2.0.

Character motion (in scene): The adult walrus performed four forward somersaults toward the foreground.

Transformational motion: The adult walrus performed a counterclockwise rotation of four full turns about the ࢞-axis. ૚૙ ૙ Matrix representation of motion: ൥૙܋ܗܛሺ૚૝૝૙ιሻ െܛܑܖሺ૚૝૝૙ιሻ൩ ࡭ for any point ࡭ on the walrus translated to ૙ܛܑܖሺ૚૝૝૙ιሻ ܋ܗܛሺ૚૚૝૙ιሻ be centered around the origin.

Procedure 2, baby walrus: this.babyWalrus turn: LEFT 2.0, asSeenBy: this, duration: 2.0.

Character motion (in scene): The baby walrus spun twice in a counterclockwise circle next to the adult.

Transformational motion: The baby walrus rotated about the ࢟-axis counterclockwise. ܋ܗܛሺૠ૛૙ιሻ ૙െܛܑܖሺૠ૛૙ιሻ Matrix representation of motion: ൥ ૙૚ ૙൩ ࡮ for any point ࡮ on the walrus translated to be ܛܑܖሺૠ૛૙ιሻ ૙܋ܗܛሺૠ૛૙ιሻ centered around the origin.

Procedure 3, dolphin: this.dolphin roll: LEFT 4.0, asSeenBy: this, duration: 2.0.

Character motion (in scene): The dolphin performed four back somersaults out of the water.

Transformational motion: The dolphin rotated counterclockwise about the ࢠ-axis.

Matrix representation of motion: ܋ܗܛሺૠ૛૙ιሻ െܛܑܖሺૠ૛૙ιሻ ૙ ൥ܛܑܖሺૠ૛૙ιሻ ܋ܗܛሺૠ૛૙ιሻ ૙൩ ࡯ for any point ࡯ on the dolphin translated to be centered around the origin. ૙૙૚

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Closing (5 minutes) Have students respond in writing to the prompt. After a few minutes, select several students to share their thoughts. If technology is not available, have students respond to the prompt with respect to rotations of 3D objects in any 3D animation situation. Make a generalization about how to represent one-step turns or rolls in ALICE using matrix operations. Use an example from your animation to support your response. à Answers will vary. An example of an acceptable response is included: One-step turns and rolls can be represented as rotations of a given point on the character once it has been translated to the origin. This rotation can be represented using a rotation matrix applied to the translated point ܣ about the axis of rotation for a rotation of ߠ degrees. For example, the adult walrus performed a rotation about the ݔ-axis, which can be represented as ͳͲ Ͳ ൥Ͳ ሺͳͶͶͲιሻ െሺͳͶͶͲιሻ൩ܣ. ͲሺͳͶͶͲιሻ ሺͳͳͶͲιሻ

Exit Ticket (5 minutes)

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Name Date

Lesson 27: Designing Your Own Game

Exit Ticket

Ͳ Consider the following set of code for the ALICE program featuring a bluebird whose center is located at ൥Ͳ൩. Ͳ this.bluebird resize 2.0 this.bluebird resizeHeight 0.5 this.bluebird turn RIGHT 0.25 this.bluebird move FORWARD 1.0 For an arbitrary point ݔ on this bluebird, write the four matrices that represent the code above, and state where the point ends after the program runs.

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Exit Ticket Sample Solutions

૙ Consider the following set of code for the ALICE program featuring a bluebird whose center is located at ൥૙൩. ૙ this.bluebird resize 2.0 this.bluebird resizeHeight 0.5 this.bluebird turn RIGHT 0.25 this.bluebird move FORWARD 1.0

For an arbitrary point ࢞ on this bluebird, write the four matrices that represent the code above, and state where the point ends after the program runs. ૛૙૙ Resize 2.0: ൥૙૛૙൩ ૙૙૛ ૚૙૙ ResizeHeight 0.5: ൥૙૙Ǥ૞૙൩ ૙૙૚ Turn RIGHT 0.25: turning right is a rotation about the ࢟-axis clockwise, so we get ܋ܗܛሺെૢ૙ιሻ ૙ െܛܑܖሺെૢ૙ιሻ ૙૙૚ ൥ ૙૚ ૙൩ ൌ ൥ ૙૚૙൩ ܛܑܖሺെૢ૙ιሻ ૙ ܋ܗܛሺૢ૙ιሻ െ૚ ૙ ૙ Move forward is a translation by ૚ unit in the positive ࢞ direction after the turn right. There is no linear transformation ૚ we can use, but we can write this as ࢞Ԣ ൅ ൥૙൩. ૙ ࢞ would end at ૙૙૚૚૙૙૛૙૙ ૚ ൥ ૙૚૙൩൥૙૙Ǥ૞૙൩൥૙૛૙൩ ࢞൅൥૙൩ െ૚૙૙૙૙૚૙૙૛ ૙ ૙૙૛ ૚ ൥ ૙૚૙൩ ࢞൅൥૙൩ െ૛૙૙ ૙ ࢇ ૛ࢉ ൅ ૚ If ࢞ൌቈ࢈቉, then the image of ࢞ at the end of the program would be ൥ ࢈ ൩. ࢉ െ૛ࢇ

Problem Set Sample Solutions

1. For the following commands, describe a matrix you can use to get the desired result. Assume the character is centered at the origin, facing in the negative ࢠ direction with positive ࢞ on its right for each command. Let ࢉ be a nonzero real number. a. move LEFT ࢉ ૙ െࢉ ൥૙൩ ൅ ቈ ૙ ቉ ૙ ૙

b. move RIGHT ࢉ ૙ ࢉ ൥૙൩ ൅ ቈ૙቉ ૙ ૙

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c. move UP ࢉ ૙ ૙ ൥૙൩ ൅ ൥ࢉ൩ ૙ ૙

d. move DOWN ࢉ ૙ ૙ ൥૙൩ ൅ ൥െࢉ൩ ૙ ૙

e. move FORWARD ࢉ ૙ ૙ ൥૙൩ ൅ ൥ ૙ ൩ ૙ െࢉ

f. move BACKWARD ࢉ ૙ ૙ ൥૙൩ ൅ ൥૙൩ ૙ ࢉ

g. turn LEFT ࢉ ܋ܗܛሺ૜૟૙ࢉιሻ ૙െܛܑܖሺ૜૟૙ࢉιሻ ൥ ૙૚ ૙൩ ܛܑܖሺ૜૟૙ࢉιሻ ૙ ܋ܗܛሺ૜૟૙ࢉιሻ

h. turn RIGHT ࢉ ܋ܗܛሺെ૜૟૙ࢉιሻ ૙െܛܑܖሺെ૜૟૙ࢉιሻ ൥ ૙૚ ૙൩ ܛܑܖሺെ૜૟૙ࢉιሻ ૙ ܋ܗܛሺെ૜૟૙ࢉιሻ

i. turn FORWARD ࢉ ૚૙ ૙ ൥૙ ܋ܗܛሺെ૜૟૙ࢉιሻ െܛܑܖሺെ૜૟૙ࢉιሻ൩ ૙ ܛܑܖሺെ૜૟૙ࢉιሻ ܋ܗܛሺെ૜૟૙ࢉιሻ

j. turn BACKWARD ࢉ ૚૙ ૙ ൥૙ ܋ܗܛሺ૜૟૙ࢉιሻ െܛܑܖሺ૜૟૙ࢉιሻ൩ ૙ ܛܑܖሺ૜૟૙ࢉιሻ ܋ܗܛሺ૜૟૙ࢉιሻ

k. roll LEFT ࢉ ܋ܗܛሺെ૜૟૙ࢉιሻ െܛܑܖሺെ૜૟૙ࢉιሻ ૙ ൥ܛܑܖሺെ૜૟૙ࢉιሻ ܋ܗܛሺെ૜૟૙ࢉιሻ ૙൩ ૙૙૚

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l. roll RIGHT ࢉ ܋ܗܛሺ૜૟૙ࢉιሻ െܛܑܖሺ૜૟૙ࢉιሻ ૙ ൥ܛܑܖሺ૜૟૙ࢉιሻ ܋ܗܛሺ૜૟૙ࢉιሻ ૙൩ ૙૙૚

m. resize ࢉ ࢉ૙૙ ൥૙ࢉ૙൩ ૙૙ࢉ

n. resizeWidth ࢉ ࢉ૙૙ ൥૙૚૙൩ ૙૙૚

o. resizeHeight ࢉ ૚૙૙ ൥૙ࢉ૙൩ ૙૙૚

p. resizeDepth ࢉ ૚૙૙ ൥૙૚૙൩ ૙૙ࢉ

2. In each of the transformations above, we have assumed that each animation will take one second of time. If ࢀ is the number of seconds an animation takes and ࢚ is the current running time of the animation, then rewrite the following commands as a function of ࢚. a. move RIGHT ࢉ duration ࢀ

૙ ࢉ ࢚ ൥૙൩ ൅ ቈ૙቉ ࢀ ૙ ૙

b. turn FORWARD ࢉ duration ࢀ

૚૙ ૙࢚ ൥૙܋ܗܛሺെ૜૟૙ࢉιሻ െܛܑܖሺെ૜૟૙ࢉιሻ൩ ૙ܛܑܖሺെ૜૟૙ࢉιሻ ܋ܗܛሺെ૜૟૙ࢉιሻ ࢀ

c. resize ࢉ duration ࢀ

ࢉ૙૙࢚ ൥૙ࢉ૙൩ ࢀ ૙૙ࢉ

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3. For computational simplicity, we have been assuming that the pivot points of our characters occur at the origin. a. If we apply a rotation matrix like we have been when the pivot point is at the origin, what happens to characters that are not located around the origin?

The characters trace out a circular path much like we saw in previous lessons when we were rotating points instead of objects.

࢞ ࢇ b. Let ࢞ൌቈ࢟቉ be the pivot point of any three-dimensional object and ࡭ൌቈ࢈቉ a point on the surface of the ࢠ ࢉ object. Moving the pivot point to the origin has what effect on ࡭? Find ࡭Ԣ, the image of ࡭ after moving the object so that its pivot point is the origin. ࡭ᇱ ൌ࡭െ࢞ ࢇ ࢞ ൌ ቈ࢈቉ െ ቈ࢟቉ ࢉ ࢠ ࢇെ࢞ ൌ ቈ࢈െ࢟቉ ࢉെࢠ

c. Apply a rotation of ࣂ about the ࢞-axis to ࡭Ԣ. Does this transformation cause a pivot or what you described in part (a)? ૚૙ ૙ ࢇെ࢞ ࢇെ࢞ ൥૙܋ܗܛሺࣂሻ െܛܑܖሺࣂሻ൩ቈ࢈െ࢟቉ ൌ ൥܋ܗܛሺࣂሻሺ࢈െ࢟ሻ െ ܛܑܖሺࣂሻሺࢉ െ ࢠሻ൩ ૙ܛܑܖሺࣂሻ ܋ܗܛሺࣂሻ ࢉെࢠ ܛܑܖሺࣂሻሺ࢈െ࢟ሻ ൅ ܋ܗܛሺࣂሻሺࢉ െ ࢠሻ Since the pivot point has been shifted to the origin, the rotation rotates the object about itself instead of through space.

࢞ d. After applying the rotation, translate the object so that its pivot point returns to ࢞ൌቈ࢟቉. Was the ࢠ ࢞-coordinate affected by the rotation? Was the pivot point? ࢇെ࢞ ࢞ ࢇ ൥܋ܗܛሺࣂሻሺ࢈െ࢟ሻ െܛܑܖሺࣂሻሺࢉെࢠሻ൩ ൅ ቈ࢟቉ ൌ ൥܋ܗܛሺࣂሻሺ࢈െ࢟ሻ െܛܑܖሺࣂሻሺࢉെࢠሻ ൅࢟൩ ܛܑܖሺࣂሻሺ࢈െ࢟ሻ ൅܋ܗܛሺࣂሻሺࢉെࢠሻ ࢠ ܛܑܖሺࣂሻሺ࢈െ࢟ሻ ൅܋ܗܛሺࣂሻሺࢉെࢠሻ ൅ࢠ No, the ࢞-coordinate stayed the same once we shifted it back to the starting point. The pivot point would also have stayed the same since we shifted it to the origin (the origin does not change during rotations) and then shifted it back in the opposite direction.

e. Summarize what you found in parts (a)–(d).

To pivot points that are not centered on the origin, we have to translate the object so that its pivot point is at the origin, then perform the rotation, and then shift the object back. If ࡾ is the rotation matrix, ࡭ is the point on the surface of the object, and ࢞ is the pivot point, then we do the following:

ࡾሺ࡭െ࢞ሻ ൅࢞ for each point ࡭ on the surface of the object.

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Extension:

4. In first-person computer games, we think of the camera moving left-right, forward-backward, and up-down. For computational simplicity, the camera and screen stays fixed and the objects in the game world move in the opposite ૚૙ direction instead. Let the camera be located at ሺ૙ǡ ૙ǡ ૙ሻ, the screen be located at ࢠൌ૚, and the point ࢜ൌ൥ ૟ ൩ ૞ represent the center of an object in the game world. If a character in ALICE is the camera, then answer the following questions. a. What are the coordinates of the projection of ࢜ on the screen? ૛ ࢜ᇱ ൌ ൥૚Ǥ ૛൩ ૚

b. What is the value of the image of ࢜ as the character moves ૝ units closer to ࢜ in the ࢞ direction? ૟ ࢜Ԣ ൌ ൥૟൩ ૞

c. What are the coordinates of the projection of ࢜Ԣ? ૚Ǥ ૛ ࢜ᇱ ൌ ൥૚Ǥ ૛൩ ૚

d. If the character jumps up ૟ units, then where does the image of ࢜ move on the screen? ૚૙ ૛ ࢜ moves to the apparent position ൥ ૙ ൩, which appears to be located at ൥૙൩ on the screen. ૞ ૚

e. In Lesson 26, you learned that if the camera is not in the standard orientation, that rotation matrices need to be applied to the camera first. In Lesson 27, you learned that rotation matrices only pivot an object if that object is located at the origin. If a camera is in a nonstandard orientation and not located at the origin, then should the rotation matrices be applied first or a translation to the origin?

The translation to the origin needs to take place before the rotation or else the camera will be tracing out a circular path around the axes instead of rotating to the correct position.

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Name Date

a. Find values for ܽ, ܾ, ܿ, ݀, and ݁ so that the following matrix product equals the ͵ൈ͵ identity matrix. Explain how you obtained these values.

ܽെ͵ͷ ͳܾ݀ ൥ܿܿ ͳ൩൥ͳܿ݁൩ ͷܾെͶʹܾܾ

,ܾ ݔ ൌܣ b. Represent the following system of linear equations as a single matrix equation of the form .is a ͵ൈ͵ matrix, and ݔ and ܾ are ͵ൈͳ column matrices ܣ where

ݔ൅͵ݕ൅ʹݖൌͺ ʹݔെݕ൅ݖൌെ ݔ ൅ ͵ݕ ൅ ͵ݖ ൌ ͹ʹ

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c. Solve the system of three linear equations given in part (b).

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2. The following diagram shows two two-dimensional vectors ܞ and ܟ in the place positioned to each have an endpoint at point ܲ.

a. On the diagram, make reasonably accurate sketches of the following vectors, again each with an endpoint at ܲ. Be sure to label your vectors on the diagram. i. ʹܞ ii. െܟ iii. ܞ ൅͵ܟ iv. ܟ െʹܞ ଵ v. ܞ ଶ

Vector ܞ has magnitudeͷ units, ܟ has magnitude ͵, and the acute angle between them is Ͷͷι.

b. What is the magnitude of the scalar multiple െͷܞ?

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c. What is the measure of the smallest angle between െͷܞ and ͵ܟ if these two vectors are placed to have a common endpoint?

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.ۄെʹǡ െͳۃ ൌ ܟ and ۄ͵ ǡʹۃ ൌ ܞ Consider the two-dimensional vectors .3

a. What are the components of each of the vectors ܞ൅ܟ and ܞെܟ?

b. On the following diagram, draw representatives of each of the vectors ܞ, ܟ, and ܞ൅ܟ, each with an endpoint at the origin.

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c. The representatives for the vectors ܞ and ܟ you drew form two sides of a parallelogram, with the vector ܞ൅ܟ corresponding to one diagonal of the parallelogram. What vector, directed from the third quadrant to the first quadrant, is represented by the other diagonal of the parallelogram? Express your answer solely in terms of ܞ and ܟ, and also give the coordinates of this vector.

d. Show that the magnitude of the vector ܞ൅ܟ does not equal the sum of the magnitudes of ܞ and of ܟ.

e. Give an example of a non-zero vector ܝ such that ԡܞ൅ܝԡ does equal ԡܞԡ ൅ ԡܝԡ.

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f. Which of the following three vectors has the greatest magnitude: ܞ൅ሺെܟሻ, ܟെܞ, or ሺെܞሻ െሺെܟሻ?

g. Give the components of a vector one-quarter the magnitude of vector ܞ and with direction opposite the direction of ܞ.

4. Vector ܉ points true north and has magnitude ͹ units. Vector ܊ points ͵Ͳι east of true north. What should the magnitude of ܊ be so that ܊െ܉ points directly east?

a. State the magnitude and direction of ܊െ܉.

b. Write ܊െ܉ in magnitude and direction form.

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5. Consider the three points ܣൌሺͳͲǡ െ͵ǡͷሻ, ܤൌሺͲǡʹǡͶሻ, and ܥൌሺʹǡͳǡͲሻ in three-dimensional space. Let ܯ be the midpoint of ܣܤതതതത and ܰ be the midpoint of ܣܥതതതത .

a. Write down the components of the three vectors ሬሬሬሬሬԦܣܤ, ሬሬሬሬሬԦܤܥ, and ܥܣሬሬሬሬሬԦ, and verify through arithmetic that their sum is zero. Also, explain why geometrically we expect this to be the case.

b. Write down the components of the vector ܯܰሬሬሬሬሬሬሬԦ . Show that it is parallel to the vector ܤܥሬሬሬሬሬԦ and half its magnitude.

Let ܩൌሺͶǡͲǡ͵ሻ.

ฮெீሬሬሬሬሬሬԦ ฮ c. What is the value of the ratio ? ฮெ஼ሬሬሬሬሬሬԦ ฮ

d. Show that the point ܩ lies on the line connecting ܯ and ܥ. Show that ܩ also lies on the line connecting ܰ and ܤ.

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6. A section of a river, with parallel banks ͻͷǤ apart, runs true north with a current of ʹȀ . Lashana, an expert swimmer, wishes to swim from point ܣ on the west bank to the point ܤ directly opposite it. In still water, she swims at an average speed of ͵Ȁ .

The diagram to the right illustrates the situation.

To counteract the current, Lashana realizes that she is to swim at some angle ߠ to the east/west direction as shown.

With the simplifying assumptions that Lashana’s swimming speed will be a constant ͵Ȁ and that the current of the water is a uniform ʹȀ flow northward throughout all regions of the river (e.g., we ignore the effects of drag at the river banks,), at what angle ߠ to east/west direction should Lashana swim in order to reach the opposite bank precisely at point ܤ? How long will her swim take?

a. What is the shape of Lashana’s swimming path according to an observer standing on the bank watching her swim? Explain your answer in terms of vectors.

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b. If the current near the banks of the river is significantly less than ʹȀ , and Lashana swims at a constant speed of ͵Ȁ at the constant angle ߠ to the east/west direction as calculated in part (a), will Lashana reach a point different from ܤ on the opposite bank? If so, will she land just north or just south of ܤ? Explain your answer.

7. A ͷ ball experiences a force due to gravity ܨԦ of magnitude Ͷͻ newtons directed vertically downward. If this ball is placed on a ramp tilted at an angle of Ͷͷι, what is the magnitude of the component of this force, in newtons, on the ball directed Ͷͷι toward the bottom of the ramp? (Assume the ball has a radius small enough that all forces are acting at the point of contact of the ball with the ramp.)

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8. Let ܣ be the point ሺͳǡ ͳǡ െ͵ሻ and ܤ be the point ሺെʹǡ ͳǡ െͳሻ in three-dimensional space.

A particle moves along the straight line through ܣ and ܤ at uniform speed in such a way that at time Let ܲሺݐሻ be the location of the .ܤ and at ݐൌͳ second the particle is at ,ܣ ݐൌͲ seconds the particle is at .(ܤand ܲሺͳሻ ൌ ܣparticle at time ݐ (so, ܲሺͲሻ ൌ

.a. Find the coordinates of the point ܲሺݐሻ each in terms of ݐ

b. Give a geometric interpretation of the point ܲሺͲǤͷሻ.

ʹͲͳ Let ܮ be the linear transformation represented by the ͵ൈ͵ matrix ൥ͳ͵Ͳ൩, and let ܣᇱ ൌܮܣ and Ͳͳͳ ܤᇱ ൌܮܤ be the images of the points ܣ and ܤ, respectively, under ܮ.

c. Find the coordinates of ܣԢ and ܤԢ.

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൫ܲሺݐሻ൯, theܮ A second particle moves through three-dimensional space. Its position at time ݐ is given by image of the location of the first particle under the transformation ܮ.

.d. Where is the second particle at times ݐൌͲ and ݐൌͳ? Briefly explain your reasoning

e. Prove that the second particle is also moving along a straight-line path at uniform speed.

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A Progression Toward Mastery STEP 1 STEP 2 STEP 3 STEP 4 Assessment Missing or Missing or A correct answer A correct answer Task Item incorrect answer incorrect answer with some supported by and little evidence but evidence of evidence of substantial of reasoning or some reasoning or reasoning or evidence of solid application of application of application of reasoning or mathematics to mathematics to mathematics to application of solve the problem. solve the problem. solve the problem, mathematics to or an incorrect solve the problem. answer with substantial evidence of solid reasoning or application of mathematics to solve the problem. 1 a Student shows little or Student attempts to Student calculates the Student calculates the no understanding of multiply matrices, correct values for ܽ, ܾ, correct values for ܽ, ܾ, matrix operations. making major mistakes, ܿ, ݀, and ݁ but does not ܿ, ݀, and ݁ and clearly but equates the terms explain or does not explains how the values to the identity matrix. clearly explain how the were obtained. values were obtained. b Student shows little or Student writes one Student writes two Student writes all three .ݔ, or ܾ) matrices correctly ,ܣ) ݔ, or ܾ) matrices ,ܣ) no understanding of matrix matrices. correctly. correctly.

c Student shows little or Student multiplies by Student multiplies by Student multiplies by no understanding of the inverse matrix but the inverse matrix but the inverse matrix matrices. makes mistakes in makes mistakes in arriving at the correct .computations leading to computations leading to values of ݔ, ݕ, and ݖ ,only one correct value two correct values of ݔ .of ݔ, ݕ, or ݖ. ݕ, or ݖ 2 a Student shows little or Student shows some Student shows Student graphs and no evidence of vectors. knowledge of vectors, reasonable knowledge labels all vectors drawing at least two of vectors, drawing at correctly. correctly. least four correctly.

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b Student shows little or Student understands Student calculates the Student calculates the no understanding of the that the magnitude is ͷ magnitude of ܞ magnitude of െͷܞ magnitude of a vector. times the magnitude of correctly but does not correctly. the vector ܞ but multiply by ͷ. calculates the magnitude of ܞ incorrectly. c Student shows little or Student draws vectors Student draws vectors Student draws vectors no understanding of with a common with a common with a common vectors and angles. endpoint but does no endpoint, but the angle endpoint and identifies calculations. identified is Ͷͷι. the angle between them as ͳ͵ͷι. 3 a Student shows little or Student identifies the Student identifies the Student identifies the no understanding of vector components but vector components and vector components and vectors. does not add or either adds or subtracts adds and subtracts the subtract the vectors. the vectors correctly. vectors correctly.

b Student shows little or Student graphs either ܞ Student graphs ܞ and ܟ Student graphs ܞ, ܟ, no understanding of or ܟ correctly but does correctly but does not and ܞ൅ܟ correctly. graphing vectors. not graph ܞ൅ܟ or graph ܞ൅ܟ or graphs graphs ܞ൅ܟ ܞ൅ܟ incorrectly. incorrectly. c Student shows little or Student draws the Student draws the Student draws the no understanding of vector correctly but vector correctly and vector, writes it in terms vectors. does not write it in either writes the vector of ܞ and ܟ, and terms of ܞ and ܟ or in terms of ܞ and ܟ identifies its identify the correctly or identifies its components correctly. components. components correctly. d Student shows little or Student finds one of the Student finds two of the Student finds the no understanding of magnitudes of ܞ, ܟ, or magnitudes of ܞ, ܟ, and magnitudes of ܞ, ܟ, and vector magnitude. ܞ൅ܟ correctly. ܞ൅ܟ correctly. ܞ൅ܟ correctly and shows that the magnitude of ܞ plus the magnitude of ܟ does not equal the magnitude of the sum of ܞ and ܟ. e Student shows little or Student identifies a Student identifies a Student identifies a no understanding of vector ܝ, but it does not vector ܝൌܞ that vector ܝൌܞ that vectors. satisfy the conditions. satisfies the conditions satisfies the conditions but does not justify the and justifies the answer. answer. f Student shows little or Student finds two of the Student finds the three Student finds the three no understanding of three magnitudes magnitudes correctly magnitudes correctly vector magnitude. correctly. but does not state that and states that the the magnitudes are all magnitudes are all the the same. same.

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g Student shows little or Student either Student multiplies ܞ by Student multiplies ܞ by no understanding of ଵ ͳ ͳ multiplies ܞ by or െͳ െ but does not െ and identifies the vectors. ସ Ͷ Ͷ ͳ identify the components of the െ but not by Ͷ. components of the resulting vector. resulting vector. 4 a Student shows little or Student draws the Student draws the Student draws the no understanding of vectors correctly and vectors correctly and vectors correctly and vectors. attempts to determine either the magnitude or determines the correct magnitude and direction is correct but magnitude and direction, but both are not both. direction for both. incorrect. b Student shows little or Student draws the Student draws the Student draws the no understanding of vectors correctly and vectors correctly and vectors correctly and vectors. either the magnitude or determines the correct determines the correct direction is correct but magnitude and magnitude and not both. direction for both but direction for both and does not write them in writes them in magnitude and magnitude and direction form. direction form. 5 a Student shows little or Student correctly Student correctly Student correctly no understanding of identifies the identifies the identifies the vectors. components of all three components of all three components of all three vectors. vectors and shows their vectors, shows their sum is Ͳ. sum is Ͳ, and explains geometrically why this is expected. b Student shows little or Student correctly Student correctly Student no understanding of determines the determines the determines the vectors. components of ሬሬሬሬሬሬሬԦܯܰ. components of ሬሬሬሬሬሬሬԦܯܰ and components of ሬሬሬሬሬሬሬԦܯܰ and either shows that it is shows that it is parallel parallel to or half the to and half the magnitude of ܤܥሬሬሬሬሬԦ . magnitude of ܤܥሬሬሬሬሬԦ . c Student shows little or Student finds the Student finds the Student finds the no understanding of components of ܯܩሬሬሬሬሬሬԦ and components of ܯܩሬሬሬሬሬሬԦ and components of ܯܩሬሬሬሬሬሬԦ and vector magnitude. ܯܥሬሬሬሬሬሬԦ . ܯܥሬሬሬሬሬሬԦ and shows that ܯܥሬሬሬሬሬሬԦ and shows the ratio ሬሬሬሬሬሬԦ ሬሬሬሬሬሬԦ ଵ ܯܥ ൌ͵ܯܩ. is . ଷ

d Student shows little or Student explains that ܩ Student explains that ܩ Student explains that ܩ no understanding of must lie on ܯܥሬሬሬሬሬሬԦ since must lie on ܯܥሬሬሬሬሬሬԦ and must lie on ܯܥሬሬሬሬሬሬԦ and also vectors. ͳ ሬሬሬሬሬሬԦܯܩ ൌ ܯܥሬሬሬሬሬሬԦ . finds that ሬሬሬሬሬሬԦܰܤ. ͵ ͳ ܰܩሬሬሬሬሬሬԦ ൌ ܰܤሬሬሬሬሬሬԦ ͵ .

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6 a Student shows little or Student states or draws Students states or Student states or draws no understanding of the the correct shape of the draws the correct shape the correct shape of the shape of the path or path but does not of the path and uses path and clearly vectors. explain using vectors. vectors to support the explains using vectors. answer but makes errors in reasoning. b Student shows little or Student states that she Student states that she Student states that she no understanding of will not land at point ܤ will land at a point will land at a point vectors. but shows little south of ܤ, but the south of ܤ and clearly supporting work and supporting work is not explains the answer. does not state the point clear or has simple will be south of point ܤ. mistakes. 7 Student shows little or Student identifies the Student identifies the Student identifies the no understanding of components of the components of the components of the vector components or vector. vector and attempts to vector and correctly magnitude. find the magnitude of calculates the the force component magnitude of the force down the ramp but component down the makes calculation ramp. mistakes. 8 a Student shows little or Student calculates the Student calculates the Student calculates the no understanding of components of ܣܤሬሬሬሬሬԦ . components of vector components of ܣܤሬሬሬሬሬԦ and vectors. ܣܤሬሬሬሬሬԦ and understands finds the coordinates of .ሺݐሻ in terms of ݐܲ ܤܣ൅ ݐ ܣ that ܲሺݐሻ ൌ but does not determine .ሺݐሻ in terms of ݐܲ b Student shows little or Students states either Students states that Students states that no understanding of ܲሺͲሻ ൌ ܣ or ܲሺͳሻ ൌ ܤ. ܲሺͲሻ ൌ ܣ and ܲሺͲሻ ൌ ܣ, ܲሺͳሻ ൌ ܤ, vectors. ܲሺͳሻ ൌ ܤ. and ܲሺͲǤͷሻ is the midpoint of segment ܣܤ. c Student shows little or Students sets up the Student sets up the Student sets up the no knowledge of matrices to determine matrices to determine matrices to determine vectors and matrices. ܣԢ and ܤԢ but makes ܣԢ and ܤԢ and calculates ܣԢ and ܤԢ and calculates errors in calculating one correctly. both correctly. both. d Student shows little or Student understands Student correctly Student correctly ൫ܲሺݐሻ൯ must be calculates either the calculates the positionܮ no knowledge of that ൫ܲሺͲሻ൯ andܮ vectors and matrices. used to find the position at ݐൌͲ as at ݐൌͲ as .൫ܲሺͳሻ൯ܮ ൫ܲሺͲሻ൯ or ݐൌͳ asܮ position of the second .൫ܲሺͳሻ൯ܮ particle. ݐൌͳ as

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e Student shows little or Student states the Student states the Student states the no understanding of location of the second location of the second location of the second ൫ܲሺݐሻ൯ andܮ ൫ܲሺݐሻ൯ and particle isܮ ൫ܲሺݐሻ൯. particle isܮ vectors and matrices. particle is that that

ԢሬሬሬሬሬሬሬሬԦܤԢܣԢ ൅ ݐܣ ൫ܲሺݐሻ൯ ൌܮ . ԢሬሬሬሬሬሬሬሬԦܤԢܣԢ ൅ ݐܣ ൫ܲሺݐሻ൯ ൌܮ and explains the particle is moving on a straight line from ܣǯ to ܤǯ at uniform velocity given by ܣǯܤǯሬሬሬሬሬሬሬԦ.

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Name Date

a. Find values for ܽ, ܾ, ܿ, ݀, and ݁ so that the following matrix product equals the ͵ൈ͵ identity matrix. Explain how you obtained these values.

ܽെ͵ͷ ͳܾ݀ ൥ܿܿ ͳ൩൥ͳܿ݁൩ ͷܾെͶʹܾܾ

The row 1, column 1 entry of the product is a-3 + 10. This should equal 1, so a = -6.

The row 2, column 1 entry of the product is c + c + 2. This should equal zero, so c=-1.

The row 3, column 1 entry of the product is 5 + b - 8. This should equal zero, so b = 3.

The row 1, column 3 entry of the product is ad - 3e + 5b = –6d - 3e + 15, and this should equal zero. So, we should have 6d + 3e = 15.

The row 2, column 3 entry of the product is cd + ce + b = –d- e + 3, and this should equal zero. So, we need d + e = 3.

Solving the two linear equations d and e gives d = 2 and e = 1.

So, in summary, we have a=–6, b=3, c = –1, d=2, and e=1.

,ܾ ݔ ൌܣ b. Represent the following system of linear equations as a single matrix equation of the form .is a ͵ൈ͵ matrix and ݔ and ܾ are ͵ൈͳ column matrices ܣ where

ݔ൅͵ݕ൅ʹݖൌͺ ʹݔെݕ൅ݖൌെ ݔ ൅ ͵ݕ ൅ ͵ݖ ൌ ͹ʹ

132 x 8 We have Ax=b with A=቎1–11቏ and x= ቈy቉ and b= ቎–2቏. 233 z 7

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c. Solve the system of three linear equations given in part (b).

ି The solution is given by x=A 1b, if the matrix inverse exists. But part (a) shows that if we set –6 –3 5 B= ൥–1 –1 1 ൩ , 53–4 then we have BA = I. We could check that AB = I, as well (in which case B is the matrix inverse of A), but even without knowing this, from 132x 8 Ax = b ൥1-11൩ቈy቉ൌ൥-2൩ , 233z 7

we get -6 -3 5 132x -6 -3 5 8 BAx = Bb ൥-1 -1 1 ൩൥1-11൩ቈy቉ൌ൥-1 -1 1 ൩൥-2൩ 53-4233z 53-47 10Ͳ x -6 -3 5 8 Ix = Bb ൥Ͳ 1 Ͳ൩ቈy቉ൌ൥-1 -1 1 ൩൥-2൩ ͲͲͳz 53-47 x -6 -3 5 8 x = Bb ቈy቉ൌ൥-1 -1 1 ൩൥-2൩ z 53-47 x -7 ቈy቉ൌ൥ 1 ൩ ; z 6

that is, that x =–7, y = 1, z = 6 is a solution to the system of equations. (In fact, it can be the only solution to the system.)

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2. The following diagram shows two two-dimensional vectors ܞ and ܟ in the place positioned to each have an endpoint at point ܲ.

a. On the diagram, make reasonably accurate sketches of the following vectors, again each with endpoint at ܲ. Be sure to label your vectors on the diagram. i. ʹܞ ii. െܟ iii. ܞ൅͵ܟ iv. ܟെʹܞ ͳ ܞ v. ʹ

Vector ܞ has magnitude ͷ units, ܟ has magnitude ͵, and the acute angle between them is Ͷͷι.

b. What is the magnitude of the scalar multiple െͷܞ?

We have ቛ–5vቛ=5ԡvԡ=5×5=25.

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c. What is the measure of the smallest angle between െͷܞ and ͵ܟ if these two vectors are placed to have a common endpoint?

The two vectors in question have an angle of smallest measure 135° between them.

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.ۄെʹǡ െͳۃൌܟ and ۄ͵ǡʹۃൌܞ Consider the two-dimensional vectors .3

a. What are the components of each of the vectors ܞ൅ܟ and ܞെܟ?

ۄ4 ,4ۃ = ۄቀ-2ቁ , 3- ቀ-1ቁ -2ۃ = and v-w ۄ2 ,0ۃ = ۄ(ቀ-2ቁ , 3+(-1 +2ۃ = v + w

b. On the following diagram, draw representatives of each of the vectors ܞ, ܟ, and ܞ൅ܟ, each with an endpoint at the origin.

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ሬሬሬሬሬሬԦ Label the points A and B as shown. The vector we seek is AB. To move from A to B we need to follow –w and then v. Thus, the vector we seek is –w + v, which is the same as

v – w.

Also, we see that to move from A to B we need to move 4 units to the right and 4 .ۄ4 ,4ۃ = ۄቀ-2ቁ , 3 - ቀ-1ቁ - 2ۃ = units upward. This is consistent with v - w

d. Show that the magnitude of the vector ܞ൅ܟ does not equal the sum of the magnitudes of ܞ and of ܟ.

2 2 2 2 ට We have ԡvԡ=ඥ2 +3 =ξ13 and ԡwԡ= ቀ-2ቁ + ቀ-1ቁ =ξ5, so ԡvԡ+ԡwԡ=ξ13+ξ5.

Now, ԡv+wԡ=ඥ02+22=2. This does not equal ξ13+ξ5.

e. Give an example of a non-zero vector ܝ such that ԡܞ൅ܝԡ does equal ԡܞԡ ൅ ԡܝԡ.

Choosing u to be the vector v works.

ԡv+uԡ=ԡv+vԡ=ԡ2vԡ=2ԡvԡ=ԡvԡ+ԡvԡ

(In fact, u=kv for any positive real number k works.)

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f. Which of the following three vectors has the greatest magnitude: ܞ൅ሺെܟሻ, ܟെܞ, or ሺെܞሻ െ ሺെܟሻ?

(-v) – (–w) = –v + w = w – v and

v + (–w) = v – w = –(w – v)Ǥ

So, each of these vectors is either w – v or the scalar multiple (–1)(w – v), which is the same vector but with opposite direction. They all have the same magnitude.

g. Give the components of a vector one-quarter the magnitude of vector ܞ and with direction opposite the direction of ܞ.

1 1 1 3 ۄ -, -ۃ = ۄ2,3ۃ -= v - 4 4 2 4

4. Vector ܉ points true north and has a magnitude of ͹. Vector ܊ points ͵Ͳι east of true north. What should the magnitude of ܊ be so that ܊െ܉ points directly east? a. State the magnitude and direction of ܊െ܉.

We hope to have the following vector diagram incorporating a right triangle.

We have ԡaԡ=7, and for this 30-60-90 triangle we need

ฮbฮ=2 ቛb-aቛ and 7=ԡaԡ=ξ3 ቛb-aቛ. This shows the magnitude 14 of b should be ฮbฮ= . ඥ3 7 The vector b–a points east and has magnitude . ξ3

b. Write ܊െ܉ in magnitude and direction form.

7 7 b – a has a magnitude of and a direction of 0° measured from the ;ۄ°0 , ۃ ඥ3 ξ3 horizontal.

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5. Consider the three points ܣൌሺͳͲǡ െ͵ǡ ͷሻ, ܤൌሺͲǡ ʹǡ Ͷሻ, and ܥൌሺʹǡ ͳǡ Ͳሻ in three-dimensional space. Let ܯ be the midpoint of ܣܤതതതത and ܰ be the midpoint of ܣܥതതതത .

a. Write down the components of the three vectors ሬሬሬሬሬԦܣܤ, ܤܥሬሬሬሬሬԦ , and ܥܣሬሬሬሬሬԦ, and verify through arithmetic that their sum is zero. Also, explain why geometrically we expect this to be the case.

We have the following: ሬሬሬሬሬሬԦ ۄ1- ,5 ,10-ۃ AB ൌ ሬሬሬሬሬሬԦ ۄ4- ,1- ,2ۃ BC ൌ ሬሬሬሬሬሬԦ ۄ5 ,4- ,8ۃ CA ൌ

.ۄ0 ,0 ,0ۃ= ۄ5+4-1- ,4-1-5 ,8+2+10-ۃ Their sum is This is to be expected as the three points A, B, and C are vertices of a triangle (even in ሬሬሬሬሬሬԦ ሬሬሬሬሬሬԦ ሬሬሬሬሬሬԦ three-dimensional space), and the vectors AB, BC, and CA, when added geometrically, traverse the sides of the triangle and have the sum effect of “returning to start.” That is, the cumulative effect of the three vectors is no vectorial shift at all.

b. Write down the components of the vector ܯܰሬሬሬሬሬሬሬԦ . Show that it is parallel to the vector ܤܥሬሬሬሬሬԦ and half its magnitude.

1 9 5 ሬሬሬሬሬሬሬሬԦ 1 .ۄ2-, -,1ۃ =We have M=ቀ5,- , ቁ and N=ቀ6,-1, ቁ. Thus, MN 2 2 2 2 ሬሬሬሬሬሬሬሬԦ 1 1 ሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬԦ BC, which shows that MN has the same direction as BC = ۄ1,4-,2ۃ =We see that MN 2 2 (and hence is parallel to it) and half the magnitude.

Let ܩൌሺͶǡͲǡ͵ሻ.

ฮெீሬሬሬሬሬሬԦ ฮ c. What is the value of the ratio ? ฮெ஼ሬሬሬሬሬሬԦ ฮ

ሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬԦ ቛMGቛ ቛMGቛ ሬሬሬሬሬሬሬԦ 1 ۄ9 3 ۃ ሬሬሬሬሬሬሬሬԦ ۄ3 1 ۃ ሬሬሬሬሬሬሬԦ MG = -1, ,- and MC = -3, ,- = 3MG. Thus, ሬሬሬሬሬሬሬԦ = ሬሬሬሬሬሬሬԦ = . 2 2 2 2 ቛMCቛ 3ቛMCቛ 3

d. Show that the point ܩ lies on the line connecting ܯ and ܥ. Show that ܩ also lies on the line connecting ܰ and ܤ.

ሬሬሬሬሬሬሬሬሬԦ 1 ሬሬሬሬሬሬሬԦ തതതതത That MG= MC means that the point G lies a third of the way along MC. 3 ሬሬሬሬሬሬሬԦ 1 1 3 1 ሬሬሬሬሬሬԦ NB = ۄ ,3 ,6-ۃ = ۄ ,1 ,2–ۃ =Check: NG 2 3 2 3 തതതതത So, G also lies on NB (and one-third of the way along, too).

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The diagram to the right illustrates the situation.

To counteract the current, Lashana realizes that she is to swim at some angle ߠ to the east/west direction as shown.

With the simplifying assumptions that Lashana’s swimming speed will be a constant ͵Ȁ and that the current of the water is a uniform ʹȀ flow northward throughout all regions of the river (e.g., we ignore the effects of drag at the river banks), at what angle ߠ to east/west direction should Lashana swim in order to reach the opposite bank precisely at point ܤ? How long will her swim take?

a. What is the shape of Lashana’s swimming path according to an observer standing on the bank watching her swim? Explain your answer in terms of vectors.

Lashana’s velocity vector v has magnitude 3 and resolves into two components as shown, a component

in the east direction vE and a component in the south

direction vS.

We see

ԡ ԡ ԡ ԡ vs = v ௘sin Ż =3௘sin Ż .

Lashana needs this component of her velocity vector to counteract the northward current of the water. This will ensure that Lashana will swim directly toward point B with no sideways deviation.

–1 2 Since the current is 2 ft/sec, we need 3௘sinŻ=2, showing that Ż= sin ቀ ቁ§41.8°. 3 ԡ ԡ Lashana will then swim at a speed of vE =3௘cosŻ ft/sec toward the opposite bank.

2 ඥ5 Since sin Ż = , Ż is part of a 2-ξ5-3 right triangle, so cos Ż = . Thus, 3 3 ԡ ԡ vE =ξ5 ft/sec.

She needs to swim an east/west distance of 95 ft. at this speed. It will take her 95 ft =19ξ5 seconds§42 seconds to do this. ξ5 ft/sec

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As noted in the previous solution, Lashana will have no sideways motion in her swim. She will swim a straight-line path from A to B.

If the current is slower than 2 ft/sec at any region of the river surface, Lashana’s

velocity vector component vS, which has magnitude 2 ft/sec, will be larger in magnitude than the magnitude of the current. Thus, she will swim slightly southward during these periods. Consequently, she will land at a point on the opposite bank south of B.

7. A ͷ ball experiences a force due to gravity ܨԦ of magnitude Ͷͻ newtons directed vertically downward. If this ball is placed on a ramp tilted at an angle of Ͷͷι, what is the magnitude of the component of this force, in newtons, on the ball directed Ͷͷιtoward the bottom of the ramp? (Assume the ball has a small enough radius that all forces are acting at the point of contact of the ball with the ramp.)

The force vector can be resolved into two components as shown:

Framp and Fperp.

We are interested in the component Framp.

We see a 45-90-45 triangle in this diagram, with hypotenuse

of magnitude 49 N. This means that the magnitude of Framp is 49 N §35 N. ξ2

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8. Let ܣ be the point ሺͳǡ ͳǡ െ͵ሻ and ܤ be the point ሺെʹǡ ͳǡ െͳሻ in three-dimensional space.

.a. Find the coordinates of the point ܲሺݐሻ each in terms of ݐ

We will write the coordinates of points as 3×1 column matrices, as is consistent for work with matrix notation.

ሬሬሬሬሬሬԦ So, its position at time t is .ۄ2 ,0 ,3-ۃ =The velocity vector of the particle is AB

ې ۍ 1 -3 1 - 3t ۑ ێ ሬሬሬሬሬሬԦ Pሺtሻ=A+tAB=቎ 1 ቏+t ቎ ቏= 1 . ۑ ێ 0 ے2t+3-ۏ 2 -3

b. Give a geometric interpretation of the point ܲሺͲǤͷሻ.

തതതതത Since P(0) = A and P(1) = B, P(0.5) is the midpoint of AB.

c. Find the coordinates of ܣԢ and ܤԢ.

ې ۍ ې ۍ 1 201 -1 201-2 -5 ۑ ێ ۑ ێ A'=൥130൩቎ 1 ቏= 4 and B'=൥130൩ 1 =቎ ቏ 1 ۑ ێ ۑ ێ 0 ے1-ۏ011 ے2–ۏ 011-3

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൫ܲሺݐሻ൯, theܮ A second particle moves through three-dimensional space. Its position at time ݐ is given by image of the location of the first particle under the transformation ܮ.

.d. Where is the second particle at times ݐൌͲ and ݐൌͳ? Briefly explain your reasoning

We see L൫Pሺ0ሻ൯=L(A)=A' and L൫Pሺ1ሻ൯=LሺBሻ=B'. Since the position of the particle at time t is given by L൫Pሺtሻ൯, to find the location at t = 0 and t = 1, evaluate L൫Pሺ0ሻ൯ and L൫Pሺ1ሻ൯.

e. Prove that the second particle is also moving along a straight-line path at uniform speed.

At time t the location of the second particle is

201 1-3t -1-4t -1 -4 L൫Pሺtሻ൯=൥130൩൥ 1 ൩=൥ 4-3t ൩=൥ 4 ൩+t ൥-3൩ . 011-3+2t -2+2t -2 2 We recognize this as ሬሬሬሬሬሬሬሬሬԦ L൫Pሺtሻ൯=A'+t A'B'.

Thus, the second particle is moving along the straight line through A' and B' at a ሬሬሬሬሬሬሬሬሬԦ .ۄ2 ,3-,4-ۃ ='uniform velocity given by the vector A'B

490 Module 2: Vectors and Matrices

A STORY OF FUNCTIONS Credits M2 PRECALCULUS AND ADVANCED TOPICS Credits

'ƌĞĂƚDŝŶĚƐΠŚĂƐŵĂĚĞĞǀĞƌǇĞīŽƌƚƚŽŽďƚĂŝŶƉĞƌŵŝƐƐŝŽŶĨŽƌƚŚĞƌĞƉƌŝŶƟŶŐŽĨĂůůĐŽƉǇƌŝŐŚƚĞĚŵĂƚĞƌŝĂů͘ /ĨĂŶǇŽǁŶĞƌŽĨĐŽƉǇƌŝŐŚƚĞĚŵĂƚĞƌŝĂůŝƐŶŽƚĂĐŬŶŽǁůĞĚŐĞĚŚĞƌĞŝŶ͕ƉůĞĂƐĞĐŽŶƚĂĐƚ'ƌĞĂƚDŝŶĚƐĨŽƌƉƌŽƉĞƌ ĂĐŬŶŽǁůĞĚŐŵĞŶƚŝŶĂůůĨƵƚƵƌĞĞĚŝƟŽŶƐĂŶĚƌĞƉƌŝŶƚƐŽĨƚŚŝƐŵŽĚƵůĞ͘ ඵ ůůŵĂƚĞƌŝĂůĨƌŽŵƚŚĞŽŵŵŽŶŽƌĞ^ƚĂƚĞ^ƚĂŶĚĂƌĚƐĨŽƌDĂƚŚĞŵĂƟĐƐΞŽƉǇƌŝŐŚƚϮϬϭϬEĂƟŽŶĂů 'ŽǀĞƌŶŽƌƐƐƐŽĐŝĂƟŽŶĞŶƚĞƌĨŽƌĞƐƚWƌĂĐƟĐĞƐĂŶĚŽƵŶĐŝůŽĨŚŝĞĨ^ƚĂƚĞ^ĐŚŽŽůKĸĐĞƌƐ͘ůůƌŝŐŚƚƐ ƌĞƐĞƌǀĞĚ͘ ඵ >ĞƐƐŽŶϮϯ͕ƉƉ͘ϯϵϬʹϯϵϭ͗DĂƉĚĂƚĂΞ'ŽŽŐůĞ ඵ >ĞƐƐŽŶϮϱ͕Ɖ͘ϰϮϰ͗;ůĞŌͿ͕^ĐĂůĂͬƌƚZĞƐŽƵƌĐĞ͕Ez͖;ƌŝŐŚƚͿ͕ůŝŶĂƌŝͬƌƚZĞƐŽƵƌĐĞ͕Ez

Module 2: Credits 491