DOCSLIB.ORG

Vector Notation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Vector Notation Trigonometry Vector Notation A vector is a line segment. It has a starting point and an ending point. A vector has direction and length. It may be easiest to think of a vector as a line on a road map. There are usually several ways to describe the route. Look at the diagram below. End Point Start Point 5 45° Telling someone to get from the beginning to the end, you could say “Go 5 miles at an angle (in standard position) of 45 degrees.” This direct route is a vector. In this case, 5 is the magnitude, or length, of the vector, and 45° is the amplitude, or direction, of the vector. Often, when we travel, we cannot go along the vector. Suppose you want to tell a neighbor how to go from your apartment to the library. In the diagram above, you may tell them to “Go east 3.5 miles then north 3.5 miles.” You are breaking the directions into components. We often break vectors into two components; the i component is the number of units in the x-axis direction and the j component is the number of units in the y-axis direction. In this example, we would write the vector: li=+3.5 3.5 j. The vector is the hypotenuse of the triangle; notice that the variable for the vector has an arrow symbol over it. Vector components do not have to be positive. The vector v = −3i − 2 j , which can also be ⎛ −3⎞ written v = , is shown below. ⎝⎜ −2⎠⎟ −3i −2 j v The length of a vector is its magnitude. The symbol for magnitude of a vector can be written v , or like an absolute value, v , or just written “magnitude”. If you are given the components of a vector, use the distance formula to find the magnitude. 2 2 The magnitude of vector v = −3i − 2 j is ()−3 +−()2 = 13 . Amplitude is the angle of the vector. We measure the angle as if the vector was the terminal side of an angle in standard position. When given the i and j components of the vector, use right triangle trigonometry to find the amplitude. Example: Find the magnitude and amplitude of vector vi=12+ 5 j. Sketch of the vector. v 5 The magnitude is ()12 2 + ()5 2 = 13 . θ 12 −1 ⎛⎞5 The amplitude is θ ==tan⎜⎟ 22.6°. ⎝⎠12 Example: Find the magnitude and amplitude of vector v = −3i − 2 j . -3 Sketch of the vector. θ 22 -2 The magnitude is ()−+−=32 () 13. v −1 ⎛⎞−2 The amplitude is θ ==tan⎜⎟ 33.7°. ⎝⎠−3 −1 ⎛⎞−2 The angleθ ==tan⎜⎟ 33.7°, but this angle is not in standard position. From the ⎝⎠−3 diagram, we can see that the angle should be a quadrant 3 angle, 180 +=33.7 213.7 The amplitude of vector v = −3i − 2 j is 213.7°. Try these: Sketch the following vectors in standard position. 1. f = 4i − 4 j ⎛ 1⎞ 2. g = ⎜ 7⎟ ⎝ ⎠ 3. amplitude=135°, magnitude = 2 cm Find the magnitude and amplitude of the following vectors. 4. f = 4i − 4 j ⎛ 1⎞ 5. g = ⎜ 7⎟ ⎝ ⎠ Vectors are line segments with length and direction. Compare the following pairs of vectors. Vectors v and w have exactly the same length and direction, we call them v equal vectors. They don’t need the same starting points, but they have the same w magnitude and amplitude. v Vectors v and w have exactly the same length and start at the same point, but w they point in opposite directions, we call them opposite vectors. v Vectors v and w have the same direction, but different lengths, we call them parallel vectors. w Label the following pairs of vectors as equal, opposite, parallel, or none of these. (Hint- sketch the vectors first) • a = 3i + 2 j • a = 3i + 2 j • a = 3i + 2 j ⎛ 3⎞ b =−3i − 2 j c = 6i + 4 j e = ⎝⎜2⎠⎟ .
Load more

Recommended publications
  • Linear Algebra I
    Linear Algebra I Martin Otto Winter Term 2013/14 Contents 1 Introduction7 1.1 Motivating Examples.......................7 1.1.1 The two-dimensional real plane.............7 1.1.2 Three-dimensional real space............... 14 1.1.3 Systems of linear equations over Rn ........... 15 1.1.4 Linear spaces over Z2 ................... 21 1.2 Basics, Notation and Conventions................ 27 1.2.1 Sets............................ 27 1.2.2 Functions......................... 29 1.2.3 Relations......................... 34 1.2.4 Summations........................ 36 1.2.5 Propositional logic.................... 36 1.2.6 Some common proof patterns.............. 37 1.3 Algebraic Structures....................... 39 1.3.1 Binary operations on a set................ 39 1.3.2 Groups........................... 40 1.3.3 Rings and fields...................... 42 1.3.4 Aside: isomorphisms of algebraic structures...... 44 2 Vector Spaces 47 2.1 Vector spaces over arbitrary fields................ 47 2.1.1 The axioms........................ 48 2.1.2 Examples old and new.................. 50 2.2 Subspaces............................. 53 2.2.1 Linear subspaces..................... 53 2.2.2 Affine subspaces...................... 56 2.3 Aside: affine and linear spaces.................. 58 2.4 Linear dependence and independence.............. 60 3 4 Linear Algebra I | Martin Otto 2013 2.4.1 Linear combinations and spans............. 60 2.4.2 Linear (in)dependence.................. 62 2.5 Bases and dimension....................... 65 2.5.1 Bases............................ 65 2.5.2 Finite-dimensional vector spaces............. 66 2.5.3 Dimensions of linear and affine subspaces........ 71 2.5.4 Existence of bases..................... 72 2.6 Products, sums and quotients of spaces............. 73 2.6.1 Direct products...................... 73 2.6.2 Direct sums of subspaces................
    [Show full text]
  • Vector Spaces in Physics
    San Francisco State University Department of Physics and Astronomy August 6, 2015 Vector Spaces in Physics Notes for Ph 385: Introduction to Theoretical Physics I R. Bland TABLE OF CONTENTS Chapter I. Vectors A. The displacement vector. B. Vector addition. C. Vector products. 1. The scalar product. 2. The vector product. D. Vectors in terms of components. E. Algebraic properties of vectors. 1. Equality. 2. Vector Addition. 3. Multiplication of a vector by a scalar. 4. The zero vector. 5. The negative of a vector. 6. Subtraction of vectors. 7. Algebraic properties of vector addition. F. Properties of a vector space. G. Metric spaces and the scalar product. 1. The scalar product. 2. Definition of a metric space. H. The vector product. I. Dimensionality of a vector space and linear independence. J. Components in a rotated coordinate system. K. Other vector quantities. Chapter 2. The special symbols ij and ijk, the Einstein summation convention, and some group theory. A. The Kronecker delta symbol, ij B. The Einstein summation convention. C. The Levi-Civita totally antisymmetric tensor. Groups. The permutation group. The Levi-Civita symbol. D. The cross Product. E. The triple scalar product. F. The triple vector product. The epsilon killer. Chapter 3. Linear equations and matrices. A. Linear independence of vectors. B. Definition of a matrix. C. The transpose of a matrix. D. The trace of a matrix. E. Addition of matrices and multiplication of a matrix by a scalar. F. Matrix multiplication. G. Properties of matrix multiplication. H. The unit matrix I. Square matrices as members of a group.
    [Show full text]
  • Vectors, Matrices and Coordinate Transformations
    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
    [Show full text]
  • An Attempt to Intuitively Introduce the Dot, Wedge, Cross, and Geometric Products
    An attempt to intuitively introduce the dot, wedge, cross, and geometric products Peeter Joot March 21, 2008 1 Motivation. Both the NFCM and GAFP books have axiomatic introductions of the gener- alized (vector, blade) dot and wedge products, but there are elements of both that I was unsatisfied with. Perhaps the biggest issue with both is that they aren’t presented in a dumb enough fashion. NFCM presents but does not prove the generalized dot and wedge product operations in terms of symmetric and antisymmetric sums, but it is really the grade operation that is fundamental. You need that to define the dot product of two bivectors for example. GAFP axiomatic presentation is much clearer, but the definition of general- ized wedge product as the totally antisymmetric sum is a bit strange when all the differential forms book give such a different definition. Here I collect some of my notes on how one starts with the geometric prod- uct action on colinear and perpendicular vectors and gets the familiar results for two and three vector products. I may not try to generalize this, but just want to see things presented in a fashion that makes sense to me. 2 Introduction. The aim of this document is to introduce a “new” powerful vector multiplica- tion operation, the geometric product, to a student with some traditional vector algebra background. The geometric product, also called the Clifford product 1, has remained a relatively obscure mathematical subject. This operation actually makes a great deal of vector manipulation simpler than possible with the traditional methods, and provides a way to naturally expresses many geometric concepts.
    [Show full text]
  • Dirac Notation 1 Vectors
    Physics 324, Fall 2001 Dirac Notation 1 Vectors 1.1 Inner product T Recall from linear algebra: we can represent a vector V as a column vector; then V y = (V )∗ is a row vector, and the inner product (another name for dot product) between two vectors is written as AyB = A∗B1 + A∗B2 + : (1) 1 2 ··· In conventional vector notation, the above is just A~∗ B~ . Note that the inner product of a vector with itself is positive definite; we can define the· norm of a vector to be V = pV V; (2) j j y which is a non-negative real number. (In conventional vector notation, this is V~ , which j j is the length of V~ ). 1.2 Basis vectors We can expand a vector in a set of basis vectors e^i , provided the set is complete, which means that the basis vectors span the whole vectorf space.g The basis is called orthonormal if they satisfy e^iye^j = δij (orthonormality); (3) and an orthonormal basis is complete if they satisfy e^ e^y = I (completeness); (4) X i i i where I is the unit matrix. (note that a column vectore ^i times a row vectore ^iy is a square matrix, following the usual definition of matrix multiplication). Assuming we have a complete orthonormal basis, we can write V = IV = e^ e^yV V e^ ;V (^eyV ) : (5) X i i X i i i i i ≡ i ≡ The Vi are complex numbers; we say that Vi are the components of V in the e^i basis.
    [Show full text]
  • The Dot Product
    The Dot Product In this section, we will now concentrate on the vector operation called the dot product. The dot product of two vectors will produce a scalar instead of a vector as in the other operations that we examined in the previous section. The dot product is equal to the sum of the product of the horizontal components and the product of the vertical components. If v = a1 i + b1 j and w = a2 i + b2 j are vectors then their dot product is given by: v · w = a1 a2 + b1 b2 Properties of the Dot Product If u, v, and w are vectors and c is a scalar then: u · v = v · u u · (v + w) = u · v + u · w 0 · v = 0 v · v = || v || 2 (cu) · v = c(u · v) = u · (cv) Example 1: If v = 5i + 2j and w = 3i – 7j then find v · w. Solution: v · w = a1 a2 + b1 b2 v · w = (5)(3) + (2)(-7) v · w = 15 – 14 v · w = 1 Example 2: If u = –i + 3j, v = 7i – 4j and w = 2i + j then find (3u) · (v + w). Solution: Find 3u 3u = 3(–i + 3j) 3u = –3i + 9j Find v + w v + w = (7i – 4j) + (2i + j) v + w = (7 + 2) i + (–4 + 1) j v + w = 9i – 3j Example 2 (Continued): Find the dot product between (3u) and (v + w) (3u) · (v + w) = (–3i + 9j) · (9i – 3j) (3u) · (v + w) = (–3)(9) + (9)(-3) (3u) · (v + w) = –27 – 27 (3u) · (v + w) = –54 An alternate formula for the dot product is available by using the angle between the two vectors.
    [Show full text]
  • 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
    LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set.
    [Show full text]
  • Low-Level Image Processing with the Structure Multivector
    Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.
    [Show full text]
  • Basics of Linear Algebra
    Basics of Linear Algebra Jos and Sophia Vectors ● Linear Algebra Definition: A list of numbers with a magnitude and a direction. ○ Magnitude: a = [4,3] |a| =sqrt(4^2+3^2)= 5 ○ Direction: angle vector points ● Computer Science Definition: A list of numbers. ○ Example: Heights = [60, 68, 72, 67] Dot Product of Vectors Formula: a · b = |a| × |b| × cos(θ) ● Definition: Multiplication of two vectors which results in a scalar value ● In the diagram: ○ |a| is the magnitude (length) of vector a ○ |b| is the magnitude of vector b ○ Θ is the angle between a and b Matrix ● Definition: ● Matrix elements: ● a)Matrix is an arrangement of numbers into rows and columns. ● b) A matrix is an m × n array of scalars from a given field F. The individual values in the matrix are called entries. ● Matrix dimensions: the number of rows and columns of the matrix, in that order. Multiplication of Matrices ● The multiplication of two matrices ● Result matrix dimensions ○ Notation: (Row, Column) ○ Columns of the 1st matrix must equal the rows of the 2nd matrix ○ Result matrix is equal to the number of (1, 2, 3) • (7, 9, 11) = 1×7 +2×9 + 3×11 rows in 1st matrix and the number of = 58 columns in the 2nd matrix ○ Ex. 3 x 4 ॱ 5 x 3 ■ Dot product does not work ○ Ex. 5 x 3 ॱ 3 x 4 ■ Dot product does work ■ Result: 5 x 4 Dot Product Application ● Application: Ray tracing program ○ Quickly create an image with lower quality ○ “Refinement rendering pass” occurs ■ Removes the jagged edges ○ Dot product used to calculate ■ Intersection between a ray and a sphere ■ Measure the length to the intersection points ● Application: Forward Propagation ○ Input matrix * weighted matrix = prediction matrix http://immersivemath.com/ila/ch03_dotprodu ct/ch03.html#fig_dp_ray_tracer Projections One important use of dot products is in projections.
    [Show full text]
  • A Guided Tour to the Plane-Based Geometric Algebra PGA
    A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra.
    [Show full text]
  • Relativistic Inversion, Invariance and Inter-Action
    S S symmetry Article Relativistic Inversion, Invariance and Inter-Action Martin B. van der Mark †,‡ and John G. Williamson *,‡ The Quantum Bicycle Society, 12 Crossburn Terrace, Troon KA1 07HB, Scotland, UK; [email protected] * Correspondence: [email protected] † Formerly of Philips Research, 5656 AE Eindhoven, The Netherlands. ‡ These authors contributed equally to this work. Abstract: A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter. Keywords: invariants; inversion; division; non-division algebra; Dirac algebra; Clifford algebra; geometric algebra; special relativity; photon interaction Citation: van der Mark, M.B.; 1. Introduction Williamson, J.G. Relativistic Inversion, Invariance and Inter-Action.
    [Show full text]
  • Chapter 3 Vectors
    Chapter 3 Vectors 3.1 Vector Analysis ....................................................................................................... 1 3.1.1 Introduction to Vectors ................................................................................... 1 3.1.2 Properties of Vectors ....................................................................................... 1 3.2 Cartesian Coordinate System ................................................................................ 5 3.2.1 Cartesian Coordinates ..................................................................................... 6 3.3 Application of Vectors ............................................................................................ 8 Example 3.1 Vector Addition ................................................................................. 12 Example 3.2 Sinking Sailboat ................................................................................ 13 Example 3.3 Vector Description of a Point on a Line .......................................... 14 Example 3.4 Rotated Coordinate Systems ............................................................ 15 Example 3.5 Vector Description in Rotated Coordinate Systems ...................... 15 Example 3.6 Vector Addition ................................................................................. 17 Chapter 3 Vectors Philosophy is written in this grand book, the universe which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and
    [Show full text]