Dot Products and Cross Products

The dot product for a pair of vectors v = a, b , c and w = d, e , f results is a scalar (numeric) value and can be calculated in two different ways : #1 - Dot Product using Component Forms vi w =ad + be + cf (multiply i,j and k components and add) ex) Calculate the dot product of these vectors a) 3,− 2i 1,5 b) (2ij+ − 4 k )i (6 jk − )

#2 - Dot Product using Magnitudes and Angle θ Between Vectors vwi = v w cos θ (multiply the magnitudes and cosine of the angle) ex) Calculate the dot product of the vector shown here. (Round to 1 decimal place)

Depending on what type of data you’re given, the dot product can be calculated in either of these two ways. You can merge these two methods together to determine the angle measure between two vectors . vi w Since vwi = v w cos θ , then cos θ = and use “ cos −1 ” to get θ v w ex) Determine the angle measure (to the nearest tenth of a degree) between the vectors v= −3 i + 5 j and w=2 j .

WHAT THE DOT PRODUCT CAN TELL YOU:

The dot product is ALWAYS a ______QUANTITY!!!

If vi w is positive then ______

If vi w is negative then ______

If vi w is zero then ______

These facts apply to ALL vectors in ALL dimensions!!! ex) Determine ANY vector orthogonal to w=2 ij + − 4 k

ex) What happens when you form a dot product with the same vector? vi v = ???

Vector Projections The vector projection of v onto w is (geometrically) found by taking the ‘head’ of the vector and projecting (or flattening) it onto the vector w.

This results in a VECTOR quantity (not a scalar) which is parallel to w. What would the projection look like if the angle θ were obtuse?

HOW TO CALCULATE VECTOR PROJECTIONS (derivation on scrap paper)

Projection of v onto w = proj Wv =

Some books denote this as v which means “ v’s projection parallel to w” ex) Calculate the vector projection of v=4,3 onto w = 7, − 2

ex ) Graph the vectors v = 4,3 , w =7, − 2 and the projection together here.

ex) Using the same vectors from the previous problem, determine the vector v⊥ = + such that v v v ⊥ . Graph them all together on the same grid.

This is a decomposition of the vector v into a sum of two other vectors: one parallel to w and another perpendicular to w.

The Cross Product of a pair of vectors is a calculation which produces a vector result. This new vector is orthogonal (perpendicular) to the two original vectors forming the product.

BEFORE we get into cross products ... I need to show you how to evaluate a determinant.

A determinant is simply a square array of numbers. Most of the determinants you’ll encounter in calculus III will be 2X2 or 3X3 determinants.

6− 2 ex) Evaluate the 2X2 determinant . 5 1

3− 1 − 6 ex) Evaluate the 3X3 (*) determinant 2 4− 2 . −5 0 5

( ) * There are other ways to evaluate ... I’m illustrating the most effective and easiest to remember. If you’re curious about the other ways of evaluating, please see me during office hours. ex) Calculate the cross product v× w for v = − 4,1,0 and w = 3,2,0

Set up the determinant first: ORDER IS IMPORTANT!!!

Carefully evaluate the determinant. REMEMBER ... THE ANSWER SHOULD BE A VECTOR!!!

ex) Calculate the cross product v× w for v =1, − 4, 3 and w =5, 3, − 2

PROPERTIES OF THE CROSS PRODUCT 1. v× w is perpendicular to both v and w .

2. The order of the product follows the ‘right hand rule’

Your index finger ‘points’ toward the first vector in the cross product. Your middle finger represents the second vector in the cross product. Your thumb always points in the direction of the vector FORMED by making the cross product.

3. v× w and w× v point in opposite directions.

4. The magnitude of the cross product vector can be found in two ways: a. Using the component form: v× w =a2 ++ b 2 c 2 b. Using magnitudes and angle: v× w = vw sin θ The magnitude of a cross product vector calculates the area of the parallelogram the two vectors span.

ex) Determine the area of the parallelogram shown below.

RECAP OF DIFFERENCES IN DOT PRODUCT AND CROSS PRODUCT

Dot Product Calculated using components by multiplying like components and adding the result OR Calculated using magnitudes and angle between vectors as vw⋅ = v w cos θ

Useful in determining the angle between two vectors

Always returns a SCALAR value.

Cross Product Calculated using a determinant

Useful in producing a vector which is perpendicular to the two vectors crossed

Always returns a VECTOR value.

Magnitude of the cross product vector is the area of the spanned parallelogram.