24. Dot Product of Vectors

What you’ll learn about

How to find the Dot Product How to find the Angle Between Vectors Projecting One Vector onto Another

… and why

Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.

Dot Product

1 2

1 2 1 1 2 2

The or of , and

, is .

u u

v v u v u v

dot product inner product

u

v u v

Vector addition and scalar multiplication gives you

another vector. The gives you a scalar (a #),

not a vector

dot product

Note that the does not mean regular multiplication!

Example Finding the Dot Product Find the dot product.

4,5 2,3

(4)(2) (5)(3)23

Example Finding the Dot Product Find the dot product.

2, 1 1,2

(2)(1) ( 1)(2) 0

Example Finding the Dot Product =<-1,3> =<2, -4> =<1,-2>

Find ( )u v wu v w

[( 1)(2) (3)( 4)] w

[( 14)] w

14,28

Angle Between Two Vectors

-1

If is the angle between the nonzero vectors and , then

cos and cos| | | |

u v

u v u v

u v u v

This formula comes from the law of cosines!!

Example Finding the Angle Between Vectors

Find the angle between the vectors 3, 2 and 1,0 . u v

1

-1

-1

cos

3,2 1,0cos

3,2 1,0

3 cos

13 1

33.7

u v

u v

Example Finding the Angle Between Vectors

Find the angle between the vectors cos sin and cos sin4 4 2 2i j i j

u v

1

-1

-1

cos

2 / 2, 2 / 2 0,1cos

2 / 2, 2 / 2 0,

2 2, , 0,

1

2 / 2 cos

1 1

12

4

2

5

u v

u v

u v

Orthogonal VectorsThe vectors u and v are orthogonal if and only if u·v = 0.

The terms orthogonal and perpendicular mean essentially the same thing – meeting at a right angle.

Example

Are vectors u = <2,-3> and v = <6,4> orthogonal?

Find the dot product.

Therefore, yes. If you graph, you will see a right angle.

2(6) ( 3)(4) 0 u v

Parallel vectors

Two vectors are parallel if Dot product = negative product of magnitude

u vu v

Example – Tell if vectors are perpendicular, parallel, or neither

1 12 ,

2 4i j i j u v

2( 1/ 2) ( 1)(1/ 4) 5 / 4 u v

Not orthogonal because 0u v

55 5 / 4

16 u v

Parallel because u v u v

Finding vector components

There are many applications in which 2 vectors are added together to find a resultant vector such as forces pulling on an object or wind resistance on a plane

There are many applications in physics and engineering where you need to do the reverse – decompose the vector into the sum of 2 vector components

Definition of vector components

Let u and v be nonzero vectors such that u = w1 and w2 Where w1 and w2 are orthogonal and w1 is parallel to v.

w1 and w2 are called vector components

To find w1 and w2 (the vector components)

W1 is the projection of u onto v and is denoted W1=projvu

W2 = u - w1

Projection of u onto v

2

If and are nonzero vectors, the projection of onto

is proj .

v

u v u

u vv u v

v

Example Find the projection u = <3,-5> onto v = <6,2>. Then

write u as the sum of two vector components.1 proj vw u

1 2

u vw v

v

1

86,2

40

w

1

6 2,

5 5w

2

9 27,

5 5 w

2

6 23, 5 ,

5 5 w

1 2

6 2 9 27, ,

5 5 5 5 u w w