Copyright © 2011 Pearson, Inc. 6.2 Dot Product of Vectors

Copyright © 2011 Pearson, Inc.

6.2Dot Product of Vectors

Copyright © 2011 Pearson, Inc. Slide 6.2 - 2

What you’ll learn about

The Dot Product Angle Between Vectors Projecting One Vector onto Another Work

… 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

The dot product or inner product of u u1,u

2

and v v1,v

2 is

uv u1v

1u

2v

2.


Properties of the Dot Product

Let u, v, and w be vectors and let c be a scalar.

1. u·v = v·u

2. u·u = |u|2

3. 0·u = 0

4. u·(v + w) = u·v + u·w

(u + v) ·w = u·w + v·w

5. (cu)·v = u·(cv) = c(u·v)


Example Finding the Dot Product

Find the dot product.

4,3 1, 2


Example Finding the Dot Product

4,3 1, 2 (4)( 1) (3)( 2) 10

Find the dot product.

4,3 1, 2


Angle Between Two Vectors

If is the angle between the nonzero vectors u and v,

then

cos uvu v

and cos-1 uvu v


Example Finding the Angle Between Vectors

Let u 2,3 and v 4, 1 .

Find the angle between the vectors u and v.


Example Finding the Angle Between Vectors

cos uvu v

= 2,3 4, 1

2,3 4, 1

5

13 17

So,

cos 1 3

13 1

70¼

Let u 2,3 and v 4, 1 .

Find the angle between the vectors u and v.


Orthogonal Vectors

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


Projection of u and v

If u and v are nonzero vectors, the projection of u

onto v is

projvu

uv

v2

v.


Work

If F is a constant force whose direction is the same as

the direction of AB, then the work W done by F in

moving an object from A to B is W | F || AB |

Documents

Copyright © 2011 Pearson, Inc. 6.2 Dot Product of Vectors