6.46.46.46.4

Dot Product of VectorsDot Product of Vectors

Quick Review

2 2

Find || ||.

1. 1,2

2. 4 3

The points and lie on the circle 4.

Find the component form of the vector .

3. (1, 3), (2,0)

4. ( 1, 3), (2,0)

5. Find a vector with the given magnitude in

A B x y

AB

A B

A B

u

u

u i j

u the

direction of . || || 3, 2,4 v u v

Quick Review Solutions

2 2

Find || ||.

1. 1,2

2. 4 3

The points and lie on the circle 4.

Find the component form of the vector .

3. (1, 3), (2,0)

4. ( 1, 3), (2,0)

5. Find a

5

5

1

vector

,

with

3

3, 3

A B x y

AB

A B

A B

u

u

u i j

u the given magnitude in the

direction of . || || 3, 2,4 1.34,2.68 v u v

What you’ll learn about• How to find the Dot Product• How to find the Angle Between Vectors• Projecting One Vector onto Another• How to use vectors to find the work done by a force

… and whyVectors 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

Properties of the Dot Product

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

1. u·v=v·u2. u·u=|u|2

3. 0·u=04. u·(v+w)=u·v+u·w (u+v)

·w=u·w+v·w5. (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

Find the dot product.

4,3 1, 2

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

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

Example Finding the Angle Between

Vectors

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

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

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

Projection of u and v

2

If and are nonzero vectors, the projection of onto

is proj .

v

u v u

u vv u v

v

Work

If is a constant force whose direction is the same as

the direction of AB, then the done by in

moving an object from to is || |||| ||

W

A B W AB

work

F

F

F