copyright © 2014, 2010, 2007 pearson education, inc. 1 section 6.7 dot product

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Section 6.7

Dot Product


Objectives:

Find the dot product of two vectors.

Find the angle between two vectors.

Use the dot product to determine if two vectors are orthogonal.

Find the projection of a vector onto another vector.

Express a vector as the sum of two orthogonal vectors.

Compute work.


The Dot Product of Two Vectors

In a previous section, we learned that the operations of vector addition and scalar multiplication result in vectors.

The dot product of two vectors results in a scalar (real number) value, rather than a vector.


Definition of the Dot Product


Example: Finding Dot Products

If v = 7i – 4j and w = 2i – j, find each of the following dot products:

a.

b.

c.

v w

w v

w w

7(2) ( 4)( 1) 14 4 18

2(7) ( 1)( 4) 14 4 18

2(2) ( 1)( 1) 4 1 5


Properties of the Dot Product


Alternative Formula for the Dot Product


Formula for the Angle between Two Vectors


Example: Finding the Angle between Two Vectors

Find the angle between the two vectors v = 4i – 3j and w = i + 2j. Round to the nearest tenth of a degree.

v wcos

v w

2 2 2 2

(4i 3j) (i 2 j)

4 ( 3) 1 2

4(1) ( 3)(2) 2

25 5 125

1 2cos

125

100.3

The angle betweenthe vectors is

100.3 .


Parallel and Orthogonal Vectors

Two vectors are parallel when the angle between the vectors is 0° or 180°. If = 0°, the vectors point in the same direction. If = 180°, the vectors point in opposite directions.


Parallel and Orthogonal Vectors (continued)

Two vectors are orthogonal when the angle between the vectors is 90°.


Example: Determining Whether Vectors are Orthogonal

Are the vectors v = 2i + 3j and w = 6i – 4j orthogonal?

2(6) 3( 4)v w 12 12 0

The dot product is 0. Thus, the given vectors are orthogonal.


The Vector Projection of v onto w


Example: Finding the Vector Projection of One Vector onto Another

If v = 2i – 5j and w = i – j, find the vector projection of v onto w.

w 2

v wproj v = w

w

22 2

(2i 5j) (i j)w

1 +( 1)

2

2(1) ( 5)( 1)w

2

7w

2 7

(i j)2

7 7

i j2 2


The Vector Components of v


Example: Decomposing a Vector into Two Orthogonal Vectors

Let v = 2i – 5j and w = i – j. Decompose v into two vectors vl and v2, where v1 is parallel to w and v2 is orthogonal to w.

1 wv = proj v

In the previous example, we found that w

7 7proj v i j

2 2

7 7i j

2 2

2 1v = v v7 7

(2i 5j) i j2 2

3 3i j

2 2


Definition of Work


Example: Computing Work

A child pulls a wagon along level ground by exerting a force of 20 pounds on a handle that makes an angle of 30° with the horizontal. How much work is done pulling the wagon 150 feet.

30

20 pounds

W F cos��AB (20)(150)cos30

2598 ft-lb

The work done is approximately 2598 foot-pounds.

copyright © 2014, 2010, 2007 pearson education, inc. 1 section 6.7 dot product

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