1. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Section 6.7 Dot Product

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 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.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 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.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Definition of the Dot Product

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Finding Dot Products If v = 7i – 4j and w = 2i – j, find each of the following dot products: a. b. c.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Properties of the Dot Product

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Alternative Formula for the Dot Product

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Formula for the Angle between Two Vectors

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 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. The angle between the vectors is

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 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.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Parallel and Orthogonal Vectors (continued) Two vectors are orthogonal when the angle between the vectors is 90°.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Determining Whether Vectors are Orthogonal Are the vectors v = 2i + 3j and w = 6i – 4j orthogonal? The dot product is 0. Thus, the given vectors are orthogonal.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 The Vector Projection of v onto w

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 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.

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 The Vector Components of v

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Decomposing a Vector into Two Orthogonal Vectors Let v = 2i – 5j and w = i – j. Decompose v into two vectors v l and v 2, where v 1 is parallel to w and v 2 is orthogonal to w. In the previous example, we found that

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Definition of Work

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 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. The work done is approximately 2598 foot-pounds.