1. 1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved Chapter 7 Applications of Trigonometric Functions

2. OBJECTIVES The Dot Product SECTION 7.6 1 2 Define the dot product of two vectors. Find the angle between two vectors. Define orthogonal vectors. Find the projection of a vector onto another vector. Decompose a vector into two orthogonal vectors. Use the definition of work. 3 4 5 6

3. 3 © 2011 Pearson Education, Inc. All rights reserved THE DOT PRODUCT For two vectors the dot product of v and w, denoted v w, is defined as

4. 4 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding the Dot Product Find the dot product v w. Solution

5. 5 © 2011 Pearson Education, Inc. All rights reserved THE DOT PRODUCT AND THE ANGLE BETWEEN TWO VECTORS If  (0° ≤  ≤ 180°) is the angle between two nonzero vectors v and w, then

6. 6 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding the Dot Product If v and w are two vectors of magnitudes 5 and 7, respectively, and the angle between them is 75º, find v · w. Round the answer to the nearest tenth. Solution

7. 7 © 2011 Pearson Education, Inc. All rights reserved PARALLEL VECTORS Two vectors v and w are parallel if there is a nonzero scalar, c, so that v = cv. The angle θ between parallel vectors is either 0º or 180º. So v and w are parallel if v · w = ±||v|| ||w||.

8. 8 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Angle Between Two Vectors Find the angle θ (in degrees) between the vectors v = 2i + 3j and w = −3i + 4j. Round the answer to the nearest tenth of a degree. Solution

9. 9 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Angle Between Two Vectors Solution continued

10. 10 © 2011 Pearson Education, Inc. All rights reserved PROPERTIES OF THE DOT PRODUCT If u, v, and w are vectors and c is a scalar, then

11. 11 © 2011 Pearson Education, Inc. All rights reserved Two vectors v and w are orthogonal (perpendicular) if and only if v · w = 0. Because 0 · w = 0 for any vector w, it follows from the definition that the zero vector is orthogonal to every vector. ORTHOGONAL VECTORS

12. 12 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Orthogonal Vectors Find the scalar c so that the vectors v = 3i + 2j and w = 4i + cj are orthogonal. Solution The vectors v = 3i + 2j =  3, 2  and w = 4i + cj =  4, c  are orthogonal if and only if

13. 13 © 2011 Pearson Education, Inc. All rights reserved Let and be representations of the nonzero vectors v and w, respectively. The projection of in the direction of is the directed line segment, where R is the foot of the perpendicular from P to the line containing PROJECTION OF A VECTOR

14. 14 © 2011 Pearson Education, Inc. All rights reserved PROJECTION OF A VECTOR The vector represented by is called the vector projection of v onto w and is denoted by proj w v. Let θ (0º ≤ θ ≤ 180º) be the measure of the angle between nonzero vectors v and w. The number ||v|| cos θ is called the scalar projection of v onto w.

15. 15 © 2011 Pearson Education, Inc. All rights reserved VECTOR AND SCALAR PROJECTIONS OF v ONTO w Let v and w be two nonzero vectors and let θ (0º ≤ θ ≤ 180º) be the angle between them. The vector projection of v onto w is The scalar projection of v onto w is

16. 16 © 2011 Pearson Education, Inc. All rights reserved DECOMPOSITION OF A VECTOR Let v and w be two nonzero vectors. The vector v can be written in the form v = v 1 + v 2, where v 1 is parallel to w and v 2 is orthogonal to w. We have The expression v = v 1 + v 2 is called the decomposition of v with respect to w. The vectors v 1 and v 2 are called components of v.

17. 17 © 2011 Pearson Education, Inc. All rights reserved DEFINITION OF WORK The work W done by a constant force F in moving an object from a point P to a point Q is defined by where θ is the angle between F and.

18. 18 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 10 Computing Work A child pulls a wagon along level ground, with a force of 40 pounds along the handle, which makes an angle of 42º with the horizontal. How much work has she done by pulling the wagon 150 feet? Solution