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

2. 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.

3. Copyright © 2011 Pearson, Inc. Slide 6.2 - 3 Dot Product

4. Copyright © 2011 Pearson, Inc. Slide 6.2 - 4 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)

5. Copyright © 2011 Pearson, Inc. Slide 6.2 - 5 Example Finding the Dot Product

6. Copyright © 2011 Pearson, Inc. Slide 6.2 - 6 Example Finding the Dot Product

7. Copyright © 2011 Pearson, Inc. Slide 6.2 - 7 Angle Between Two Vectors

8. Copyright © 2011 Pearson, Inc. Slide 6.2 - 8 Example Finding the Angle Between Vectors

9. Copyright © 2011 Pearson, Inc. Slide 6.2 - 9 Example Finding the Angle Between Vectors

10. Copyright © 2011 Pearson, Inc. Slide 6.2 - 10 Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0.

11. Copyright © 2011 Pearson, Inc. Slide 6.2 - 11 Projection of u and v

12. Copyright © 2011 Pearson, Inc. Slide 6.2 - 12 Work