1. 6.46.4 Dot Product of Vectors

2. Quick Review

3. Quick Review Solutions

4. 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 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.

5. Dot Product

6. 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)

7. Example Finding the Dot Product

9. Angle Between Two Vectors

10. Example Finding the Angle Between Vectors

12. Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0.

13. Projection of u and v

14. Work