1. 6.2
Dot Product of Vectors

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. Dot Product

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. Example Finding the Dot Product

6. Example Finding the Dot Product

7. Angle Between Two Vectors

8. Example Finding the Angle Between Vectors

9. Example Finding the Angle Between Vectors

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

11. Projection of u and v

12. Work