1. 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.
u·v=v·u
u·u=||u||2
0·u=0
u·(v+w)=u·v+u·w (u+v)·w=u·w+v·w
(cu) ·v=u·(cv)=c(u·v)

7. Example Finding the Dot Product

8. Example Finding the Dot Product

9. Example Finding the Dot Product

10. Example
The dot product of u with itself is 5. What is the magnitude of u.

11. Angle Between Two Vectors
This formula comes from the law of cosines!!

12. Example Finding the Angle Between Vectors

13. Example Finding the Angle Between Vectors

14. Example Finding the Angle Between Vectors

15. Orthogonal Vectors
The vectors u and v are orthogonal if and only if u·v = 0. The terms orthogonal and perpendicular mean essentially the same thing – meeting at a right angle.

16. Example Are vectors u = <2,-3> and v = <6,4> orthogonal?
Find the dot product.
Therefore, yes. If you graph, you will see a right angle.

17. Projection of u and v

18. Work