1. 6.2
Dot Product of Vectors

2. 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.

3. Dot Product

4. Example Finding the Dot Product

5. Example Finding the 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
5. (cu) ·v = u·(cv) = c(u·v)

7. Angle Between Two Vectors

8. Find the angle between the vectors u = <2, 3> and v = <-2, 5>

9. You Try!

10. Example Finding the Angle Between Vectors

11. Orthogonal Vectors
The vectors u and v are orthogonal if and only if u·v = 0.
The terms “perpendicular” and “orthogonal” almost mean the same thing.
The zero vector has no direction angle, so technically speaking, the zero vector is not perpendicular to any vector
However, the zero vector is orthogonal to every vector.
Except for this special case, orthogonal and perpendicular are the same

12. Prove that the vectors u = <2, 3> and v = <-6, 4> are orthogonal

13. Projection of u and v

14. Find the vector projection of u = <5, 2> onto v = <10, 0>.

15. Find the vector projection of u = <6, 2> onto v = <5, -5>
Find the vector projection of u = <6, 2> onto v = <5, -5>. Then write u as the sum of two orthogonal vectors, one of which is projvu.

16. Work

17. Find the work done by a 10 pound force acting in the direction <1, 2> in moving an object 3 feet from (0, 0) to (3, 0).