1. Dot Products Objectives of this Section Find the Dot Product of Two Vectors Find the Angle Between Two Vectors Determine Whether Two Vectors and Parallel or Orthogonal Decompose a Vector into Two Orthogonal Vectors Compute Work

3. Properties of Dot Product If u, v, and w are vectors, then Commutative Property Distributive Property

4. Angle between Vectors

7. Determine whether the vectors v = -3i + 2j and w = -9i + 6j are parallel. The vectors are parallel since w = 3v.

8. Two vectors v and w are orthogonal if and only if Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal. The vectors v and w are orthogonal.

9. w = 2i + 8j v = 4i - j x y

14. The work W done by a constant force F in moving an object from A to B is defined as

15. Find the work done by a force of 30 pounds acting in the direction 3i + j in moving an object 20 feet from (0, 0) to (20, 0). (3, 1) (20, 0)