1. 8.5 The Dot Product

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

5. Theorem Angle between Vectors

8. v = 4i + 3j
v = 2i - j

9. Two vectors v and w are said to be parallel if there is a nonzero scalar a so that v = aw. In this case, the angle q between v and w is 0 or p.

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

11. Theorem
Two vectors u and v are orthogonal if and only if

12. Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal.

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

14. Theorem

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

21. Find the work done by a force of 50 pounds acting in the direction 3i + j in moving an object 20 feet from (0, 0) to (20, 0).

22. ||F|| = 50
(3, 1)
(20, 0)

23. Work is approximately 948.68 foot-pounds.