Honors Pre-Calculus

12-4 The Dot ProductPage: 441

Objective: To define and apply the dot product

1 1 2 2( , ) ( , ),If v a b and w a b then the dot product of the vectors is defined by:

If v = (-2,5) and w = (4, 1), find

a) v۰w b) v۰v

Theorem Properties of Dot ProductIf u, v, and w are vectors, then

k(u•v) = (ku) •v

Theorem Angle between Vectors

Find the angle θ between u = (2, -1) and v = (4, 3)

Two vectors v and w are said to be parallel if there is a nonzero scalar so that v = w. In this case, the anglebetween v and w is 0 or

The vectors are parallel since w = 3v.

Determine whether the vectors v = (-3, 2) and w = (-9, 6) are parallel.

Theorem

Two vectors u and v are orthogonal if and only if

Determine whether the vectors v = (4, -1) and w = (2, 8) are orthogonal.

Theorem

v1

v2

v

w

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

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).

(3, 1)

(20, 0)

||F|| = 50

Work is approximately 948.68 foot-pounds.

TB p. 444 #3, 13-21 odd, 25