Review: 6.4-Vectors and Dot Products

One of the first review presentations was Darron's on vectors and dot products. Here is an overview this topic. 

Dot Product of Two Vectors
-The dot product of u = <a,b> and v = <c,d> is u • v =  ac + bd 

-The properties of the dot products of two vectors are as follows: Let u, v, and w be vectors in a plane or in space and let c be a scalar.

1. u • v = v • u 2. 0 • v = 0 3. u • (v + w) = u • v + u • w 4. v • v = ll v ll ²  5. c(u • v) = cu • v = u • c


The Angle Between Two Vectors 

If ø is the angle between two nonzero vectors u and v, then cos ø = u • v / ll u ll ll v ll

Vectors u and v  are orthogonal if u • v = 0 

Vector Components
Let u and v be nonzero vectors such that u = w + twhere w and t are orthogonal and w is parallel to v. The vectors w and t are called vector components of u. 

The vector w is the projection of u onto v and is denoted byw = proj v uThe vector t is given by t = u - w
Let u and v be nonzero vectors. The projection of u onto v isproj v u = (u • v/ ll v ll²)v
Work

W = ll proj PQ F ll ll PQ ll                      Projection Form W = (cos ø) ll F ll ll PQ ll = F • PQ              Dot Product Form

Math Joke of the day:
Q: Why didn't the quarter roll down the hill with the nickel?
A: Because it had more cents.