1. 6.2 - Dot Product of Vectors
HW: Pg #1-18e, 24

2. The Dot Product DEF: The dot product or inner product of
u = <u1,u2> and v = <v1,v2> is
u • v = u1v1 + u2v2
Used to calculate the angle between two vectors

3. 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 =
u • (v + w) =
(cu) • v = u • (cv) = c(u • v)

4. Find each dot product <3,4> • <5,2>
<1,-2> • <-4,3>
(2i - j) •(3i - 5j)

5. Use the dot product to find the length of the vector

6. Angle Between Two Vectors
If ө is the angle between the nonzero vectors u and v, then
Cos = (u • v)/( |u| |v| )
And  = cos-1((u • v)/ (|u| |v|))

7. Find the angle between the vector u and v
u = <2,3>, v = <-2,5>
u = <2,1>, v = <-1,-3>

8. Orthogonal Vectors
The vectors u and v are orthogonal if and only if u • v=0

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

10. Projecting One Vector onto Another
The vector projection of u = PQ onto a nonzero vector v = PS is the vector PR determined by dropping a perpendicular from Q to the line PS.
u = PR + RQ
PR and RQ are perpendicular
The standard notation for PR = projvu

11. Projection of u onto v
If u and v are nonzero vectors, the projection of u onto v is
Projvu = ((u • v)/(|v|2))v

12. Decomposing a vector into perpendicular components
Find the vector projection of u = <6,2> onto v = <5,-5>. Write u as the sum of two orthogonal vectors.

13. HW: Pg. 520 #25-28, 33-38