DOT Product!

Hello,I hope everyone had a great weekend. Im currently stressing over the trig exam this week; I'm sure I am not the only one. 
Anyway,

DOT Product:
u*v=<x1, y1> * <x2, y2> = x1x2+y1y2

-- If the dot product equals zero, then the vectors are orthogonal, which means perpendicular.
-- If the vectors are multiples of each other, then they are parallel.
-- To find the angle between two vectors, use the formula:
cos(theta)= u*v/magnitudeU * magnitudeV

-- Properties of the DOT product:
1.  commutative:
u * v = v * u
2.  squared
u * u does not equal u^2
u * u = magnitude of u^2
3.  K(u * v)= Ku * v
doesn't distribute
4.  u * (v+w) = u * v + u * w

Example 1:
Find (2,3) * (4,-5)
2(4)+3(-5)
=-7

Example 2:
Find (3,-5) * (7,4)
3(7)+-5(4)
=1

Example 3:
Find (-3,0) * (5,7)
-3(5)+0(7)
=-15

Example 4:
Find (3/5,4/5) * (1/2,-3/2)
3/5(1/2)+4/5(-3/2)
=-9/10

Example 5:
Find the value of a if the vectors (6,-8) and (4,a) are parallel.
a/4=3/6
6a=12
divide both sides by 6
a=2

Example 6:
If u=(-2,3), find u*u
-2(-2)+3(3)
=13

Example 7:
If u=(5,-3) and v=(3,7), verify that u * v = v * u
5(3)+-3(7)=-6
3(5)+7(-3)=-6
-6=-6

Example 8:
If u=(5,-3) and v=(3,7), verify that 2(u * v) = (2u) * v
2(-6)= -12
10(3)+ -6(7)=-12
-12=-12