DOT Product: u*v= * =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 = 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 (3/5,4/5) * (1/2,-3/2) 3/5(1/2)+4/5(-3/2) =-9/10 Example 3: 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 4:  u(3, -6) v(4,2) w(-12,-6) -Find the u*v and v*w. Show that u and v are orthogonal and v and w are parallel.   u*v = 3(4) + -6(2) = 0 which means they are perpendicular because it equals 0  v*w = -12(2) + 2(-6) = -60 w/v = -12/4 = -3 -6/2 = -3 : They are multiples of each other therefore they are parallel