Prove that if u and v are nonzero vectors, and theta is the angle between them then u dot product v = ||u|| ||v|| cos (theta). Consider the triangle with sides u ,v , and u-v. The Law of Cosines implies that ||u-v||^2 = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos(theta). On the other hand, ||u-v||^2 = (u-v) dot product (u-v)... Simplify this last expression, then equate the two formulas for ||u-v||^2.
The Attempt at a Solution
I have no idea how to start! I thought of doing
cos(theta)= uxvx + uyvy / sqrt[(uxvy- vxuy)^2] + (uxvx + uyvy)^2] When x and y are subscripts, because
u= uxi + uyj
v= vxi + vyj