Pick any numbers that add to x + y + z = 0. Find the angle between your vector v = (x, y, z) and the vector w = ( z, x, y). Challenge question: explain why v.w/|v||w| is always -1/2.
The Attempt at a Solution
I chose (1, -2, 1) for the first part, which is straightforward. The second bit has me somewhat flummoxed. Using x, y, and z, I get:
(xz + xy + yz)/(x^2 + y^2 + z^2)
I can see that the numerator will always be less than the denominator because either one or two of the components must be negative. The explanation proper eludes me though, and I feel missing something simple here. Please assist me in understanding this one.