If two vectors u and v fit the equation (u − v) • (u − v) = u•u+v•v, how must these vectors u and v be related? What familiar theorem does this equation represent?

Answer 1

(u − v) • (u − v) = u•u + v•v -2u•v
Is what you get by multiplying out the dot products.

For the equation
(u − v) • (u − v) = u•u + v•v

To be valid, u•v must be zero, so u and v must be perpendicular. In that case the triangle formed by u, v and the hypotenuse u -v is a right triangle and the left side is the square of the hypotenuse, as required by the Pythagorian theorem.