I'm familiar with this proof using cosine law, but happened to find this other one in Lang's Introduction to Linear Algebra. Pages 23 and then 24. He finds a scalar value c to the scale be base vector. c = (A dot B) / ||B||.
Then he says plug that in cos(t) = c||B||/||A||, which yields (A dot B) = ||A|| ||B|| cos(t).
Am I getting crazy?? Have you guys ever checked that?