How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized)
Thinking about how to prove this in the most intuitive way resulted in proving a trigonometric identity: $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$.
But even after proving this successfully, the connection between and cosine and dot product does not immediately stick out and instead I rely on remembering that this is valid while taking comfort in the fact that I've seen the proof in the past.
My questions are:
How do you see this connection?
How do you extend the notion of dot product vs. angle to higher dimensions - 4 and higher?