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?
