>>12096332Here's a way I like thinking about the dot product:
Say you have some vector A, as an arrow in 3D Euclidean space. What is the set of all vectors B such that A.B is a constant value? It always forms a plane which is normal to A. For instance, the set of all vectors B such that A.B=0 forms a plane normal to A, which passes through the origin. Meanwhile, the set of all vectors B such that A.B=|A|^2 forms a plane normal to A which touches the tip of A's arrow.
We can plot out the planes for A.B= 1, 2, 3, etc. and find that they form planes, normal to A, which are increasingly translated forward in the direction of A. Likewise, A.B= -1, -2, -3, etc. form planes which are translated backwards from the vector A's direction. So the value of the dot product A.B tells us in which of these planes B lies, or in other words, how far forward or backward B is with respect to A's direction. But this is exactly what projection is - how far a vector goes with respect to another vector's direction.
This picture doesn't make it immediately clear why the dot product is commutative, but you can figure it out with some clever geometrical thinking.
