Question

What is the physical interpretation of dot and cross products?

Answer 1

A dot product is the projection of one vector on another. If a light shines perpendicularly onto one vector A, the dot product of A with another vector B is the length of shadow that B casts on A. Another way to think of it: let A define the horizontal ground. Then B is a "sundial pointer" sticking out of it, and the dot product A . B equals the length of "shadow" that B casts on the "ground" (along A).

A cross product is the parallelogram defined by two vectors (or if they are more than 90 degrees apart, by one vector and the inverse of the other). The magnitude of the cross product is the area of the parallelogram, and the direction of the cross product is perpendicular to the parallelogram.