Intuitive

Concrete

• Dot product, the interactions between similar dimensions (x*x y*y, z*z)
• Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.)

The dot product (vec(a) · vec(b)) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.

The cross product (written vec(a) times vec(b)) has to measure a half-dozen “cross interactions”. The calculation looks complex but the concept is simple: accumulate 6 individual differences for the total.

Vector Calculus: Understanding the Dot Product by Kalid Azad

Recommended Resources:

• Vector Calculus: Understanding the Dot Product by Kalid Azad
• A Student's Guide to Vectors and Tensors by Daniel A. Fleisch

Abstract

Why is it interesting?

The dot product is a tool that we can use to combine two vectors and get another vector out. This number describes the plane spanned by the two vectors. The magnitude of the resulting vector is the area of the plane, and the direction denotes the orientation of the plane, because it is perpendicular to it.

This is an extremely useful concept and used in almost any physical theory, like for example, electrodynamics. Moreover, many other important tools, like the Curl are defined with the help of the cross product.