First, a simple explanation of dot product and cross product
But why are dot product and cross product defined as they are? Why are they sort-of connected, despite being very different?
They are both come out of quaternion multiplication.
Quaternions are used to represent rotations in three dimensions (i.e. rotations which are combinations of turning simultaneously around three axes) roughly as complex numbers of the type (cos θ and i sin θ) can be used to represent rotations in two dimensions (i.e. around one axis).
These videos, produced to teach maths to computer games developers, explain how:
Just the first six minutes of this one (the rest is about computer coding of quaternions):
This is “quaternions in a minute” for games developers
Another video clip about quaternions from James Grime
In this video I explain how dot-product and cross-product come from quaternion multiplication.
This explains how dot-product and cross-product are sort-of-connected, despite being so different. It also explains why they can be useful despite not being at all “proper” multiplications in the sense of obeying the rules of arithmetic that work with real numbers, with complex numbers, or even with matrices.
Please work through this video with a pen and paper, pausing from time to time. Forgive me when I say in the video that in quaternions there are three square roots of −1, i, j and k. There are six, ±i, ±j, ±k, as in complex numbers there are two square roots of −1, ±i.
You can also read this introduction to quaternions: