Theory

Introduction

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One can combine 3d vectors and 1d scalers into 4d quantities called quaternions, and written $\mathbb{H}$. Quaternions can be added and multiplied and divided.

Quaternions have real number coefficients for each of four basic units:

$$a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$$

The coefficient $a$ is called the scalar part and the coefficients $(b,c,d)$ are together called the vector part of the quaternion.

Multiplication rules
Multiplication for $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ follows the cross product rules (written formerly using ${\mathbf{e}}_{\mathbf{i}}$ etc. instead), with an additional rule for squares:

$${\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1$$

Using these rules, one can multiply any quaternion sums.

Multiplication charts

$$\begin{array}{rl}& \\ \mathbf{i}\cdot \mathbf{j}\cdot \mathbf{k}& =-1\\ {\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2& =-1\end{array}$$

If two vector quaternions (with zero scalar part) are multiplied, then the resulting scalar part is the negative dot product, and the resulting vector part is the cross product, of the vector parts considered as vectors using their components.

Quaternions can also be divided:

$$\frac{1}{a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}}=\frac{a-b\mathbf{i}-c\mathbf{j}-d\mathbf{k}}{a^2+b^2+c^2+d^2}$$