Quaternions are similar to complex and hyper complex numbers, but their multiplication is not cummutative .
They have 3 non real parts, that are marked here with the letters j, k and l.
The real part will not have an own letter here.
The multiplication between the non real parts is defined in the following way:
Multiplication:
Division:
Addition of quaternions:
Subtraction:
conjunct:
Abs:
Inverse
Standard quaternion
Description of rotation:
corresponds to the rotation angle.
corresponds to the rotation axis.
Power
A quaternion can also be written like a complexe 2x2-matrix
or like a real 4x4-matrix
Quaternions are mainly used in computer graphics to descibe rotations in the space.
Also they are used in quant mechanics to describe spins. Doing this, put instead of the marks (e,i,j,k) the Pauli spin matrices
; ; ;