These are just some personal notes and musings about rotations in three dimensions. Some properties of the 3D rotation group, its generating Lie algebra, the way that \(\mathrm{SU}(2,\mathbb{C})\) acts as the double-cover of the rotation groups and the origin of spinors, and how quaternions and rotors fit into all this.

Active and passive rotations

Transformations can be either active or passive. An active transformation is one that transforms an object. A passive transformation is one that transforms a coordinate system. As an example, suppose we have a vector in a plane: $$ V = \begin{bmatrix} x \\ y \end{bmatrix} \,. $$ An active rotation is one that rotates the vector itself. By convention, rotation by a positive angle is counter-clockwise. Under a rotation by an angle \(\theta\), a vector transforms as: $$ V \mapsto \begin{bmatrix} x\cos\theta - y\sin\theta \\ y\cos\theta + x\sin\theta \end{bmatrix} $$ This is represented graphically below:

By contrast, a passive rotation by the same angle \(\theta\) will rotate the coordinate axes instead: The projections of the vector onto the \(x\) and \(y\) axes change as if the vector itself were actively rotated by an angle \(-\theta\), i.e., by a rotation in the opposite direction: $$ V \mapsto \begin{bmatrix} x\cos\theta + y\sin\theta \\ y\cos\theta - x\sin\theta \end{bmatrix} \,. $$ I typically prefer to use active transformations. This is because I think of transformations as acting on specific physical objects. Throughout this (and the other) pages, assume all transformations are active unless I say otherwise.

Rotation matrices

A rotation of a vector in a plane can be represented using a matrix: $$ R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \,. $$ This acts on a vector through matrix multiplication: $$ V = \begin{bmatrix} x \\ y \end{bmatrix} \mapsto R(\theta) V = \begin{bmatrix} x\cos\theta - y\sin\theta \\ y\cos\theta + x\sin\theta \end{bmatrix} \,. $$ Written out in terms of components, the transformation law can also be written: $$ V_a = R_{ab}(\theta) V_b \,, $$ where I'm using the Einsteim summation convention for repeated indices. Rotations can be compounded through matrix multiplication. A rotation by \(\theta_1\) followed by another rotation by \(\theta_2\) is: $$ R(\theta_2) R(\theta_1) \,. $$ The set of all rotations in the plane forms a group. Formally, a group obeys the following axioms:

1. For every pair of group elements \(g_1\) and \(g_2\), \(g_1 \circ g_2\)... <open>