Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: From Vector Algebra to Geometric Algebra

Abstract

This article starts by presenting and comparing four methods for computing the rotation of a point about an axis by an angle in $\mathbb{R}^3$. We illustrate these methods by computing, by hand, the rotation of point $P=(1,0,1)^T$ about axis $\mathbf{a}=(1,1,1)^T$ by angle $θ=60^\circ$ (following the right-hand rule). The four methods considered are: (1) an ad hoc geometric method exploiting a symmetry in the situation; (2) a projection method that sets up a new coordinate system using the dot and cross products; (3) a matrix method which rotates the standard basis and uses matrix-vector multiplication; (4) a Geometric (Clifford) Algebra method that represents the rotation as a double reflection via a rotor. We provide a brief introduction to Geometric Algebra. Subsequently, we also address rotations in other dimensions and explain how these can be handled with Geometric Algebra to provide deeper insights.