Comparison of Methods for Rotating a Point in ${\mathbb{R}}^3$: A Case Study

This article presents and compares four approaches 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 $\theta ={60}^{\circ }$ (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. All methods yield the same exact result: $P^{\mathrm{\prime }}={(\frac{4}{3},\frac{1}{3},\frac{1}{3})}^T$.