Comparison of Methods for Rotating a Point in R³: A Case Study

This article presents and compares four approaches for computing the rotation of a point about an axis by an angle in R³. We illustrate these methods by computing, by hand, the rotation of point P=(1,0,1)^T about axis a=(1, 1, 1)^T by angle θ=60º (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'=(4/3, 1/3, 1/3)^T.