What if we generalize? That is, given any two rotations that fix the origin, what is the single rotation that is equivalent to their composition? Can we solve this just as easily?

Yes! With geometric algebra, we can solve the first problem comfortably with pen and paper, and apply the same method to its generalization.

We’ll take a whirlwind tour of A Survey of Geometric Algebra and Geometric Calculus by Alan Macdonald. We’ll also assume our constructions are well-defined; for proofs, refer to An elementary construction of the geometric algebra by the same author. See also: