Geometric algebra

Summary

Geometric algebra is the Clifford algebra of a real vector space with a nondegenerate quadratic form, used as a language for geometry. Its geometric product of two vectors combines the inner product and the outer product: $uv = u \cdot v + u \wedge v$. Its elements, the multivectors, represent scalars, vectors, oriented plane areas, oriented volumes and so on, all in one algebra. Rotations act by conjugation, $x \mapsto R x R^{-1}$, and the reflection in the hyperplane orthogonal to a vector $n$ acts by $x \mapsto -n x n^{-1}$, in every dimension. This makes it popular in physics, computer graphics and robotics.