---
tag: N8H460W3E2
kind: concept
name: Geometric algebra
msc: [15A66]
specializes: [A38B1QFK20]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_algebra
same_as:
  wikidata: Q1186649
  wikipedia: Geometric algebra
---

## Summary

Geometric algebra is the [[Clifford algebra|A38B1QFK20]] 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.

## Explanations

### L1 human

Geometric algebra lets you multiply vectors.
The product of two vectors has a number part, which measures how much they point the same way.
It also has a plane part, which measures the oriented area of the parallelogram they span.
In the plane, the product of the two unit axis vectors squares to $-1$, so the complex numbers appear inside the algebra by themselves.
Rotations then become products, just as multiplying by $e^{i\theta}$ rotates the complex plane.