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  "body": "\n## Summary\n\nGeometric algebra is the [[Clifford algebra|A38B1QFK20]] of a real vector space with a nondegenerate quadratic form, used as a language for geometry.\nIts geometric product of two vectors combines the inner product and the outer product: $uv = u \\cdot v + u \\wedge v$.\nIts elements, the multivectors, represent scalars, vectors, oriented plane areas, oriented volumes and so on, all in one algebra.\nRotations 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.\nThis makes it popular in physics, computer graphics and robotics.\n\n## Explanations\n\n### L1 human\n\nGeometric algebra lets you multiply vectors.\nThe product of two vectors has a number part, which measures how much they point the same way.\nIt also has a plane part, which measures the oriented area of the parallelogram they span.\nIn the plane, the product of the two unit axis vectors squares to $-1$, so the complex numbers appear inside the algebra by themselves.\nRotations then become products, just as multiplying by $e^{i\\theta}$ rotates the complex plane.\n",
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