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  "body": "\n## Summary\n\nLet $V$ be a vector space with a [[quadratic form|0RBQS1JJCR]] $Q$.\nThe Clifford algebra of $(V, Q)$ is the associative algebra generated by $V$ in which each vector $v$ squares to the scalar $Q(v)$.\nIt contains the complex numbers, the quaternions and the [[exterior algebra|0MDMZ965VE]] as special cases.\nIt is used to build spinors and spin groups, and it gives an algebraic language for rotations and reflections in geometry and physics.\n\n## Explanations\n\n### L1 human\n\nStart with vectors and allow to multiply them, with one rule: a vector times itself is the number $Q(v)$.\nOn the line $\\R$ with $Q(x) = -x^2$, the basis vector $e$ satisfies $e^2 = -1$, so it behaves like the imaginary unit $i$.\nIn the plane with $Q(x, y) = -(x^2 + y^2)$, the basis vectors $e_1, e_2$ square to $-1$ and anticommute.\nThen $e_1$, $e_2$ and $e_1 e_2$ behave like the quaternion units $i$, $j$ and $k$.\n",
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