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  "body": "\n## Definition\n\nLet $V$ be a finite-dimensional real [[vector space|000AYPA9AJ]] with a nondegenerate symmetric [[bilinear form|0SQZMXB6GP]] $u \\cdot v$.\nThe *geometric algebra* $\\mathcal{G}(V)$ is the [[Clifford algebra|A38B1QFK20]] of $V$ for the quadratic form $Q(v) = v \\cdot v$.\nSo it is the associative unital real algebra generated by $V$ subject to $v v = v \\cdot v$ for all $v \\in V$.\nIts product is the *geometric product*.\nFor vectors $u, v \\in V$, the *outer product* is $u \\wedge v = \\tfrac{1}{2}(uv - vu)$, and then $uv = u \\cdot v + u \\wedge v$.\nAn element of $\\mathcal{G}(V)$ is a *multivector*.\nA product of $k$ pairwise orthogonal vectors is a $k$-blade, and the span of the $k$-blades is the grade $k$ part $\\mathcal{G}^k(V)$.\n\n## Remarks\n\nPolarizing $v v = v \\cdot v$ gives $uv + vu = 2\\, u \\cdot v$, which is why $u \\cdot v = \\tfrac{1}{2}(uv + vu)$.\nAs a graded vector space, $\\mathcal{G}(V) = \\bigoplus_{k=0}^{n} \\mathcal{G}^k(V)$ is isomorphic to the [[exterior algebra|0MDMZ965VE]] $\\Lambda V$, of dimension $2^n$ for $n = \\dim V$.\nExample: $V = \\R^2$ with the dot product and orthonormal basis $e_1, e_2$.\nThen $e_1 e_2 = -e_2 e_1$, and $I = e_1 e_2$ satisfies $I^2 = e_1 e_2 e_1 e_2 = -e_1 e_1 e_2 e_2 = -1$, so $\\set{a + bI}$ is a copy of $\\C$.\nWith $R = \\cos(\\theta/2) - I \\sin(\\theta/2)$ and $\\tilde R = \\cos(\\theta/2) + I \\sin(\\theta/2)$, one checks $R e_1 \\tilde R = e_1 (\\cos \\theta + I \\sin \\theta) = \\cos\\theta \\, e_1 + \\sin\\theta \\, e_2$, a rotation by $\\theta$.\nNon-example: the exterior algebra $\\Lambda V$ with the wedge product alone is not a geometric algebra in this sense, because there $v \\wedge v = 0$ for every $v$, while $v v = v \\cdot v \\ne 0$ for a non-null vector.\nThe notation $\\mathcal{G}(p, q)$ or $\\mathrm{Cl}_{p,q}(\\R)$ is used when the form has $p$ positive and $q$ negative squares.\n",
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