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  "body": "\n## Statement\n\nLet $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\\mathbb{R}^n$, with dot product $u \\cdot v = u_1 v_1 + \\dots + u_n v_n$.\nLet $\\theta$ be the [[angle between $u$ and $v$|0ZEHZY85E6]].\nThen $u \\cdot v = 0$ if and only if $\\theta = \\pi / 2$.\n\n## Remarks\n\nThe proof uses the [[geometric formula for the dot product|070TXS9AXM]]: $u \\cdot v = \\|u\\| \\, \\|v\\| \\cos \\theta$ with $\\|u\\| \\neq 0$ and $\\|v\\| \\neq 0$, and $\\cos \\theta = 0$ for $0 \\leq \\theta \\leq \\pi$ only when $\\theta = \\pi / 2$.\nTwo vectors $u$ and $v$ with $u \\cdot v = 0$ are called orthogonal. With this word, the zero vector is orthogonal to each vector.\nIn Mathlib the angle with the zero vector is $\\pi / 2$, so the equivalence holds there for all vectors.\n\n## Explanations\n\n### L1 human\n\nThis gives a quick test for a right angle. The vectors $(2, 1)$ and $(-1, 2)$ have dot product $-2 + 2 = 0$, so they are perpendicular. You do not need to draw them or measure an angle.\n",
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