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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $x = (x_1, \\dots, x_n)$ and $y = (y_1, \\dots, y_n)$ be vectors of $\\R^n$.\nThe *dot product* of $x$ and $y$ is the real number\n$$x \\cdot y = \\sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \\cdots + x_n y_n.$$\n\n## Remarks\n\nIn matrix notation, $x \\cdot y = x^T y$ for column vectors $x$ and $y$.\nIt is symmetric, bilinear and positive definite, so it is an [[inner product|0NB0AWGJBW]] on $\\R^n$.\nGeometrically, $x \\cdot y = \\norm{x} \\norm{y} \\cos\\theta$, where $\\theta$ is the angle between $x$ and $y$.\nThe same formula defines a symmetric bilinear form on $F^n$ for every field $F$.\n\n## Explanations\n\n### L1 human\n\nThe dot product multiplies two vectors coordinate by coordinate and adds the results.\nFor example, $(1, 2, 3) \\cdot (4, 5, 6) = 4 + 10 + 18 = 32$.\nIt is zero exactly when the two vectors are perpendicular, or when one of them is zero.\n",
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