{
  "aliases": [
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  "body": "\n## Statement\n\nLet $a = (a_1, a_2, a_3)$, $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$ be vectors in $\\mathbb{R}^3$. Let $\\times$ be the [[cross product|0N5N15HWZ7]] and let $\\cdot$ be the dot product.\nThe scalar triple product of $a$, $b$ and $c$ is the real number $a \\cdot (b \\times c)$. It is equal to the determinant of the matrix whose rows are $a$, $b$ and $c$:\n$$a \\cdot (b \\times c) = \\det \\begin{pmatrix} a_1 \u0026 a_2 \u0026 a_3 \\\\ b_1 \u0026 b_2 \u0026 b_3 \\\\ c_1 \u0026 c_2 \u0026 c_3 \\end{pmatrix}.$$\n\n## Remarks\n\nA cyclic shift of the three vectors does not change the value: $a \\cdot (b \\times c) = b \\cdot (c \\times a) = c \\cdot (a \\times b)$. A swap of two vectors changes its sign.\nThe absolute value $|a \\cdot (b \\times c)|$ is the volume of the [[parallelepiped|0SQV2VJTGY]] with the edges $a$, $b$ and $c$.\nThe scalar triple product is $0$ if and only if $a$, $b$ and $c$ are linearly dependent. This happens exactly when they lie in one plane through $0$.\n\n## Explanations\n\n### L1 human\n\nThe scalar triple product measures the volume of the slanted box that three arrows make. For the unit vectors $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$ the box is a unit cube, and the product is $1$. If the three arrows lie flat in one plane, the box has no volume, and the product is $0$.\n",
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  "msc": [
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  "name": "Scalar triple product",
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