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  "body": "\n## Definition\n\nLet $p$ be a point of $\\mathbb{R}^3$, and let $a$, $b$ and $c$ be linearly independent [[vectors|0QN3D80AWP]] in $\\mathbb{R}^3$.\nThe parallelepiped with the vertex $p$ and the edges $a$, $b$ and $c$ is the set\n$$\\{\\, p + s a + t b + u c \\;:\\; s, t, u \\in [0, 1] \\,\\}.$$\n\n## Remarks\n\nThe parallelepiped has 8 vertices $p + s a + t b + u c$ with $s, t, u \\in \\{0, 1\\}$, 12 edges and 6 faces. Opposite faces are congruent parallelograms in parallel planes.\nIts volume is the absolute value of the [[scalar triple product|059Q2XWJDC]] $a \\cdot (b \\times c)$.\nIn Mathlib, the parallelepiped of a finite family of vectors $v_i$ is the set of the sums $\\sum_i t_i v_i$ with each $t_i \\in [0, 1]$. It does not ask for linear independence.\nA cube and a rectangular cuboid are special parallelepipeds.\n\n## Explanations\n\n### L1 human\n\nTake a cube and push it sideways so that its faces lean, but stay flat. The result is a parallelepiped. Each face is a parallelogram, and opposite faces are copies of each other.\n",
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