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  "body": "\n## Definition\n\nLet $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be [[vectors|0QN3D80AWP]] in $\\mathbb{R}^3$.\nThe cross product of $a$ and $b$ is the vector\n$$a \\times b = (a_2 b_3 - a_3 b_2, \\; a_3 b_1 - a_1 b_3, \\; a_1 b_2 - a_2 b_1).$$\n\n## Remarks\n\nFor the standard basis vectors $e_1$, $e_2$, $e_3$ of $\\mathbb{R}^3$: $e_1 \\times e_2 = e_3$, $e_2 \\times e_3 = e_1$ and $e_3 \\times e_1 = e_2$.\nIf $\\theta$ is the angle between nonzero vectors $a$ and $b$, then $\\|a \\times b\\| = \\|a\\| \\, \\|b\\| \\sin \\theta$. This is the area of the parallelogram with the sides $a$ and $b$.\nThe cross product is [[perpendicular to its factors|0T4QHJXSD4]] and [[anticommutative|07ZS685JJC]]. It is bilinear, but it is not associative.\nIn the usual right-handed coordinates, $a$, $b$ and $a \\times b$ follow the right-hand rule. The physics notation $a \\wedge b$ is also used.\n\n## Explanations\n\n### L1 human\n\nThe cross product of two arrows in space gives a third arrow that sticks out at right angles to both. For example, $(1, 0, 0) \\times (0, 1, 0) = (0, 0, 1)$. The longer and the more perpendicular the two arrows are, the longer their cross product.\n",
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