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  "body": "\n## Statement\n\nLet $a$ and $b$ be vectors in $\\mathbb{R}^3$, and let $a \\times b$ be their [[cross product|0N5N15HWZ7]]. Let $u \\cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3$ be the dot product.\nThen\n$$a \\cdot (a \\times b) = 0 \\quad \\text{and} \\quad b \\cdot (a \\times b) = 0.$$\n\n## Remarks\n\nBy the [[test with the dot product|03NK6RSRQ1]], $a \\times b$ is perpendicular to $a$ and to $b$ when all three vectors are nonzero.\nThe proof is a direct computation. For example, $a \\cdot (a \\times b) = a_1(a_2 b_3 - a_3 b_2) + a_2(a_3 b_1 - a_1 b_3) + a_3(a_1 b_2 - a_2 b_1)$, and the six terms cancel in pairs.\nIf $a$ and $b$ are linearly independent, then $a \\times b$ is a normal vector of the plane through $0$ that contains $a$ and $b$.\n\n## Explanations\n\n### L1 human\n\nLay two pencils flat on a table, starting at the same point. Their cross product points straight up from the table, or straight down. So it makes a right angle with each pencil.\n",
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