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  "body": "\n## Definition\n\nLet $n$ be a positive integer, let $a$ be a vector in $\\mathbb{R}^n$, and let $b$ be a nonzero vector in $\\mathbb{R}^n$. Let $a \\cdot b = a_1 b_1 + \\dots + a_n b_n$ be the dot product.\nThe vector projection of $a$ onto $b$ is the vector\n$$\\operatorname{proj}_b a = \\frac{a \\cdot b}{b \\cdot b} \\, b.$$\nThe scalar projection of $a$ onto $b$ is the real number $\\dfrac{a \\cdot b}{\\|b\\|}$, where $\\|b\\|$ is the [[length|0XKBKGTCPC]] of $b$.\n\n## Remarks\n\nThe vector $a - \\operatorname{proj}_b a$ is the vector rejection of $a$ from $b$. Its dot product with $b$ is $0$, so it is [[perpendicular to $b$|03NK6RSRQ1]].\nAmong all multiples $c b$ of $b$, the projection is the one that makes $\\|a - c b\\|$ smallest.\nIf $\\theta$ is the angle between $a \\neq 0$ and $b$, then the scalar projection is $\\|a\\| \\cos \\theta$.\n\n## Explanations\n\n### L1 human\n\nThe projection answers the question \"how much of $a$ points along $b$?\". For $a = (3, 4)$ and $b = (1, 0)$ the projection is $(3, 0)$. The rest, $(0, 4)$, is perpendicular to $b$.\n",
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