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  "body": "\n## Definition\n\nLet $H$ be a [[line|0MP9BCF6DP]] in $\\mathbb{R}^2$ or a [[plane|0V3XAMF3C1]] in $\\mathbb{R}^3$. Let $\\cdot$ be the dot product.\nA normal vector of $H$ is a nonzero vector $n$ such that $n \\cdot (x - y) = 0$ for all points $x$ and $y$ of $H$.\n\n## Remarks\n\nIf $n$ is a normal vector of $H$, then the normal vectors of $H$ are exactly the vectors $c n$ with $c \\neq 0$.\nThe plane $\\{(x, y, z) : a x + b y + c z = d\\}$ with $(a, b, c) \\neq (0, 0, 0)$ has the normal vector $(a, b, c)$. The line $\\{(x, y) : a x + b y = c\\}$ with $(a, b) \\neq (0, 0)$ has the normal vector $(a, b)$.\nFor the plane $\\{p + s u + t v : s, t \\in \\mathbb{R}\\}$, the [[cross product|0N5N15HWZ7]] $u \\times v$ is a normal vector.\nA unit normal vector is a normal vector of length 1.\n\n## Explanations\n\n### L1 human\n\nA normal vector sticks straight out of a flat object. Think of a pencil standing upright on a table: it is normal to the table. Each arrow that lies in the table makes a right angle with the pencil, so its dot product with the pencil is $0$.\n",
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