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  "body": "\n## Definition\n\nLet $n$ be a positive integer, and let $p = (p_1, \\dots, p_n)$ and $q = (q_1, \\dots, q_n)$ be points of $\\mathbb{R}^n$.\nThe Euclidean distance between $p$ and $q$ is the real number\n$$d(p, q) = \\|p - q\\| = \\sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2 + \\dots + (p_n - q_n)^2},$$\nwhere $\\|p - q\\|$ is the [[length|0XKBKGTCPC]] of the vector $p - q$.\n\n## Remarks\n\nThe function $d$ is a metric on $\\mathbb{R}^n$. For all points $p$, $q$ and $r$: $d(p, q) \\geq 0$, $d(p, q) = 0$ if and only if $p = q$, $d(p, q) = d(q, p)$, and $d(p, r) \\leq d(p, q) + d(q, r)$.\nFor $n = 1$ the formula is $d(p, q) = |p - q|$.\nSome applications use the squared distance $d(p, q)^2$, because it has no square root. The squared distance is not a metric, because it does not satisfy the triangle inequality.\n\n## Explanations\n\n### L1 human\n\nTo find the distance between two points in the plane, draw the right triangle with horizontal and vertical legs between them. The legs have lengths $|p_1 - q_1|$ and $|p_2 - q_2|$. The distance is the hypotenuse, so the Pythagorean theorem gives the formula. For $(1, 1)$ and $(4, 5)$ the distance is $\\sqrt{9 + 16} = 5$.\n",
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