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  "body": "\n## Description\n\nLet $n$ be a positive integer.\nThe Euclidean $n$-space is the set $\\mathbb{R}^n$ of the $n$-tuples $x = (x_1, \\dots, x_n)$ of real numbers, with the dot product\n$$x \\cdot y = x_1 y_1 + x_2 y_2 + \\dots + x_n y_n.$$\nThe elements of $\\mathbb{R}^n$ are its points, and also its vectors.\nThe dot product gives the length $\\|x\\| = \\sqrt{x \\cdot x}$ of a vector $x$.\nIt also gives the distance $\\|x - y\\|$ between two points $x$ and $y$, and the angle between two nonzero vectors.\nFor $n = 2$ this is the [[Euclidean plane|0XW7V9ZZ05]], and for $n = 3$ it is the space of elementary solid geometry.\n\n## Remarks\n\nIn a more abstract form, a Euclidean space is a real [[affine space|09XNAPTJQ7]] of finite dimension whose space of translations has an inner product. Each such space of dimension $n$ is isometric to $\\mathbb{R}^n$ with the dot product.\nThe notation $\\mathbb{E}^n$ is also common for the Euclidean $n$-space.\n\n## Explanations\n\n### L1 human\n\nThe Euclidean $n$-space is the space of ordinary geometry in $n$ dimensions. A point is a list of $n$ coordinates. The dot product lets you measure lengths and angles, so the familiar facts of school geometry hold in each dimension.\n",
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