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  "body": "\n## Definition\n\nLet $c$ be a point of $\\mathbb{R}^n$, and let $A$ be an orthogonal $n \\times n$ matrix with $\\det A = 1$. The rotation about $c$ with the matrix $A$ is the map\n$$R \\colon \\mathbb{R}^n \\to \\mathbb{R}^n, \\qquad R(p) = c + A\\,(p - c).$$\nIn the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$, the rotation about $c$ through the angle $\\theta \\in \\mathbb{R}$ is the case\n$$A = \\begin{pmatrix} \\cos \\theta \u0026 -\\sin \\theta \\\\ \\sin \\theta \u0026 \\cos \\theta \\end{pmatrix}.$$\n\n## Remarks\n\nEach rotation is a direct [[Euclidean isometry|0J1N3WTKM4]] that fixes $c$. In the plane, each orthogonal matrix with determinant $1$ has this form, so each plane rotation has an angle.\nIn the plane, rotations about one centre commute, and their angles add modulo $2\\pi$. Rotations about different centres do not commute in general.\nIn the plane, the composite of the reflections in two lines that meet at $c$ is a rotation about $c$.\nMathlib's `Orientation.rotation` is the rotation through an angle of an oriented Euclidean plane, as a linear map.\n\n## Explanations\n\n### L1 human\n\nA plane rotation turns the whole plane about one fixed point through one angle. With $\\theta \u003e 0$, the turn is anticlockwise. The rotation about $(0, 0)$ through $\\pi/2$ sends $(1, 0)$ to $(0, 1)$ and $(0, 1)$ to $(-1, 0)$.\n",
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