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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Rotation_(mathematics)


00:01:45 1 Related definitions and terminology
00:03:26 2 Definitions and representations
00:03:37 2.1 In Euclidean geometry
00:08:57 2.2 Linear and multilinear algebra formalism
00:11:08 2.2.1 Two dimensions
00:16:15 2.2.2 Three dimensions
00:18:32 2.2.3 Quaternions
00:20:56 2.2.4 Further notes
00:22:43 2.2.5 More alternatives to the matrix formalism
00:23:50 2.3 In non-Euclidean geometries
00:24:49 2.4 In relativity
00:26:59 2.5 Discrete rotations
00:27:09 3 Importance
00:28:14 4 Generalizations
00:29:10 5 See also



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SUMMARY
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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations.