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In 2-dimensional and 3-dimensional Euclidean space, a rotation is a type of distance-preserving linear transformation that has a fixed point and preserves orientation. In terms of the usual inner product $\langle\cdot,\cdot\rangle$ on $\Bbb R^2$ and $\Bbb R^3$, rotations are the transformations $T$ such that $\langle Tx,Ty\rangle=\langle x,y\rangle$ for all $x,y$, and the determinant of $T$ is 1.
Geometrically, a rotation in $\Bbb R^2$, "spins" the plane around a point without flipping the plane or sliding it. In $\Bbb R^3$, a rotation fixes a line (called the axis of rotation) and "spins" the space around this line (without reflecting or sliding).
More generally, given an inner product space $V$ over a field $\Bbb F$, anything in the part of the orthogonal group connected to the identity can be considered a "rotation." This allows rotations to be defined for $\Bbb R^n$ for $n$ greater than 2 and 3, as well as vector spaces over fields other than $\Bbb R$.
As an example, the space $\Bbb R^4$ with a non-Euclidean metric $(1,1,1,-1)$ is Minkowski space which is a model for special relativity. Rotations still play an important role here beyond that of rotations in the spacial coordinates. For example, Lorentz transformations are rotations which move the time coordinate.