About

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.

Rotations can also be represented in terms of matrices and the tag often goes with this tag, as questions can pertain to rotation matrices.

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