A rotation about the origin is a linear transformation that preserves lengths and angles while preserving orientation.

In two dimensions, rotating counterclockwise by angle $\theta$ is represented by

$$R(\theta)= \begin{bmatrix} \cos\theta&-\sin\theta\ \sin\theta&\cos\theta \end{bmatrix}.$$

Orthogonal matrices

A rotation matrix $R$ satisfies

$$R^TR=I,$$

so

$$R^{-1}=R^T.$$

Its determinant is

$$\det R=1,$$

distinguishing rotations from reflections, which can also preserve lengths but reverse orientation.

Three-dimensional rotations

In three dimensions, rotations can be represented by $3\times3$ orthogonal matrices with determinant one.

Rotations about the coordinate axes have simple matrix forms, but a general orientation can be obtained by composing several rotations.

Composition is order-dependent

Matrix multiplication represents composition. In three dimensions,

$$R_1R_2\ne R_2R_1$$

in general, so applying rotations in a different order can produce a different final orientation.

Rotation matrices provide a direct linear-algebra representation of orientation but use nine components to encode only three rotational degrees of freedom.