Improper rotation

An improper rotation is an ordinary rotation of real 3-dimensional space R³ that keeps the origin fixed, followed by an inversion (x goes to -x). An improper rotation of an object thus produces a rotation of its mirror image. Improper rotations are described by 3-by-3 orthogonal matrices with a determinant of -1.

(Its antonym, a proper rotation, is simply an ordinary rotation, which has a determinant of 1.)

The product (composition) of two improper rotations is a proper rotation, and the product of an improper and a proper rotation is a improper rotation.

When studying the symmetry of a physical system under an improper rotation (e.g. if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, etcetera), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).