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In geometry, an improper rotation is the combination of an
ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with an inversion (x goes to −x).
Improper rotations are described by 3×3 orthogonal matrices
with a determinant of −1. 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 an improper rotation.
An improper rotation of an object thus produces a rotation of its mirror
image.
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, and in general; between tensors and pseudotensors), since the latter
transform differently under proper and improper rotations (pseudovectors are invariant under inversion).
See also: Isometry,Orthogonal group
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