is an axial vector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). The conceptual opposite of a pseudovector is a (true) vector or a polar vector.

A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b:

p = a × b

A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.