pseudovector

A pseudovector, also called an axial vector, is a quantity that behaves as a vector under proper rotations but picks up an extra sign under improper rotations (mirror reflections). In three-dimensional space, a pseudovector A can be formed as the cross product of two polar vectors, A = u × v. Equivalently, it is the axial dual to an antisymmetric rank-2 tensor; its components can be obtained from the Levi-Civita symbol εijk times the antisymmetric tensor components.

Under a rotation represented by a proper orthogonal matrix R with det(R) = +1, A′ = R A. Under a transformation that includes a reflection, with det(R) = −1, A′ = det(R) R A = − R A. Thus axial vectors are invariant under inversion in the sense that they do not flip sign as polar vectors do.

Common examples: angular momentum L = r × p is a pseudovector; torque τ = r × F is also a pseudovector; magnetic field B is an axial vector in classical electromagnetism, whereas the electric field E is a polar vector.

Pseudovectors clarify how quantities respond to spatial symmetries and parity. They are distinct from polar vectors in transformation rules under improper rotations, and they often arise as duals of antisymmetric tensors or as cross products of polar vectors.

In mathematical physics, the concept generalizes to higher dimensions via duality with antisymmetric tensors and the use of Levi-Civita symbols.