Overview. In both physics and mathematics, a pseudovector—also called an axial vector—is an object that responds to coordinate changes like an ordinary vector under proper rotations, but acquires an extra sign flip under certain orientation-reversing transformations. This distinctive behaviour distinguishes pseudovectors from ordinary (or polar) vectors and influences how they appear in formulas and physical laws.

Transformation properties

A pseudovector transforms like a standard vector when the coordinate system is rotated without changing the handedness of the axes. However, under an improper rotation—such as spatial inversion or reflection, which can be viewed as an inversion combined with a proper rotation—it gains an additional factor of −1. That is why some authors say it is a vector times the sign of the coordinate system's orientation. The distinction is tied to whether a transformation preserves or flips the orientation of space; many algebraic treatments express this using the determinant of the transformation matrix.

Common examples and physical meaning

Examples that are typically classified as pseudovectors in three dimensions include angular momentum, torque, and the magnetic field. These quantities often arise from a cross product of two polar vectors (for instance, r × p gives angular momentum), and the cross product of two ordinary vectors is itself an axial vector. Because of their behavior under reflections, pseudovectors play a central role in distinguishing physical processes that are sensitive to parity or handedness.

Mathematical context and generalizations

In three dimensions a close relationship exists between pseudovectors and antisymmetric second-rank tensors: a 3×3 antisymmetric matrix can be identified with a pseudovector via the isomorphism given by the Levi-Civita symbol. In higher-dimensional settings the concept generalizes through the Hodge dual: what is an axial vector in three dimensions corresponds to certain (n−1)-forms or dual tensors in n dimensions. For formal discussions see treatments in linear algebra and tensor calculus.

Uses, distinctions, and notable facts

  • Practical: recognizing a quantity as axial or polar helps ensure equations respect symmetry under reflections.
  • Distinction: polar (true) vectors do not change sign under inversion, while axial vectors do.
  • Historical/terminology: both names—"pseudovector" and "axial vector"—are used; the latter emphasizes their origin from rotations about an axis.

For introductory treatments and more formal definitions consult standard texts in mechanics and vector analysis or accessible online references: see entries in physics and mathematics resources on rotations and on improper transformations.