Pseudovectors

Pseudovectors, also known as axial vectors, are quantities that behave like ordinary vectors under proper rotations but respond differently to improper rotations such as reflections. In other words, under a pure rotation they transform as vectors, but under a parity transformation they may retain their direction while polar vectors would reverse sign. This distinction is important in physics and geometry because it affects how these quantities transform when coordinate systems are mirrored or inverted.

A standard way to identify a pseudovector is through its construction: the cross product of two ordinary vectors, A = u × v, yields an axial vector. More generally, pseudovectors in three dimensions can be viewed as the duals of antisymmetric rank-2 tensors. This dual relationship helps connect cross products to tensor algebra and clarifies why certain physical quantities behave as axial vectors.

Common examples include angular momentum L = r × p, torque τ = r × F, and the magnetic field B. These quantities are axial vectors because they arise from cross products and transform differently under reflections than polar vectors. Conversely, quantities like displacement, velocity, and force are polar vectors, which change sign under parity. Vorticity, ω = ∇ × v, is another pseudovector, reflecting its origin from a curl operation.

In practical terms, recognizing pseudovectors helps avoid errors when performing coordinate changes or analyzing symmetry. They obey the same transformation laws as ordinary vectors under rotations but maintain their sign under reflections, in contrast to polar vectors. This concept is essential across mechanics, electromagnetism, and continuum physics.