Pseudovector

A pseudovector, also called an axial vector, transforms as a polar vector under rotations but acquires an extra sign under orientation-reversing orthogonal transformations (Morse and Feshbach 1953; Arfken 1985, p. 128). Given an orthogonal matrix with entries acting on a physical configuration relative to a fixed right-handed coordinate system, a pseudovector transforms as

(1)

where Einstein summation is used. A polar vector instead transforms as . Inversion of all three spatial directions therefore reverses polar vectors but leaves pseudovectors unchanged.

The cross product of two polar vectors

(2)

is a pseudovector, whereas the vector triple product

(3)

is a polar vector. Cross products involving pseudovectors also have the following types:

(4)

(5)

Examples of pseudovectors include angular velocity , angular momentum , torque , auxiliary magnetic field , and magnetic dipole moment .

A passive switch to a left-handed coordinate system instead changes the component formula for the cross product through its handedness factor. It does not describe the physical inversion used to distinguish polar vectors from pseudovectors.