Pseudovector

A typical vector (i.e., a vector such as the radius vector ) is transformed to its negative under inversion of its coordinate axes. Such "proper" vectors are known as polar vectors. A vector-like object which is invariant under inversion is called a pseudovector, also called an axial vector (as a result of such vectors frequently arising as vectors describing rotation; Arfken 1985, p. 128; Morse and Feshbach 1953). The cross product

(1)

is a pseudovector, whereas the vector triple product

(2)

is a polar vector. (Polar) vectors and pseudovectors are interrelated in the following ways under application of the cross product,

(3)

(4)

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

Given a transformation matrix ,

(5)

where Einstein summation has been used.