Pseudovector

A typical Vector is transformed to its Negative under inversion. A Vector which is invariant under inversion is called a pseudovector, also called an Axial Vector in older literature (Morse and Feshbach 1953). The Cross Product

$${\bf A}\times {\bf B}$$ (1)

is a pseudovector, whereas the Vector Triple Product

$${\bf A}\times ({\bf B}\times {\bf C})$$ (2)

is a Vector.

$$\hbox{[pseudovector]} \times \hbox{[pseudovector]} = \hbox{[pseudovector]}$$ (3)

$$\hbox{[vector]} \times \hbox{[pseudovector]} = \hbox{[vector]}.$$ (4)

Given a transformation Matrix ${\hbox{\sf A}}$,

$${C_i}' = {\rm det}\vert{\hbox{\sf A}}\vert a_{ij}C_j.$$ (5)

See also Pseudoscalar, Tensor, Vector

References

Arfken, G. ``Pseudotensors, Dual Tensors.'' §3.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 128-137, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 46-47, 1953.