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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}
\end{displaymath} (1)

is a pseudovector, whereas the Vector Triple Product
{\bf A}\times ({\bf B}\times {\bf C})
\end{displaymath} (2)

is a Vector.
\hbox{[pseudovector]} \times \hbox{[pseudovector]} = \hbox{[pseudovector]}
\end{displaymath} (3)

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

Given a transformation Matrix ${\hbox{\sf A}}$,
{C_i}' = {\rm det}\vert{\hbox{\sf A}}\vert a_{ij}C_j.
\end{displaymath} (5)

See also Pseudoscalar, Tensor, Vector


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.

© 1996-9 Eric W. Weisstein