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Pseudovector


A typical vector (i.e., a vector such as the radius vector r) 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

 AxB
(1)

is a pseudovector, whereas the vector triple product

 Ax(BxC)
(2)

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

 [pseudovector]x[pseudovector]=[pseudovector]
(3)
 [vector]x[pseudovector]=[vector].
(4)

Examples of pseudovectors therefore include the angular velocity vector omega, angular momentum L, torque tau, auxiliary magnetic field H, and magnetic dipole moment m.

Given a transformation matrix A,

 C_i^'=det|A|a_(ij)C_j,
(5)

where Einstein summation has been used.


See also

Polar Vector, Pseudoscalar, Tensor, Vector

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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.

Referenced on Wolfram|Alpha

Pseudovector

Cite this as:

Weisstein, Eric W. "Pseudovector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pseudovector.html

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