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)
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is a pseudovector, whereas the vector triple product
(2)
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is a polar vector. (Polar) vectors and pseudovectors are interrelated in the following ways under application of the cross product,
(3)
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(4)
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Examples of pseudovectors therefore include the angular velocity vector , angular momentum , torque , auxiliary magnetic field , and magnetic dipole moment .
Given a transformation matrix ,
(5)
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where Einstein summation has been used.