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In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation,[needs to be explained] but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation).[1]

Physical examples

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Physical examples of pseudovectors include the magnetic field, torque, vorticity, and the angular momentum.[2]


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  1. A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, y to −y and z to −z. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.
  2. Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems.
  • George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001). (ISBN 0-12-059815-9)
  • John David Jackson, Classical Electrodynamics (Wiley: New York, 1999). (ISBN 0-471-30932-X)
  • Susan M. Lea, "Mathematics for Physicists" (Thompson: Belmont, 2004) (ISBN 0-534-37997-4)