# Pseudovector

A pseudovector, also called an axial vector, is a quantity with magnitude and direction that transforms like a vector under rotations but gains an extra sign change under reflections and spatial inversion (parity). Under a parity transformation, in which all three coordinate axes are inverted, an ordinary vector changes sign, while a pseudovector does not.<sup>[1](https://hepweb.ucsd.edu/ph110b/110b_notes/node4.html)</sup> Ordinary vectors are called polar vectors to distinguish them.<sup>[2](https://www.feynmanlectures.caltech.edu/I_20.html)</sup>

| Key fact | Detail |
|---|---|
| Defining behavior | Does not change sign under spatial inversion or reflection, unlike polar vectors, which do<sup>[1](https://hepweb.ucsd.edu/ph110b/110b_notes/node4.html)</sup> |
| Transformation law | \( \mu^{\ast}_i = \det\vert M\vert\, M_{ij}\mu_j \): the polar-vector rule multiplied by the determinant of the transformation matrix<sup>[3](https://circle-test.iucr.org/what-we-do/education/pamphlets/teaching-crystallographic-and-magnetic-point-group-symmetry-using)</sup> |
| Typical examples | Magnetic field \( \mathbf{B} \), angular momentum \( \mathbf{L} = \mathbf{x} \times \mathbf{p} \), torque \( \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \), angular velocity \( \boldsymbol{\omega} \)<sup>[2](https://www.feynmanlectures.caltech.edu/I_20.html)</sup><sup> • </sup><sup>[4](https://www.damtp.cam.ac.uk/user/tong/sm/standardmodel1.pdf)</sup> |
| Origin | The cross product of two polar vectors, which carries one factor of the handedness of the coordinate system<sup>[5](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.05%3A_Appendix_-_Coordinate_transformations)</sup> |
| True mathematical nature | An antisymmetric second-rank tensor; only in three dimensions does it have three independent components and can be cast as a vector<sup>[1](https://hepweb.ucsd.edu/ph110b/110b_notes/node4.html)</sup> |
| Companion concept | The scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) of polar vectors changes sign under inversion and is a pseudoscalar<sup>[6](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_44_Symmetries_Monopole.html)</sup> |

## Why cross products behave differently

The distinction appears as soon as both factors of a cross product change sign. If \( \mathbf{A} \) and \( \mathbf{B} \) are polar vectors, spatial inversion sends \( \mathbf{A} \mapsto -\mathbf{A} \) and \( \mathbf{B} \mapsto -\mathbf{B} \), so their cross product obeys \( (-\mathbf{A}) \times (-\mathbf{B}) = \mathbf{A} \times \mathbf{B} \): it maps to itself rather than to its negative.<sup>[7](https://sites.science.oregonstate.edu/physics/coursewikis/GDF/_export/raw/book/gdf/pseudo)</sup> [Angular momentum](https://www.edgepedia.org/angular-momentum) illustrates the point directly: \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \), and since both \( \mathbf{r} \) and \( \mathbf{p} \) change sign under parity inversion, \( \mathbf{L} \) does not.<sup>[1](https://hepweb.ucsd.edu/ph110b/110b_notes/node4.html)</sup>

The general transformation rule follows from the determinant. For an orthogonal transformation with matrix \( \lambda \), a cross product transforms as \( c^{\prime}_i = \vert\lambda\vert \sum_j \lambda_{ij} c_j \). For all proper rotations \( \vert\lambda\vert = +1 \), so the cross product acts like a proper vector; for improper rotations, which include a reflection, \( \vert\lambda\vert = -1 \) and the extra sign appears.<sup>[5](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.05%3A_Appendix_-_Coordinate_transformations)</sup> [Crystallography](https://www.edgepedia.org/crystallography) states the same rule for any axial vector \( \boldsymbol{\mu} \) under a symmetry operator with matrix \( M \): \( \mu^{\ast}_i = \det\vert M\vert\, M_{ij}\mu_j \), an additional sign change for operators of odd spatial parity.<sup>[3](https://circle-test.iucr.org/what-we-do/education/pamphlets/teaching-crystallographic-and-magnetic-point-group-symmetry-using)</sup>

**Handedness is built in.** Pseudovectors are defined using the [right-hand rule](https://www.edgepedia.org/right-hand-rule) and therefore carry a handedness: in a right-handed system \( \mathbf{C}_R = \mathbf{A} \times \mathbf{B} \), while switching to a left-handed system gives \( \mathbf{C}_L = \mathbf{B} \times \mathbf{A} = -\mathbf{A} \times \mathbf{B} \). Proper orthogonal transformations preserve the chirality (handedness) of a coordinate system; improper ones reverse it.<sup>[5](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.05%3A_Appendix_-_Coordinate_transformations)</sup>

## The tensor behind the vector

The vector disguise works only in three dimensions. The cross product is really an antisymmetric second-rank tensor, \( L_{ij} = r_i p_j - r_j p_i \), which has just three independent components in three dimensions and can therefore be cast as a vector, with \( L_k = r_i p_j \epsilon_{ijk} \) relating the two forms through the [Levi-Civita symbol](https://www.edgepedia.org/levi-civita-symbol).<sup>[1](https://hepweb.ucsd.edu/ph110b/110b_notes/node4.html)</sup> Feynman makes the same point about torque: the components \( \tau_{xy} = xF_y - yF_x \) and their cyclic partners form an antisymmetric tensor \( \tau_{ij} = r_i F_j - r_j F_i \), whose three nonzero terms transform like vector components "almost by accident," something that happens only in three dimensions. In four dimensions an antisymmetric second-rank tensor has up to six nonzero terms and cannot be replaced by a four-component vector.<sup>[8](https://www.feynmanlectures.caltech.edu/II_31.html)</sup>

[Differential geometry](https://www.edgepedia.org/differential-geometry) identifies the same object as a 2-form: over Euclidean \( \mathbb{R}^3 \), 0-forms are scalars, 1-forms are vectors, 2-forms are pseudovectors, and 3-forms are pseudoscalars, with even-rank forms unchanged under parity and odd-rank forms picking up a minus sign.<sup>[7](https://sites.science.oregonstate.edu/physics/coursewikis/GDF/_export/raw/book/gdf/pseudo)</sup>

## Electromagnetism

The electric and magnetic fields split across the two classes. Under parity, \( \mathbf{E} \mapsto -\mathbf{E} \) while \( \mathbf{B} \mapsto +\mathbf{B} \); vectors like \( \mathbf{E} \), which transform as \( \mathbf{x} \) does, keep the name vector, while \( \mathbf{B} \) is a pseudovector.<sup>[4](https://www.damtp.cam.ac.uk/user/tong/sm/standardmodel1.pdf)</sup> The magnetic field's status can be seen from the Biot-Savart law, in which \( \mathbf{B} \) depends on the cross product of a current element and a radius vector, both of which change sign under parity.<sup>[9](https://sbhep.physics.sunysb.edu/~rijssenbeek/PHY557/PHY557_F09_Lectures_LS.html)</sup> The curl of a proper vector field is likewise a pseudovector field.<sup>[10](https://farside.ph.utexas.edu/teaching/em/lectures/node120.html)</sup>

Consistency of the [Lorentz force](https://www.edgepedia.org/lorentz-force) \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \) requires this assignment: force and velocity are proper vectors, so \( \mathbf{B} \) must be a pseudovector for the equation to hold under reflection. The same physical results follow whether a right- or left-handed definition of \( \mathbf{B} \) is used, provided the convention is applied consistently.<sup>[5](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.05%3A_Appendix_-_Coordinate_transformations)</sup>

## Related quantities and physical significance

The classification extends to scalars and tensors. Scalars such as \( \mathbf{a} \cdot \mathbf{b} \) do not change sign under spatial inversion, but the triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) of polar vectors does, so it is called a pseudoscalar. A tensor of order \( N \) transforms under inversion with a factor \( (-1)^N \), while a pseudotensor carries \( (-1)^{N+1} \).<sup>[6](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_44_Symmetries_Monopole.html)</sup>

Time reversal acts differently from parity: position is unchanged while momentum changes sign, so angular momentum reverses, \( \mathbf{L}^{\prime} = -\mathbf{L} \). Magnetic fields and magnetization are odd under time reversal because they are generated by circulating currents or by spin, itself an angular momentum.<sup>[6](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_44_Symmetries_Monopole.html)</sup>

Whether nature distinguishes left from right is a physical question, not a matter of convention. Gravitational, electromagnetic, and strong interactions are symmetric under spatial inversion, a property called conservation of parity, while the weak force violates parity and favors handedness.<sup>[5](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.05%3A_Appendix_-_Coordinate_transformations)</sup> In mirror reflection, a counter-clockwise current becomes a clockwise one, so the mirror image of a magnetic moment parallel to a mirror plane points opposite to the original vector, whereas a polar vector parallel to the plane keeps its direction.<sup>[3](https://circle-test.iucr.org/what-we-do/education/pamphlets/teaching-crystallographic-and-magnetic-point-group-symmetry-using)</sup>

## References

1. [Cross Products and Axial Vectors, UC San Diego Physics 110B notes](https://hepweb.ucsd.edu/ph110b/110b_notes/node4.html)
2. [The Feynman Lectures on Physics, Vol. I Ch. 20: Rotation in space](https://www.feynmanlectures.caltech.edu/I_20.html)
3. [Teaching crystallographic and magnetic point group symmetry using three-dimensional rendered visualizations, IUCr](https://circle-test.iucr.org/what-we-do/education/pamphlets/teaching-crystallographic-and-magnetic-point-group-symmetry-using)
4. [David Tong, Lectures on the Standard Model, University of Cambridge](https://www.damtp.cam.ac.uk/user/tong/sm/standardmodel1.pdf)
5. [Coordinate transformations, Physics LibreTexts (Cline, Variational Principles in Classical Mechanics)](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/19%3A_Mathematical_Methods_for_Classical_Mechanics/19.05%3A_Appendix_-_Coordinate_transformations)
6. [Symmetries and Monopole, University of Virginia (after Jackson Ch. 6.10)](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_44_Symmetries_Monopole.html)
7. [Pseudovectors and Pseudoscalars, Geometry of Differential Forms, Oregon State University](https://sites.science.oregonstate.edu/physics/coursewikis/GDF/_export/raw/book/gdf/pseudo)
8. [The Feynman Lectures on Physics, Vol. II Ch. 31: Tensors](https://www.feynmanlectures.caltech.edu/II_31.html)
9. [PHY557 lecture notes, Stony Brook University](https://sbhep.physics.sunysb.edu/~rijssenbeek/PHY557/PHY557_F09_Lectures_LS.html)
10. [Tensors and pseudo-tensors, University of Texas electrodynamics lectures](https://farside.ph.utexas.edu/teaching/em/lectures/node120.html)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Motion, forces, and dynamics › Forces, moments, and equilibrium › Moments and torque › Torque vectors, axes, and 3D moments*

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