Inphysics andmathematics, apseudovector (oraxial vector)^[2] is a quantity that behaves like avector in many situations, but itsdirection does not conform when the object isrigidly transformed byrotation,translation,reflection, etc. This can also happen when theorientation of the space is changed. For example, theangular momentum is a pseudovector because it is often described as a vector, but by just changing the position of reference (and therefore changing theposition vector), angular momentum can reverse direction, which is not supposed to happen with true vectors (also known aspolar vectors).^[3]

One example of a pseudovector is thenormal to anorientedplane. Anoriented plane can be defined by two non-parallel vectors,a andb,^[4] that span the plane. The vectora ×b is a normal to the plane (there are two normals, one on each side – theright-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics, where it has to be considered whentransforming surface normals.In three dimensions, thecurl of a polarvector field at a point and thecross product of two polar vectors are pseudovectors.^[5]

A number of quantities in physics behave as pseudovectors rather than polar vectors, includingmagnetic field andangular velocity. In mathematics, in three dimensions, pseudovectors are equivalent tobivectors, from which the transformation rules of pseudovectors can be derived. More generally, inn-dimensionalgeometric algebra, pseudovectors are the elements of the algebra with dimensionn − 1, written ⋀ⁿ⁻¹Rⁿ. The label "pseudo-" can be further generalized topseudoscalars andpseudotensors, both of which gain an extra sign-flip under improper rotations compared to a truescalar ortensor.

Physical examples of pseudovectors includetorque,^[4]angular velocity,angular momentum,^[4]magnetic field,^[4]vorticity andmagnetic dipole moment.

Consider the pseudovectorangular momentumL = Σ(r ×p). Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but theactual angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left, corresponding to the extra sign flip in the reflection of a pseudovector.

The distinction between polar vectors and pseudovectors becomes important in understandingthe effect of symmetry on the solution to physical systems. Consider an electric current loop in thez = 0 plane that inside the loop generates a magnetic field oriented in thez direction. This system issymmetric (invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged.

In physics, pseudovectors are generally the result of taking thecross product of two polar vectors or thecurl of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced by using (left-handed) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.

While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g.${\displaystyle \mathbf{a}=(a_x,a_y,a_z)}$, and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by theexterior product of the two vectors, which yields abivector which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstractvector space). Under the physics definition, a "vector" is required to havecomponents that "transform" in a certain way under aproper rotation: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective ofactive transformations.) Mathematically, if everything in the universe undergoes a rotation described by arotation matrixR, so that adisplacement vectorx is transformed tox′ =Rx, then any "vector"v must be similarly transformed tov′ =Rv. This important requirement is what distinguishes avector (which might be composed of, for example, thex-,y-, andz-components ofvelocity) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular boxcannot be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)

(In the language ofdifferential geometry, this requirement is equivalent to defining avector to be atensor ofcontravariant rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within theEinstein summation convention.)

A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield thedyadic product, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.

The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also considerimproper rotations, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation isinversion through a point in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrixR, so that a position vectorx is transformed tox′ =Rx. If the vectorv is a polar vector, it will be transformed tov′ =Rv. If it is a pseudovector, it will be transformed tov′ = −Rv.

The transformation rules for polar vectors and pseudovectors can be compactly stated as

where the symbols are as described above, and the rotation matrixR can be either proper or improper. The symbol det denotesdeterminant; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.

Supposev₁ andv₂ are known pseudovectors, andv₃ is defined to be their sum,v₃ =v₁ +v₂. If the universe is transformed by a rotation matrixR, thenv₃ is transformed to

Sov₃ is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any real number yields another pseudovector.

On the other hand, supposev₁ is known to be a polar vector,v₂ is known to be a pseudovector, andv₃ is defined to be their sum,v₃ =v₁ +v₂. If the universe is transformed by an improper rotation matrixR, thenv₃ is transformed to

${\displaystyle {{\mathbf{v}}_{\mathbf{3}}}^{\prime }={{\mathbf{v}}_{\mathbf{1}}}^{\prime }+{{\mathbf{v}}_{\mathbf{2}}}^{\prime }=(R{\mathbf{v}}_{\mathbf{1}})+(detR)(R{\mathbf{v}}_{\mathbf{2}})=R({\mathbf{v}}_{\mathbf{1}}+(detR){\mathbf{v}}_{\mathbf{2}}).}$

Therefore,v₃ is neither a polar vector nor a pseudovector (although it is still a vector, by the physics definition). For an improper rotation,v₃ does not in general even keep the same magnitude:

${\displaystyle |{\mathbf{v}}_{\mathbf{3}}|=|{\mathbf{v}}_{\mathbf{1}}+{\mathbf{v}}_{\mathbf{2}}|,\text{ but }\left|{{\mathbf{v}}_{\mathbf{3}}}^{\prime }\right|=\left|{{\mathbf{v}}_{\mathbf{1}}}^{\prime }-{{\mathbf{v}}_{\mathbf{2}}}^{\prime }\right|}$.

If the magnitude ofv₃ were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in theweak interaction: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. (Seeparity violation.)

For a rotation matrixR, either proper or improper, the following mathematical equation is always true:

${\displaystyle (R{\mathbf{v}}_{\mathbf{1}})\times (R{\mathbf{v}}_{\mathbf{2}})=(detR)(R({\mathbf{v}}_{\mathbf{1}}\times {\mathbf{v}}_{\mathbf{2}}))}$,

wherev₁ andv₂ are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.)

Supposev₁ andv₂ are known polar vectors, andv₃ is defined to be their cross product,v₃ =v₁ ×v₂. If the universe is transformed by a rotation matrixR, thenv₃ is transformed to

${\displaystyle {{\mathbf{v}}_{\mathbf{3}}}^{\prime }={{\mathbf{v}}_{\mathbf{1}}}^{\prime }\times {{\mathbf{v}}_{\mathbf{2}}}^{\prime }=(R{\mathbf{v}}_{\mathbf{1}})\times (R{\mathbf{v}}_{\mathbf{2}})=(detR)(R({\mathbf{v}}_{\mathbf{1}}\times {\mathbf{v}}_{\mathbf{2}}))=(detR)(R{\mathbf{v}}_{\mathbf{3}}).}$

Sov₃ is a pseudovector. Similarly, one can show:

• polar vector × polar vector = pseudovector
• pseudovector × pseudovector = pseudovector
• polar vector × pseudovector = polar vector
• pseudovector × polar vector = polar vector

This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.

From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a polar vector. Angular momentum is the cross product of a displacement (a polar vector) and momentum (a polar vector), and is therefore a pseudovector. Torque is angular momentum (a pseudovector) divided by time (a scalar), so is also a pseudovector. Continuing this way, it is straightforward to classify any of the common vectors in physics as either a pseudovector or polar vector. (There are the parity-violating vectors in the theory of weak-interactions, which are neither polar vectors nor pseudovectors. However, these occur very rarely in physics.)

Above, pseudovectors have been discussed usingactive transformations. An alternate approach, more along the lines ofpassive transformations, is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" everywhere in math and physics, including in the definition of thecross product and thecurl. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Nevertheless, there would be no physical consequences, apart from in theparity-violating phenomena such as certainradioactive decays.^[6]

One way to formalize pseudovectors is as follows: ifV is ann-dimensional vector space, then apseudovector ofV is an element of the (n − 1)-thexterior power ofV: ⋀ⁿ⁻¹(V). The pseudovectors ofV form a vector space with the same dimension asV.

This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces. In particular, whenn iseven, such a pseudovector does not experience a sign flip, and when thecharacteristic of the underlyingfield ofV is 2, a sign flip has no effect. Otherwise, the definitions are equivalent, though it should be borne in mind that without additional structure (specifically, either avolume form or anorientation), there is no natural identification of ⋀ⁿ⁻¹(V) withV.

Another way to formalize them is by considering them as elements of a representation space for${\displaystyle \text{O}(n)}$. Vectors transform in thefundamental representation of${\displaystyle \text{O}(n)}$ with data given by${\displaystyle ({\mathbb{R}}^n,{\rho }_{\text{fund}},\text{O}(n))}$, so that for any matrix${\displaystyle R}$ in${\displaystyle \text{O}(n)}$, one has${\displaystyle {\rho }_{\text{fund}}(R)=R}$. Pseudovectors transform in a pseudofundamental representation${\displaystyle ({\mathbb{R}}^n,{\rho }_{\text{pseudo}},\text{O}(n))}$, with${\displaystyle {\rho }_{\text{pseudo}}(R)=det(R)R}$. Another way to view this homomorphism for${\displaystyle n}$ odd is that in this case${\displaystyle \text{O}(n)\cong \text{SO}(n)\times {\mathbb{Z}}_2}$. Then${\displaystyle {\rho }_{\text{pseudo}}}$ is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on${\displaystyle \text{SO}(n)}$ with the trivial homomorphism on${\displaystyle {\mathbb{Z}}_2}$.

Ingeometric algebra the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.

The basic multiplication in the geometric algebra is thegeometric product, denoted by simply juxtaposing two vectors as inab. This product is expressed as:

${\displaystyle \mathbf{a}\mathbf{b}=\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\wedge \mathbf{b},}$

where the leading term is the customary vectordot product and the second term is called thewedge product or exterior product. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, amultivector is a summation ofk-fold wedge products of variousk-values. Ak-fold wedge product also is referred to as ak-blade.

In the present context thepseudovector is one of these combinations. This term is attached to a different multivector depending upon thedimensions of the space (that is, the number oflinearly independent vectors in the space). In three dimensions, the most general 2-blade orbivector can be expressed as the wedge product of two vectors and is a pseudovector.^[7] In four dimensions, however, the pseudovectors aretrivectors.^[8] In general, it is a(n − 1)-blade, wheren is the dimension of the space and algebra.^[9] Ann-dimensional space hasn basis vectors and alson basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of then basis vectors. For instance, in four dimensions where the basis vectors are taken to be {e₁,e₂,e₃,e₄}, the pseudovectors can be written as: {e₂₃₄,e₁₃₄,e₁₂₄,e₁₂₃}.

The transformation properties of the pseudovector in three dimensions has been compared to that of thevector cross product by Baylis.^[10] He says: "The termsaxial vector andpseudovector are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual." To paraphrase Baylis: Given two polar vectors (that is, true vectors)a andb in three dimensions, the cross product composed froma andb is the vector normal to their plane given byc =a ×b. Given a set of right-handed orthonormalbasis vectors{e_ℓ }, the cross product is expressed in terms of its components as:

${\displaystyle \mathbf{a}\times \mathbf{b}=\left(a^2{b}^3-a^3{b}^2\right){\mathbf{e}}_1+\left(a^3{b}^1-a^1{b}^3\right){\mathbf{e}}_2+\left(a^1{b}^2-a^2{b}^1\right){\mathbf{e}}_3,}$

where superscripts label vector components. On the other hand, the plane of the two vectors is represented by theexterior product or wedge product, denoted bya ∧b. In this context of geometric algebra, thisbivector is called a pseudovector, and is theHodge dual of the cross product.^[11] Thedual ofe₁ is introduced ase₂₃ ≡e₂e₃ =e₂ ∧e₃, and so forth. That is, the dual ofe₁ is the subspace perpendicular toe₁, namely the subspace spanned bye₂ ande₃. With this understanding,^[12]

${\displaystyle \mathbf{a}\wedge \mathbf{b}=\left(a^2{b}^3-a^3{b}^2\right){\mathbf{e}}_{23}+\left(a^3{b}^1-a^1{b}^3\right){\mathbf{e}}_{31}+\left(a^1{b}^2-a^2{b}^1\right){\mathbf{e}}_{12}.}$

For details, seeHodge star operator § Three dimensions. The cross product and wedge product are related by:

${\displaystyle \mathbf{a}\wedge \mathbf{b}=\mathit{i}\mathbf{a}\times \mathbf{b},}$

wherei =e₁ ∧e₂ ∧e₃ is called theunit pseudoscalar.^[13][14] It has the property:^[15]

${\displaystyle {\mathit{i}}^2=-1.}$

Using the above relations, it is seen that if the vectorsa andb are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectorse_ℓ are inverted, then the pseudovector is invariant, but the cross product changes sign. This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.

As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product.^[16] However, because the cross product does not generalize to other than three dimensions,^[17]the notion of pseudovector based upon the cross product also cannot be extended to a space of any other number of dimensions. The pseudovector as a(n – 1)-blade in ann-dimensional space is not restricted in this way.

Another important note is that pseudovectors, despite their name, are "vectors" in the sense of being elements of avector space. The idea that "a pseudovector is different from a vector" is only true with a different and more specific definition of the term "vector" as discussed above.

• Exterior algebra
• Clifford algebra
• Antivector, a generalization of pseudovector in Clifford algebra
• Orientability — discussion about non-orientable spaces.
• Tensor density

• Linear algebra
• Vector calculus
• Vectors (mathematics and physics)

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