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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its opposite if the orientation of the space is changed, or an improper rigid transformation such as a reflection is applied to the whole figure. Geometrically, the direction of a reflected pseudovector is opposite to its
mirror image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
, but with equal magnitude. In contrast, the reflection of a ''true'' (or polar) vector is exactly the same as its mirror image. In three dimensions, the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fir ...
of a polar vector field at a point and the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
of two polar vectors are pseudovectors. One example of a pseudovector is the normal to an oriented
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
. An oriented plane can be defined by two non-parallel vectors, a and b,RP Feynman: §52-5 Polar and axial vectors, Feynman Lectures in Physics, Vol. 1
that span the plane. The vector is a normal to the plane (there are two normals, one on each side – the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
will determine which), and is a pseudovector. This has consequences in computer graphics where it has to be considered when transforming surface normals. A number of quantities in physics behave as pseudovectors rather than polar vectors, including
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
and
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
. In mathematics, in three-dimensions, pseudovectors are equivalent to
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
s, from which the transformation rules of pseudovectors can be derived. More generally in ''n''-dimensional
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
pseudovectors are the elements of the algebra with dimension , written ⋀''n''−1R''n''. The label "pseudo" can be further generalized to
pseudoscalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. Th ...
s and
pseudotensor In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordinat ...
s, both of which gain an extra sign flip under improper rotations compared to a true
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
or
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
.


Physical examples

Physical examples of pseudovectors include
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of the ...
,
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
,
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, and
magnetic dipole moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
. Consider the pseudovector
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
. 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 the ''actual'' 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 understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the plane that inside the loop generates a magnetic field oriented in the ''z'' direction. This system is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
(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 the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
of two polar vectors or the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fir ...
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. \mathbf=(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 the exterior product of the two vectors, which yields a
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
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.


Details

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 abstract
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
). Under the physics definition, a "vector" is required to have
components Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit.   In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assemb ...
that "transform" in a certain way under a
proper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
: 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 of active transformations.) Mathematically, if everything in the universe undergoes a rotation described by a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end ...
''R'', so that a
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a s ...
x is transformed to , then any "vector" v must be similarly transformed to . This important requirement is what distinguishes a ''vector'' (which might be composed of, for example, the ''x''-, ''y''-, and ''z''-components of
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box ''cannot'' be considered the three components of a vector, since rotating the box does not appropriately transform these three components.) (In the language of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
, this requirement is equivalent to defining a ''vector'' to be a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of contravariant 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 the Einstein 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 the dyadic 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 consider
improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
s, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is
inversion through a point In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix ''R'', so that a position vector x is transformed to . If the vector v is a polar vector, it will be transformed to . If it is a pseudovector, it will be transformed to . The transformation rules for polar vectors and pseudovectors can be compactly stated as : \begin \mathbf' & = R\mathbf & & \text \\ \mathbf' & = (\det R)(R\mathbf) & & \text \end where the symbols are as described above, and the rotation matrix ''R'' can be either proper or improper. The symbol det denotes
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.


Behavior under addition, subtraction, scalar multiplication

Suppose v1 and v2 are known pseudovectors, and v3 is defined to be their sum, . If the universe is transformed by a rotation matrix ''R'', then v3 is transformed to : \begin \mathbf' = \mathbf'+\mathbf' & = (\det R)(R\mathbf) + (\det R)(R\mathbf) \\ & = (\det R)(R(\mathbf+\mathbf))=(\det R)(R\mathbf). \end So v3 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, suppose v1 is known to be a polar vector, v2 is known to be a pseudovector, and v3 is defined to be their sum, . If the universe is transformed by an improper rotation matrix ''R'', then v3 is transformed to : \mathbf' = \mathbf'+\mathbf' = (R\mathbf) + (\det R)(R\mathbf) = R(\mathbf+(\det R) \mathbf). 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: : , \mathbf, = , \mathbf+\mathbf, , \text \left, \mathbf'\ = \left, \mathbf'-\mathbf'\. If the magnitude of v3 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 the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong 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. (See
parity violation In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
.)


Behavior under cross products

For a rotation matrix ''R'', either proper or improper, the following mathematical equation is always true: :(R\mathbf)\times(R\mathbf) = (\det R)(R(\mathbf\times\mathbf)), where v1 and v2 are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.) Suppose v1 and v2 are known polar vectors, and v3 is defined to be their cross product, . If the universe is transformed by a rotation matrix ''R'', then v3 is transformed to :\mathbf' = \mathbf' \times \mathbf' = (R\mathbf) \times (R\mathbf) = (\det R)(R(\mathbf \times \mathbf)) = (\det R)(R\mathbf). So v3 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.


Examples

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. 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.)


The right-hand rule

Above, pseudovectors have been discussed using
active transformation Active may refer to: Music * ''Active'' (album), a 1992 album by Casiopea * Active Records, a record label Ships * ''Active'' (ship), several commercial ships by that name * HMS ''Active'', the name of various ships of the British Royal ...
s. An alternate approach, more along the lines of
passive transformation Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive * Passive language, a language from which an interpreter works * Passivity (behavior), the condition of submitting to the influence of on ...
s, is to keep the universe fixed, but switch "
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
" with "left-hand rule" everywhere in math and physics, including in the definition of the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
and the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fir ...
. 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 the parity-violating phenomena such as certain
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consi ...
s.Se
Feynman Lectures, 52-7, "Parity is not conserved!"


Formalization

One way to formalize pseudovectors is as follows: if ''V'' is an ''n''-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
vector space, then a ''pseudovector'' of ''V'' is an element of the (''n'' − 1)-th
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of ''V'': ⋀''n''−1(''V''). The pseudovectors of ''V'' form a vector space with the same dimension as ''V''. This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces. In particular, when ''n'' is
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
, such a pseudovector does not experience a sign flip, and when the characteristic of the underlying
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of ''V'' 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 a volume form or an orientation), there is no natural identification of ⋀''n''−1(''V'') with ''V''. Another way to formalize them is by considering them as elements of a representation space for \text(n). Vectors transform in the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defin ...
of \text(n) with data given by (\mathbb^n, \rho_, \text(n)), so that for any matrix R in \text(n), one has \rho_(R) = R. Pseudovectors transform in a pseudofundamental representation (\mathbb^n, \rho_, \text(n)), with \rho_(R) = \det(R)R. Another way to view this homomorphism for n odd is that in this case \text(n) \cong \text(n)\times \mathbb_2. Then \rho_ is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on \text(n) with the trivial homomorphism on \mathbb_2.


Geometric algebra

In
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
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 the
geometric product In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
, denoted by simply juxtaposing two vectors as in ab. This product is expressed as: : \mathbf = \mathbf +\mathbf \ , where the leading term is the customary vector
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
and the second term is called the wedge 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, a
multivector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (a ...
is a summation of ''k''-fold wedge products of various ''k''-values. A ''k''-fold wedge product also is referred to as a ''k''-blade. In the present context the ''pseudovector'' is one of these combinations. This term is attached to a different multivector depending upon the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
s of the space (that is, the number of
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...
vectors in the space). In three dimensions, the most general 2-blade or
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
can be expressed as the wedge product of two vectors and is a pseudovector. In four dimensions, however, the pseudovectors are trivectors. In four dimensions, such as a
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of th ...
, the pseudovectors are trivectors.
In general, it is a -blade, where ''n'' is the dimension of the space and algebra. An ''n''-dimensional space has ''n'' basis vectors and also ''n'' basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of the ''n'' basis vectors. For instance, in four dimensions where the basis vectors are taken to be , the pseudovectors can be written as: .


Transformations in three dimensions

The transformation properties of the pseudovector in three dimensions has been compared to that of the
vector cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
by Baylis. He says: "The terms ''axial vector'' and ''pseudovector'' 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 and b in three dimensions, the cross product composed from a and b is the vector normal to their plane given by . Given a set of right-handed orthonormal
basis vector In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
s , the cross product is expressed in terms of its components as: :\mathbf \times \mathbf = \left(a^2b^3 - a^3b^2\right) \mathbf _1 + \left(a^3b^1 - a^1b^3\right) \mathbf _2 + \left(a^1b^2 - a^2b^1\right) \mathbf _3 , where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by . In this context of geometric algebra, this
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
is called a pseudovector, and is the ''
Hodge dual In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
'' of the cross product. In three dimensions, a dual may be ''right-handed'' or ''left-handed''; see The ''dual'' of e1 is introduced as , and so forth. That is, the dual of e1 is the subspace perpendicular to e1, namely the subspace spanned by e2 and e3. With this understanding, : \mathbf \wedge \mathbf = \left(a^2b^3 - a^3b^2\right) \mathbf _ + \left(a^3b^1 - a^1b^3\right) \mathbf _ + \left(a^1b^2 - a^2b^1\right) \mathbf _ \ . For details, see '. The cross product and wedge product are related by: :\mathbf \ \wedge \ \mathbf = \mathit i \ \mathbf \ \times \ \mathbf \ , where is called the '' unit pseudoscalar''. It has the property: :\mathit^2 = -1 \ . Using the above relations, it is seen that if the vectors a and b 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 vectors e 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.


Note on usage

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. For example, However, because the cross product does not generalize to other than three dimensions, 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 -blade in an ''n''-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 a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
. 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.


See also

*
Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
*
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
*
Antivector An antivector is an element of grade in an ''n''-dimensional exterior algebra. An antivector is always a blade, and it gets its name from the fact that its components each involve a combination of all except one basis vector, thus being the oppos ...
, a generalization of pseudovector in Clifford algebra *
Orientability In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
— Discussion about non-orientable spaces. *
Tensor density In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is ...


Notes


References

* * *
''Axial vector'' at Encyclopaedia of Mathematics
* * * : The dual of the wedge product is the cross product . {{Authority control Linear algebra Vector calculus Vectors (mathematics and physics)