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In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
and
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a pseudovector (or axial vector) is a quantity that is defined as a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
of some
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
or other
geometric shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...
s, that resembles a vector, and behaves like a vector in many situations, but is changed into its opposite if the
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
of the space is changed, or an improper rigid transformation such as a
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
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 Optics is the branch of physics Physics is ...

mirror image
, 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 or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
of a polar
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
at a point and the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
of two polar vectors are pseudovectors. One example of a pseudovector is the normal to an oriented
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

right-hand rule
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 In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

magnetic field
and
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

angular velocity
. In mathematics, in three-dimensions, pseudovectors are equivalent to
bivector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

bivector
s, from which the transformation rules of pseudovectors can be derived. More generally in ''n''-dimensional
geometric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
pseudovectors are the elements of the algebra with dimension , written ⋀''n''−1 R''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. ...
s and
pseudotensor In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
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 as ...
or
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

tensor
.


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, moment of force, rotational force or turning effect, depending on the field of study. The concept originated with the studies ...

torque
,
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

angular velocity
,
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

angular momentum
,
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

magnetic field
, and
magnetic dipole moment The magnetic moment is the magnetic strength and orientation of a magnet A magnet is a material or object that produces a magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assi ...

magnetic dipole moment
. Consider the pseudovector
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

angular momentum
. 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 Greek συμμετρία ''symmetria'' "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 pre ...

symmetric
(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 , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
of two polar vectors or the
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
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 In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
of the two vectors, which yields a
bivector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

bivector
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
). Under the physics definition, a "vector" is required to have
components Component may refer to: In engineering, science, and technology Generic systems *System A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounde ...
that "transform" in a certain way under a proper 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 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 (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \th ...
''R'', so that a
displacement vector In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
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 The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

velocity
) 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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, this requirement is equivalent to defining a ''vector'' to be a
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

tensor
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is inversion through a point 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
; 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 Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ...
: 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 quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ph ...
.)


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 In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτ ...
s. An alternate approach, more along the lines of
passive transformation In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτ ...
s, is to keep the universe fixed, but switch "
right-hand rule In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

right-hand rule
" with "left-hand rule" everywhere in math and physics, including in the definition of the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
. 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 ...

radioactive decay
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 File:Dimension levels.svg, thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum numb ...
vector space, then a ''pseudovector'' of ''V'' is an element of the (''n'' − 1)-th
exterior power In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
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)''Even'' is a Norwegian given name coming from Old Norse ''Eivindr'' (existing as ''Eivindur'' in Iceland). Another common name derived from Old Norse ''Eivindr'' is the Norwegianized ''Eivind''. ''Ei ...
, such a pseudovector does not experience a sign flip, and when the
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
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 grassl ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
or an
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
), there is no natural identification of ⋀''n''−1(''V'') with ''V''.


Geometric algebra

In
geometric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
and the second term is called the
wedge product A wedge is a triangular A triangle is a polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branche ...

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 algebra, graded, associative algebra, associative and alternating algebra, alternating, and ...
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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
s of the space (that is, the number of
linearly independent In the theory of vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are con ...
vectors in the space). In three dimensions, the most general 2-blade or
bivector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

bivector
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 Cℓ4(C), which may be thought of as Cℓ1,3(C). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
or wedge product, denoted by . In this context of geometric algebra, this
bivector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

bivector
is called a pseudovector, and is the ''
Hodge dual In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
'' 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. 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

*
Grassmann algebra In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, ...
*
Clifford algebra In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
* Antivector, a generalization of pseudovector in Clifford algebra *Orientation (mathematics) — Description of oriented spaces, necessary for pseudovectors. *Orientability — Discussion about non-orientable spaces. *Tensor density


Notes


References

* George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'' (Harcourt: San Diego, 2001). () * Chris Doran and Anthony Lasenby, ''Geometric Algebra for Physicists'' (Cambridge University Press: Cambridge, 2007) () * Richard Feynman, ''Feynman Lectures on Physics'', Vol. 1 Chap. 52
See §52-5: Polar and axial vectors, pp. 52-6

''Axial vector'' at Encyclopaedia of Mathematics
* John David Jackson (physicist), J. D. Jackson, ''Classical Electrodynamics'' (Wiley: New York, 1999). () * Susan M. Lea, "Mathematics for Physicists" (Thompson: Belmont, 2004) () * : The dual of the wedge product is the cross product . {{Authority control Linear algebra Vector calculus Vectors (mathematics and physics)