Inmultilinear algebra, amultivector, sometimes calledClifford number ormultor,^[1] is an element of theexterior algebraΛ(V) of avector spaceV. This algebra isgraded,associative andalternating, and consists oflinear combinations ofsimplek-vectors^[2] (also known asdecomposablek-vectors^[3] ork-blades) of the form

${\displaystyle v_1\wedge \cdots \wedge v_k,}$

where${\displaystyle v_1,\dots ,v_k}$ are inV.

Ak-vector is such a linear combination that ishomogeneous of degreek (all terms arek-blades for the samek). Depending on the authors, a "multivector" may be either ak-vector or any element of the exterior algebra (any linear combination ofk-blades with potentially differing values ofk).^[4]

Indifferential geometry, ak-vector is usually a vector in the exterior algebra of thetangent vector space of asmooth manifold; that is, it is an antisymmetrictensor obtained by taking linear combinations of theexterior product ofktangent vectors, for some integerk ≥ 0. Adifferentialk-form is ak-vector in the exterior algebra of thedual of the tangent space, which is also the dual of the exterior algebra of the tangent space.

Fork = 0, 1, 2 and3,k-vectors are often called respectivelyscalars,vectors,bivectors andtrivectors; they are respectively dual to0-forms, 1-forms, 2-forms and 3-forms.^[5][6]

The exterior product (also called the wedge product) used to construct multivectors is multilinear (linear in each input), associative and alternating. This means for vectorsu,v andw in a vector spaceV and for scalarsα,β, the exterior product has the properties:

• Linear in an input:${\displaystyle \mathbf{u}\wedge (\alpha \mathbf{v}+\beta \mathbf{w})=\alpha \mathbf{u}\wedge \mathbf{v}+\beta \mathbf{u}\wedge \mathbf{w};}$
• Associative:${\displaystyle (\mathbf{u}\wedge \mathbf{v})\wedge \mathbf{w}=\mathbf{u}\wedge (\mathbf{v}\wedge \mathbf{w});}$
• Alternating:${\displaystyle \mathbf{u}\wedge \mathbf{u}=0.}$

The exterior product ofk vectors or a sum of such products (for a singlek) is called a gradek multivector, or ak-vector. The maximum grade of a multivector is the dimension of the vector spaceV.

Linearity in either input together with the alternating property implies linearity in the other input. The multilinearity of the exterior product allows a multivector to be expressed as a linear combination of exterior products of basis vectors ofV. The exterior product ofk basis vectors ofV is the standard way of constructing each basis element for the space ofk-vectors, which has dimension(ⁿ
_k) in the exterior algebra of ann-dimensional vector space.^[2]

Thek-vector obtained from the exterior product ofk separate vectors in ann-dimensional space has components that define the projected(k − 1)-volumes of thek-parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of thek-parallelotope.^[2][7]

The following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures the area of a parallelogram. Similarly, a three-vector in three dimensions measures the volume of a parallelepiped.

It is easy to check that the magnitude of a three-vector in four dimensions measures the volume of the parallelepiped spanned by these vectors.

Properties of multivectors can be seen by considering the two-dimensional vector spaceV =R². Let the basis vectors bee₁ ande₂, sou andv are given by

${\displaystyle \mathbf{u}=u_1{\mathbf{e}}_1+u_2{\mathbf{e}}_2,\mathbf{v}=v_1{\mathbf{e}}_1+v_2{\mathbf{e}}_2,}$

and the multivectoru ∧v, also called a bivector, is computed to be

The vertical bars denote the determinant of the matrix, which is the area of the parallelogram spanned by the vectorsu andv. The magnitude ofu ∧v is the area of this parallelogram. Notice that becauseV has dimension two the basis bivectore₁ ∧e₂ is the only multivector in ΛV.

The relationship between the magnitude of a multivector and the area or volume spanned by the vectors is an important feature in all dimensions. Furthermore, the linear functional version of a multivector that computes this volume is known as a differential form.

More features of multivectors can be seen by considering the three-dimensional vector spaceV =R³. In this case, let the basis vectors bee₁,e₂, ande₃, sou,v andw are given by

and the bivectoru ∧v is computed to be

The components of this bivector are the same as the components of the cross product. The magnitude of this bivector is the square root of the sum of the squares of its components.

This shows that the magnitude of the bivectoru ∧v is the area of the parallelogram spanned by the vectorsu andv as it lies in the three-dimensional spaceV. The components of the bivector are the projected areas of the parallelogram on each of the three coordinate planes.

Notice that becauseV has dimension three, there is one basis three-vector in ΛV. Compute the three-vector

This shows that the magnitude of the three-vectoru ∧v ∧w is the volume of the parallelepiped spanned by the three vectorsu,v andw.

In higher-dimensional spaces, the component three-vectors are projections of the volume of a parallelepiped onto the coordinate three-spaces, and the magnitude of the three-vector is the volume of the parallelepiped as it sits in the higher-dimensional space.

In this section, we consider multivectors on aprojective spacePⁿ, which provide a convenient set of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, calledGrassmann coordinates.^[8]

Points in a real projective spacePⁿ are defined to be lines through the origin of the vector spaceRⁿ⁺¹. For example, the projective planeP² is the set of lines through the origin ofR³. Thus, multivectors defined onRⁿ⁺¹ can be viewed as multivectors onPⁿ.

A convenient way to view a multivector onPⁿ is to examine it in anaffine component ofPⁿ, which is the intersection of the lines through the origin ofRⁿ⁺¹ with a selected hyperplane, such asH:x_n+1 = 1. Lines through the origin ofR³ intersect the planeE:z = 1 to define an affine version of the projective plane that only lacks the points for whichz = 0, called the points at infinity.

Points in the affine componentE:z = 1 of the projective planeP² have coordinatesx = (x,y, 1). A linear combination of two pointsp = (p₁,p₂, 1) andq = (q₁,q₂, 1) defines a plane inR³ that intersects E in the line joiningp andq. The multivectorp ∧q defines a parallelogram inR³ given by

${\displaystyle \mathbf{p}\wedge \mathbf{q}=(p_2-q_2)({\mathbf{e}}_2\wedge {\mathbf{e}}_3)+(p_1-q_1)({\mathbf{e}}_1\wedge {\mathbf{e}}_3)+(p_1{q}_2-q_1{p}_2)({\mathbf{e}}_1\wedge {\mathbf{e}}_2).}$

Notice that substitution ofαp +βq forp multiplies this multivector by a constant. Therefore, the components ofp ∧q are homogeneous coordinates for the plane through the origin ofR³.

The set of pointsx = (x,y, 1) on the line throughp andq is the intersection of the plane defined byp ∧q with the planeE:z = 1. These points satisfyx ∧p ∧q = 0, that is,

${\displaystyle \mathbf{x}\wedge \mathbf{p}\wedge \mathbf{q}=(x{\mathbf{e}}_1+y{\mathbf{e}}_2+{\mathbf{e}}_3)\wedge ((p_2-q_2)({\mathbf{e}}_2\wedge {\mathbf{e}}_3)+(p_1-q_1)({\mathbf{e}}_1\wedge {\mathbf{e}}_3)+(p_1{q}_2-q_1{p}_2)({\mathbf{e}}_1\wedge {\mathbf{e}}_2))=0,}$

which simplifies to the equation of a line

${\displaystyle \lambda :x(p_2-q_2)+y(p_1-q_1)+(p_1{q}_2-q_1{p}_2)=0.}$

This equation is satisfied by pointsx =αp +βq for real values of α and β.

The three components ofp ∧q that define the lineλ are called theGrassmann coordinates of the line. Because three homogeneous coordinates define both a point and a line, the geometry of points is said to be dual to the geometry of lines in the projective plane. This is called theprinciple of duality.

Three-dimensional projective spaceP³ consists of all lines through the origin ofR⁴. Let the three-dimensional hyperplane,H:w = 1, be the affine component of projective space defined by the pointsx = (x,y,z, 1). The multivectorp ∧q ∧r defines a parallelepiped inR⁴ given by

Notice that substitution ofαp + βq + γr forp multiplies this multivector by a constant. Therefore, the components ofp ∧q ∧r are homogeneous coordinates for the 3-space through the origin ofR⁴.

A plane in the affine componentH:w = 1 is the set of pointsx = (x,y,z, 1) in the intersection of H with the 3-space defined byp ∧q ∧r. These points satisfyx ∧p ∧q ∧r = 0, that is,

${\displaystyle \mathbf{x}\wedge \mathbf{p}\wedge \mathbf{q}\wedge \mathbf{r}=(x{\mathbf{e}}_1+y{\mathbf{e}}_2+z{\mathbf{e}}_3+{\mathbf{e}}_4)\wedge \mathbf{p}\wedge \mathbf{q}\wedge \mathbf{r}=0,}$

which simplifies to the equation of a plane

This equation is satisfied by pointsx =αp +βq +γr for real values ofα,β andγ.

The four components ofp ∧q ∧r that define the planeλ are called theGrassmann coordinates of the plane. Because four homogeneous coordinates define both a point and a plane in projective space, the geometry of points is dual to the geometry of planes.

A line as the join of two points: In projective space the lineλ through two pointsp andq can be viewed as the intersection of the affine spaceH:w = 1 with the planex =αp +βq inR⁴. The multivectorp ∧q provides homogeneous coordinates for the line

These are known as thePlücker coordinates of the line, though they are also an example of Grassmann coordinates.

A line as the intersection of two planes: A lineμ in projective space can also be defined as the set of pointsx that form the intersection of two planesπ andρ defined by grade three multivectors, so the pointsx are the solutions to the linear equations

${\displaystyle \mu :\mathbf{x}\wedge \pi =0,\mathbf{x}\wedge \rho =0.}$

In order to obtain the Plucker coordinates of the lineμ, map the multivectorsπ andρ to their dual point coordinates using the right complement, denoted by an overline, as in^[9]

${\displaystyle {\mathbf{e}}_1=\overline{{\mathbf{e}}_2\wedge {\mathbf{e}}_3\wedge {\mathbf{e}}_4},{\mathbf{e}}_2=\overline{{\mathbf{e}}_3\wedge {\mathbf{e}}_1\wedge {\mathbf{e}}_4},{\mathbf{e}}_3=\overline{{\mathbf{e}}_1\wedge {\mathbf{e}}_2\wedge {\mathbf{e}}_4},{\mathbf{e}}_4=\overline{{\mathbf{e}}_1\wedge {\mathbf{e}}_2\wedge {\mathbf{e}}_3},}$

then

${\displaystyle \overline{\pi }={\pi }_1{\mathbf{e}}_1+{\pi }_2{\mathbf{e}}_2+{\pi }_3{\mathbf{e}}_3+{\pi }_4{\mathbf{e}}_4,\overline{\rho }={\rho }_1{\mathbf{e}}_1+{\rho }_2{\mathbf{e}}_2+{\rho }_3{\mathbf{e}}_3+{\rho }_4{\mathbf{e}}_4.}$

So, the Plücker coordinates of the lineμ are given by

where the underline denotes the left complement. The left complement of the wedge product of right complements is called the antiwedge product, denoted by a downward pointing wedge, allowing us to write${\displaystyle \mu =\pi \vee \rho .}$

W. K. Clifford combined multivectors with theinner product defined on the vector space, in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and Hamilton'squaternions.^[10][11]

The Clifford product between two vectorsu andv is bilinear and associative like the exterior product, and has the additional property that the multivectoruv is coupled to the inner productu ⋅v by Clifford's relation,

${\displaystyle \mathbf{u}\mathbf{v}+\mathbf{v}\mathbf{u}=2\mathbf{u}\cdot \mathbf{v}.}$

Clifford's relation retains the anticommuting property for vectors that are perpendicular. This can be seen from the mutually orthogonal unit vectorse_i,i = 1, ...,n inRⁿ: Clifford's relation yields

${\displaystyle {\mathbf{e}}_i{\mathbf{e}}_j+{\mathbf{e}}_j{\mathbf{e}}_i=2{\mathbf{e}}_i\cdot {\mathbf{e}}_j={\delta }_{i,j},}$

which shows that the basis vectors mutually anticommute,

${\displaystyle {\mathbf{e}}_i{\mathbf{e}}_j=-{\mathbf{e}}_j{\mathbf{e}}_i,i\ne j=1,\dots ,n.}$

In contrast to the exterior product, the Clifford product of a vector with itself is not zero. To see this, compute the product

${\displaystyle {\mathbf{e}}_i{\mathbf{e}}_i+{\mathbf{e}}_i{\mathbf{e}}_i=2{\mathbf{e}}_i\cdot {\mathbf{e}}_i=2,}$

which yields

${\displaystyle {\mathbf{e}}_i{\mathbf{e}}_i=1,i=1,\dots ,n.}$

The set of multivectors constructed using Clifford's product yields an associative algebra known as aClifford algebra. Inner products with different properties can be used to construct different Clifford algebras.^[12][13]

The termk-blade was used inClifford Algebra to Geometric Calculus (1984)^[14]

Multivectors play a central role in the mathematical formulation of physics known as geometric algebra. According toDavid Hestenes,

[Non-scalar]k-vectors are sometimes calledk-blades or, merelyblades, to emphasize the fact that, in contrast to 0-vectors (scalars), they have "directional properties".^[15]

In 2003 the termblade for a multivector that can be written as the exterior product of [a scalar and] a set of vectors was used by C. Doran and A. Lasenby. Here, by the statement "Any multivector can be expressed as the sum of blades", scalars are implicitly defined as 0-blades.^[16]

Ingeometric algebra, a multivector is defined to be the sum of different-gradek-blades, such as the summation of ascalar, avector, and a 2-vector.^[17] A sum of onlyk-grade components is called ak-vector,^[18] or ahomogeneous multivector.^[19]

The highest grade element in a space is called apseudoscalar.

If a given element is homogeneous of a gradek, then it is ak-vector, but not necessarily ak-blade. Such an element is ak-blade when it can be expressed as the exterior product ofk vectors. A geometric algebra generated by a four-dimensional vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of graden elements in a real exterior algebra forn = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product ofn vectors can be visualized as anyn-dimensional shape (e.g.n-parallelotope,n-ellipsoid); with magnitude (hypervolume), andorientation defined by that on its(n − 1)-dimensional boundary and on which side the interior is.^[20][21]

• 0-vectors are scalars;
• 1-vectors are vectors;
• 2-vectors arebivectors;
• (n − 1)-vectors arepseudovectors;
• n-vectors arepseudoscalars.

In the presence of avolume form (such as given aninner product and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine invector calculus, but without a volume form this cannot be done without making an arbitrary choice.

In thealgebra of physical space (the geometric algebra of Euclidean 3-space, used as a model of (3+1)-spacetime), a sum of a scalar and a vector is called aparavector, and represents a point in spacetime (the vector the space, the scalar the time).

Abivector is an element of theantisymmetrictensor product of atangent space with itself.

Ingeometric algebra, also, abivector is a grade 2 element (a 2-vector) resulting from thewedge product of two vectors, and so it is geometrically anoriented area, in the same way avector is an oriented line segment. Ifa andb are two vectors, the bivectora ∧b has

• anorm which is its area, given by

  ${\displaystyle \Vert \mathbf{a}\wedge \mathbf{b}\Vert =\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \sin ({\varphi }_{a,b})}$

• a direction: the plane where that area lies on, i.e., the plane determined bya andb, as long as they are linearly independent;
• an orientation (out of two), determined by the order in which the originating vectors are multiplied.

Bivectors are connected topseudovectors, and are used to represent rotations in geometric algebra.

As bivectors are elements of a vector space Λ²V (whereV is a finite-dimensional vector space withdimV =n), it makes sense to define aninner product on this vector space as follows. First, write any elementF ∈ Λ²V in terms of a basis(e_i ∧e_j)_{1 ≤i <j ≤n} of Λ²V as

where theEinstein summation convention is being used.

Now define a mapG : Λ²V × Λ²V →R by insisting that

${\displaystyle G(F,H):=G_{abcd}{F}^{ab}{H}^{cd},}$

where${\displaystyle G_{abcd}}$ are a set of numbers.

Bivectors play many important roles in physics, for example, in theclassification of electromagnetic fields.

• Blade (geometry)
• Paravector

• Multilinear algebra
• Tensors
• Differential geometry
• Geometric algebra

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