Multivector
In multilinear algebra, a multivector, sometimes called Clifford number,^{[1]} is an element of the exterior algebra Λ(V) of a vector space V. This algebra is graded, associative and alternating, and consists of linear combinations of simple kvectors (also known as decomposable kvectors or kblades) of the form
where are in V.
"Multivector" may mean either homogeneous elements (all terms of the linear combination have the same grade or degree k, that is are the product of the same number k of vectors), which are referred to as kvectors or pvectors,^{[2]} or may allow sums of terms in different degrees.
In differential geometry, a pvector is the antisymmetric tensor obtained by taking linear combinations of the wedge product of p tangent vectors, for some integer p ≥ 0. It is the dual concept to a pform.
For p = 0, 1, 2 and 3, these are often called respectively scalars, vectors, bivectors and trivectors; they are respectively dual to 0forms, 1forms, 2forms and 3forms.^{[3]}^{[4]}
Contents
Wedge product
The wedge product operation used to construct multivectors is linear, associative and alternating, which reflect the properties of the determinant. This means for vectors u, v and w in a vector space V and for scalars α, β, the wedge product has the properties,
 Linear:
 Associative:
 Alternating:
The product of p vectors is called a grade p multivector, or a pvector. The maximum grade of a multivector is the dimension of the vector space V.
The linearity of the wedge product allows a multivector to be defined as the linear combination of basis multivectors. There are (^{n}
_{p}) basis pvectors in an ndimensional vector space.^{[5]}
Area and volume
The pvector obtained from the wedge product of p separate vectors in an ndimensional space has components that define the projected (p − 1)volumes of the pparallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the pparallelotope.^{[5]}^{[6]}
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 threevector in three dimensions measures the volume of a parallelepiped.
It is easy to check that the magnitude of a threevector in four dimensions measures the volume of the parallelepiped spanned by these vectors.
Multivectors in R^{2}
Properties of multivectors can be seen by considering the two dimensional vector space V = R^{2}. Let the basis vectors be e_{1} and e_{2}, so u and v are given by
and the multivector u ∧ 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 vectors u and v. The magnitude of u ∧ v is the area of this parallelogram. Notice that because V has dimension two the basis bivector e_{1} ∧ e_{2} 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.
Multivectors in R^{3}
More features of multivectors can be seen by considering the three dimensional vector space V = R^{3}. In this case, let the basis vectors be e_{1}, e_{2}, and e_{3}, so u, v and w are given by
and the bivector u ∧ 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 bivector u ∧ v is the area of the parallelogram spanned by the vectors u and v as it lies in the threedimensional space V. The components of the bivector are the projected areas of the parallelogram on each of the three coordinate planes.
Notice that because V has dimension three, there is one basis threevector in ΛV. Compute the threevector
This shows that the magnitude of the threevector u ∧ v ∧ w is the volume of the parallelepiped spanned by the three vectors u, v and w.
In higherdimensional spaces, the component threevectors are projections of the volume of a parallelepiped onto the coordinate threespaces, and the magnitude of the threevector is the volume of the parallelepiped as it sits in the higherdimensional space.
Grassmann coordinates
In this section, we consider multivectors on a projective space P^{n}, which provide a convenient set of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, called Grassmann coordinates.^{[7]}
Points in a real projective space P^{n} are defined to be lines through the origin of the vector space R^{n+1}. For example, the projective plane P^{2} is the set of lines through the origin of R^{3}. Thus, multivectors defined on R^{n+1} can be viewed as multivectors on P^{n}.
A convenient way to view a multivector on P^{n} is to examine it in an affine component of P^{n}, which is the intersection of the lines through the origin of R^{n+1} with a selected hyperplane, such as H: x_{n+1} = 1. Lines through the origin of R^{3} intersect the plane E: z = 1 to define an affine version of the projective plane that only lacks the points z = 0, called the points at infinity.
Multivectors on P^{2}
Points in the affine component E: z = 1 of the projective plane have coordinates x = (x, y, 1). A linear combination of two points p = (p_{1}, p_{2}, 1) and q = (q_{1}, q_{2}, 1) defines a plane in R^{3} that intersects E in the line joining p and q. The multivector p ∧ q defines a parallelogram in R^{3} given by
Notice that substitution of αp + βq for p multiplies this multivector by a constant. Therefore, the components of p ∧ q are homogeneous coordinates for the plane through the origin of R^{3}.
The set of points x = (x, y, 1) on the line through p and q is the intersection of the plane defined by p ∧ q with the plane E: z = 1. These points satisfy x ∧ p ∧ q = 0, that is,
which simplifies to the equation of a line
This equation is satisfied by points x = αp + βq for real values of α and β.
The three components of p ∧ q that define the line λ are called the Grassmann 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 the principle of duality.
Multivectors on P^{3}
Three dimensional projective space, P^{3} consists of all lines through the origin of R^{4}. Let the three dimensional hyperplane, H: w = 1, be the affine component of projective space defined by the points x = (x, y, z, 1). The multivector p ∧ q ∧ r defines a parallelepiped in R^{4} given by
Notice that substitution of αp + βq + γr for p multiplies this multivector by a constant. Therefore, the components of p ∧ q ∧ r are homogeneous coordinates for the 3space through the origin of R^{4}.
A plane in the affine component H: w = 1 is the set of points x = (x, y, z, 1) in the intersection of H with the 3space defined by p ∧ q ∧ r. These points satisfy x ∧ p ∧ q ∧ r = 0, that is,
which simplifies to the equation of a plane
This equation is satisfied by points x = αp + βq + γr for real values of α, β and γ.
The four components of p ∧ q ∧ r that define the plane λ are called the Grassmann 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 line λ through two points p and q can be viewed as the intersection of the affine space H: w = 1 with the plane x = αp + βq in R^{4}. The multivector p ∧ q provides homogeneous coordinates for the line
These are known as the Plücker coordinates of the line, though they are also an example of Grassmann coordinates.
A line as the intersection of two planes: A line μ in projective space can also be defined as the set of points x that form the intersection of two planes π and ρ defined by grade three multivectors, so the points x are the solutions to the linear equations
In order to obtain the Plucker coordinates of the line μ, map the multivectors π and ρ to their dual point coordinates using the Hodge star operator,^{[5]}
then
So, the Plücker coordinates of the line μ are given by
Because the six homogeneous coordinates of a line can be obtained from the join of two points or the intersection of two planes, the line is said to be self dual in projective space.
Clifford product
W. K. Clifford combined multivectors with the inner product defined on the vector space, in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and Hamilton's quaternions.^{[8]}^{[9]}
The Clifford product between two vectors u and v is linear and associative like the wedge product, and has the additional property that the multivector uv is coupled to the inner product u · v by Clifford's relation,
Clifford's relation preserves the alternating property for the product of vectors that are perpendicular. This can be seen for the orthogonal unit vectors e_{i}, i = 1, ..., n in R^{n}. Clifford's relation yields
therefore the basis vectors are alternating,
In contrast to the wedge product, the Clifford product of a vector with itself is no longer zero. To see this compute the product,
which yields
The set of multivectors constructed using Clifford's product yields an associative algebra known as a Clifford algebra. Inner products with different properties can be used to construct different Clifford algebras.^{[10]}^{[11]}
Geometric algebra
Multivectors play a central role in the mathematical formulation of physics known as geometric algebra. The term geometric algebra was used by E. Artin for matrix methods in projective geometry.^{[12]} It was D. Hestenes who used geometric algebra to describe the application of Clifford algebras to classical mechanics,^{[13]} This formulation was expanded to geometric calculus by D. Hestenes and G. Sobczyk,^{[14]} who provided new terminology for a variety of features in this application of Clifford algebra to physics. C. Doran and A. Lasenby show that Hestene's geometric algebra provides a convenient formulation for modern physics.^{[15]}
In geometric algebra, a multivector is defined to be the sum of differentgrade kblades, such as the summation of a scalar, a vector, and a 2vector.^{[16]} A sum of only kgrade components is called a kvector,^{[17]} or a homogeneous multivector.^{[18]}
The highest grade element in a space is called a pseudoscalar.
If a given element is homogeneous of a grade k, then it is a kvector, but not necessarily a kblade. Such an element is a kblade when it can be expressed as the wedge product of k vectors. A geometric algebra generated by a 4dimensional Euclidean vector space illustrates the point with an example: The sum of any two blades with one taken from the XYplane and the other taken from the ZWplane will form a 2vector that is not a 2blade. In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2blades may be written as a single 2blade.
Examples
 0vectors are scalars;
 1vectors are vectors;
 2vectors are bivectors;
 (n − 1)vectors are pseudovectors;
 nvectors are pseudoscalars.
In the presence of a volume form (such as given an inner product and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine in vector calculus, but without a volume form this cannot be done without a choice.
In the algebra of physical space (the geometric algebra of Euclidean 3space, used as a model of (3+1)spacetime), a sum of a scalar and a vector is called a paravector, and represents a point in spacetime (the vector the space, the scalar the time).
Bivectors
A bivector is therefore an element of the antisymmetric tensor product of a tangent space with itself.
In geometric algebra, also, a bivector is a grade 2 element (a 2vector) resulting from the wedge product of two vectors, and so it is geometrically an oriented area, in the same way a vector is an oriented line segment. If a and b are two vectors, the bivector a ∧ b has
 a norm which is its area, given by
 a direction: the plane where that area lies on, i.e., the plane determined by a and b, 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 to pseudovectors, and are used to represent rotations in geometric algebra.
As bivectors are elements of a vector space Λ^{2}V (where V is a finitedimensional vector space with dim V = n), it makes sense to define an inner product on this vector space as follows. First, write any element F ∈ Λ^{2}V in terms of a basis (e_{i} ∧ e_{j})_{1 ≤ i < j ≤ n} of Λ^{2}V as
where the Einstein summation convention is being used.
Now define a map G : Λ^{2}V × Λ^{2}V → R by insisting that
where are a set of numbers.
Applications
Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.
See also
References
 ^ John Snygg (2012), A New Approach to Differential Geometry Using Clifford’s Geometric Algebra, Birkhäuser, p.5 §2.12
 ^ Élie Cartan, The theory of spinors, p. 16, considers only homogeneous vectors, particularly simple ones, referring to them as "multivectors" (collectively) or pvectors (specifically).

^
William M Pezzaglia Jr. (1992). "Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations". In Julian Ławrynowicz. Deformations of mathematical structures II. Springer. p. 131 ff. ISBN 0792325761.
Hence in 3D we associate the alternate terms of pseudovector for bivector, and pseudoscalar for the trivector
 ^ Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 081763715X.
 ^ ^{a} ^{b} ^{c} H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, NY, 1963
 ^ G. E. Shilov, Linear Algebra, (trans. R. A. Silverman), Dover Publications, 1977.
 ^ W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge Univ. Press, 1947
 ^ W. K. Clifford, "Preliminary sketch of biquaternions," Proc. London Math. Soc. Vol. 4 (1873) pp. 381395
 ^ W. K. Clifford, Mathematical Papers, (ed. R. Tucker), London: Macmillan, 1882.
 ^ J. M. McCarthy, An Introduction to Theoretical Kinematics, pp. 62–5, MIT Press 1990.
 ^ O. Bottema and B. Roth, Theoretical Kinematics, North Holland Publ. Co., 1979
 ^ E. Artin, Geometric algebra, Interscience Publ., 1957
 ^ D. Hestenes, New Foundations for Classical Mechanics, Kluwer Academic Publishers, 1986.
 ^ D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Springer Verlag, 1987
 ^ C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge Univ. Press, 2007.
 ^ Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline". Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 9810242786.
 ^ R Wareham, J Cameron & J Lasenby (2005). "Applications of conformal geometric algebra in computer vision and graphics". In Hongbo Li; Peter J. Olver; Gerald Sommer. Computer algebra and geometric algebra with applications. Springer. p. 330. ISBN 3540262962.
 ^ Eduardo BayroCorrochano (2004). "Clifford geometric algebra: A promising framework for computer vision, robotics and learning". In Alberto Sanfeliu; José Francisco Martínez Trinidad; Jesús Ariel Carrasco Ochoa. Progress in pattern recognition, image analysis and applications. Springer. p. 25. ISBN 3540235272.
 ^ R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0679776311.
 ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 83. ISBN 0716703440.