In multilinear algebra, a multivector, sometimes called Clifford number or multor,[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 k-vectors[2] (also known as decomposable k-vectors[3] or k-blades) of the form where
are in V. A k-vector is such a linear combination that is homogeneous of degree k (all terms are k-blades for the same k).
Depending on the authors, a "multivector" may be either a k-vector or any element of the exterior algebra (any linear combination of k-blades with potentially differing values of k).
A differential k-form is a k-vector in the exterior algebra of the dual of the tangent space, which is also the dual of the exterior algebra of the tangent space.
For k = 0, 1, 2 and 3, k-vectors are often called respectively scalars, vectors, bivectors and trivectors; they are respectively dual to 0-forms, 1-forms, 2-forms and 3-forms.
[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.
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.
Let the basis vectors be e1 and e2, 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.
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.
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 three-dimensional space V. The components of the bivector are the projected areas of the parallelogram on each of the three coordinate planes.
This shows that the magnitude of the three-vector u ∧ v ∧ w is the volume of the parallelepiped spanned by the three vectors u, v and w. 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.
A linear combination of two points p = (p1, p2, 1) and q = (q1, q2, 1) defines a plane in R3 that intersects E in the line joining p and q.
Let the three-dimensional hyperplane, H: w = 1, be the affine component of projective space defined by the points x = (x, y, z, 1).
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 R4.
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 right complement, denoted by an overline, as in[9] then So, the Plücker coordinates of the line μ are given by where the underline denotes the left complement.
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.
[12][13] The term k-blade was used in Clifford Algebra to Geometric Calculus (1984)[14] Multivectors play a central role in the mathematical formulation of physics known as geometric algebra.
According to David Hestenes, In 2003 the term blade 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] In geometric algebra, a multivector is defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector.
[17] A sum of only k-grade components is called a k-vector,[18] or a homogeneous multivector.
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.
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 making an arbitrary choice.
In the algebra 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 a paravector, and represents a point in spacetime (the vector the space, the scalar the time).
A bivector is 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 2-vector) 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.
First, write any element F ∈ Λ2V in terms of a basis (ei ∧ ej)1 ≤ i < j ≤ n of Λ2V as where the Einstein summation convention is being used.
Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.