A multivector is the result of a product defined for elements in a vector space V. A vector space with a linear product operation between elements of the space is called an algebra; examples are matrix algebra and vector algebra. The algebra of multivectors is constructed using the wedge product ∧ and is related to the exterior algebra of differential forms.
The set of multivectors on a vector space V is graded by the number of basis vectors that form a basis multivector. A multivector that can be expressed as a linear combination of products, each product being of p vectors, is called a grade p multivector, or a p-vector. The space of p-vectors forms the pth exterior power, denoted Λp(V). The dimension of the vector space V determines the maximum grade of a multivector.
The product of a p-vector and a k-vector is a (k + p)-vector so the set of linear combinations of all multivectors on V is an associative algebra, which is closed with respect to the wedge product. This algebra, denoted by Λ(V), is called the exterior algebra of V.
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,
The product of p vectors is called a grade p multivector, or a p-vector. 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 p-vectors in an n-dimensional vector space.
Area and volume
The p-vector obtained from the wedge product of p separate vectors in an n-dimensional space has components that define the projected (p − 1)-volumes of the p-parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the p-parallelotope.
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.
Multivectors in R2
Properties of multivectors can be seen by considering the two dimensional vector space V = R2. 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. Notice that because V has dimension two the basis bivector e1 ∧ 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.
Multivectors in R3
More features of multivectors can be seen by considering the three dimensional vector space V = R3. In this case, let the basis vectors be e1, e2, and e3, 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 three-dimensional 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 three-vector in ΛV. Compute the three-vector
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.
In this section, we consider multivectors on a projective space Pn, 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.
Points in a real projective space Pn are defined to be lines through the origin of the vector space Rn+1. For example, the projective plane P2 is the set of lines through the origin of R3. Thus, multivectors defined on Rn+1 can be viewed as multivectors on Pn.
A convenient way to view a multivector on Pn is to examine it in an affine component of Pn, which is the intersection of the lines through the origin of Rn+1 with a selected hyperplane, such as H: xn+1 = 1. Lines through the origin of R3 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 P2
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 = (p1, p2, 1) and q = (q1, q2, 1) defines a plane in R3 that intersects E in the line joining p and q. The multivector p ∧ q defines a parallelogram in R3 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 R3.
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 P3
Three dimensional projective space, P3 consists of all lines through the origin of R4. 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 R4 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 3-space through the origin of R4.
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 3-space 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 R4. 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
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.
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.
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 ei, i = 1, ..., n in Rn. 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,
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.
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. It was D. Hestenes who used geometric algebra to describe the application of Clifford algebras to classical mechanics, This formulation was expanded to geometric calculus by D. Hestenes and G. Sobczyk, 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.
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. A sum of only k-grade components is called a k-vector, or a homogeneous multivector.
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 k-vector, but not necessarily a k-blade. Such an element is a k-blade when it can be expressed as the wedge product of k vectors. A geometric algebra generated by a 4-dimensional Euclidean 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 Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.
- 0-vectors are scalars;
- 1-vectors are vectors;
- 2-vectors are bivectors;
- (n − 1)-vectors are pseudovectors;
- n-vectors 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 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).
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. 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 Λ2V (where V is a finite-dimensional 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 ∈ Λ2V in terms of a basis (ei ∧ ej)1 ≤ i < j ≤ n of Λ2V as
where the Einstein summation convention is being used.
Now define a map G : Λ2V × Λ2V → R by insisting that
where are a set of numbers.
Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.
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