Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing directed line segments. The bivector a ∧ b has an attitude (or direction) of the plane spanned by a and b, has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, it has a magnitude equal to the area of the parallelogram with edges a and b), and has an orientation being the side of a on which b lies within the plane spanned by a and b. In layman terms, any surface defines the same bivector if it is parallel to the same plane (same attitude), has the same area, and same orientation (see figure). Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product a ∧ b is a bivector, as is any sum of bivectors. Not all bivectors can be expressed as an exterior product without such summation. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product of two vectors is alternating, so a ∧ a is the zero bivector, and b ∧ a = −a ∧ b, producing the opposite orientation. Concepts directly related to bivectors are rank-2 antisymmetric tensor and skew-symmetric matrix.

History The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors. Just the previous year, in Ireland, William Rowan Hamilton had discovered quaternions. Hamilton coined both vector and bivector, the latter in his Lectures on Quaternions (1853) as he introduced biquaternions, which have bivectors for their vector parts. It was not until English mathematician William Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector of this article arose. Henry Forder used the term bivector to develop exterior algebra in 1941. In the 1890s Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication. The success of vector calculus, and of the book Vector Analysis by Gibbs and Wilson, had the effect that the insights of Hamilton and Clifford were overlooked for a long time, since much of 20th century mathematics and physics was formulated in vector terms. Gibbs used vectors to fill the role of bivectors in three dimensions, and used bivector in Hamilton's sense, a use that has sometimes been copied. Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a quadratic form. Its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics.

Derivation For this article, the bivector will be considered only in real geometric algebras, which may be applied in most areas of physics. Also unless otherwise stated, all examples have a Euclidean metric and so a positive-definite quadratic form.

Geometric algebra and the geometric product The bivector arises from the definition of the geometric product over a vector space with an associated quadratic form sometimes called the metric. For vectors a, b and c, the geometric product satisfies the following properties:

Associativity

( a b ) c = a ( b c ) {\displaystyle (\mathbf {ab} )\mathbf {c} =\mathbf {a} (\mathbf {bc} )}

Left and right distributivity

a ( b + c ) = a b + a c ( b + c ) a = b a + c a {\displaystyle {\begin{aligned}\mathbf {a} (\mathbf {b} +\mathbf {c} )&=\mathbf {ab} +\mathbf {ac} \\(\mathbf {b} +\mathbf {c} )\mathbf {a} &=\mathbf {ba} +\mathbf {ca} \end{aligned}}}

Scalar square ⁠ a 2 = Q ( a ) {\displaystyle \mathbf {a} ^{2}=Q(\mathbf {a} )} ⁠, where Q is the quadratic form, which need not be positive-definite.

Scalar product From associativity, a(ab) = a2b, is a scalar times b. When b is not parallel to and hence not a scalar multiple of a, ab cannot be a scalar. But

1 2 ( a b + b a ) = 1 2 ( ( a + b ) 2 − a 2 − b 2 ) {\displaystyle {\tfrac {1}{2}}(\mathbf {ab} +\mathbf {ba} )={\tfrac {1}{2}}\left((\mathbf {a} +\mathbf {b} )^{2}-\mathbf {a} ^{2}-\mathbf {b} ^{2}\right)}

is a sum of scalars and so a scalar. From the law of cosines on the triangle formed by the vectors its value is |a| |b| cos θ, where θ is the angle between the vectors. It is therefore identical to the scalar product between two vectors, and is written the same way,

a ⋅ b = 1 2 ( a b + b a ) . {\displaystyle \mathbf {a} \cdot \mathbf {b} ={\tfrac {1}{2}}(\mathbf {ab} +\mathbf {ba} ).}

It is symmetric, scalar-valued, and can be used to determine the angle between two vectors: in particular if a and b are orthogonal the product is zero.

Exterior product Just as the scalar product can be formulated as the symmetric part of the geometric product of another quantity, the exterior product (sometimes known as the "wedge" or "progressive" product) can be formulated as its antisymmetric part:

a ∧ b = 1 2 ( a b − b a ) {\displaystyle \mathbf {a} \wedge \mathbf {b} ={\tfrac {1}{2}}(\mathbf {ab} -\mathbf {ba} )}

It is antisymmetric in a and b

b ∧ a = 1 2 ( b a − a b ) = − 1 2 ( a b − b a ) = − a ∧ b {\displaystyle \mathbf {b} \wedge \mathbf {a} ={\tfrac {1}{2}}(\mathbf {ba} -\mathbf {ab} )=-{\tfrac {1}{2}}(\mathbf {ab} -\mathbf {ba} )=-\mathbf {a} \wedge \mathbf {b} }

and by addition:

a ⋅ b + a ∧ b = 1 2 ( a b + b a ) + 1 2 ( a b − b a ) = a b {\displaystyle \mathbf {a} \cdot \mathbf {b} +\mathbf {a} \wedge \mathbf {b} ={\tfrac {1}{2}}(\mathbf {ab} +\mathbf {ba} )+{\tfrac {1}{2}}(\mathbf {ab} -\mathbf {ba} )=\mathbf {ab} }

That is, the geometric product is the sum of the symmetric scalar product and alternating exterior product. To examine the nature of a ∧ b, consider the formula

( a ⋅ b ) 2 − ( a ∧ b ) 2 = a 2 b 2 , {\displaystyle (\mathbf {a} \cdot \mathbf {b} )^{2}-(\mathbf {a} \wedge \mathbf {b} )^{2}=\mathbf {a} ^{2}\mathbf {b} ^{2},}

which using the Pythagorean trigonometric identity gives the value of (a ∧ b)2

( a ∧ b ) 2 = ( a ⋅ b ) 2 − a 2 b 2 = | a | 2 | b | 2 ( cos 2 ⁡ θ − 1 ) = − | a | 2 | b | 2 sin 2 ⁡ θ {\displaystyle (\mathbf {a} \wedge \mathbf {b} )^{2}=(\mathbf {a} \cdot \mathbf {b} )^{2}-\mathbf {a} ^{2}\mathbf {b} ^{2}=\left|\mathbf {a} \right|^{2}\left|\mathbf {b} \right|^{2}(\cos ^{2}\theta -1)=-\left|\mathbf {a} \right|^{2}\left|\mathbf {b} \right|^{2}\sin ^{2}\theta }

With a negative square, it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector. It has magnitude |a| |b| |sin θ|, where θ is the angle between the vectors, and so is zero for parallel vectors. To distinguish them from vectors, bivectors are written here with bold capitals, for example:

A = a ∧ b = − b ∧ a , {\displaystyle \mathbf {A} =\mathbf {a} \wedge \mathbf {b} =-\mathbf {b} \wedge \mathbf {a} \,,}

although other conventions are used, in particular as vectors and bivectors are both elements of the geometric algebra.

Properties The algebra generated by the geometric product (that is, all objects formed by taking repeated sums and geometric products of scalars and vectors) is the geometric algebra over the vector space. For an Euclidean vector space, this algebra is written G n {\displaystyle {\mathcal {G}}_{n}} or Cln(R), where n is the dimension of the vector space Rn. Cln(R) is both a vector space and an algebra, generated by all the products between vectors in Rn, so it contains all vectors and bivectors. More precisely, as a vector space it contains the vectors and bivectors as linear subspaces, though not as subalgebras (since the geometric product of two vectors is not generally another vector).

The space ⋀2Rn The space of all bivectors has dimension ⁠1/2⁠n(n − 1) and is written ⋀2Rn, and is the second exterior power of the original vector space.

Even subalgebra The subalgebra generated by the bivectors is the even subalgebra of the geometric algebra, written Cl[0]n(R). This algebra results from considering all repeated sums and geometric products of scalars and bivectors. It has dimension 2n−1, and contains ⋀2Rn as a linear subspace. In two and three dimensions the even subalgebra contains only scalars and bivectors, and each is of particular interest. In two dimensions, the even subalgebra is isomorphic to the complex numbers, C, while in three it is isomorphic to the quaternions, H. The even subalgebra contains the rotations in any dimension.

Magnitude As noted in the previous section the magnitude of a simple bivector, that is one that is the exterior product of two vectors a and b, is |a| |b| sin θ, where θ is the angle between the vectors. It is written |B|, where B is the bivector. For general bivectors, the magnitude can be calculated by taking the norm of the bivector considered as a vector in the space ⋀2Rn. If the magnitude is zero then all the bivector's components are zero, and the bivector is the zero bivector which as an element of the geometric algebra equals the scalar zero.

Unit bivectors A unit bivector is one with unit magnitude. Such a bivector can be derived from any non-zero bivector by dividing the bivector by its magnitude, that is

B | B | . {\displaystyle {\frac {\mathbf {B} }{\left\vert \mathbf {B} \right\vert }}.}

Of particular utility are the unit bivectors formed from the products of the standard basis of the vector space. If ei and ej are distinct basis vectors then the product ei ∧ ej is a bivector. As ei and ej are orthogonal, ei ∧ ej = eiej, written eij, and has unit magnitude as the vectors are unit vectors. The set of all bivectors produced from the basis in this way form a basis for ⋀2Rn. For instance, in four dimensions the basis for ⋀2R4 is (e1e2, e1e3, e1e4, e2e3, e2e4, e3e4) or (e12, e13, e14, e23, e24, e34).

Simple bivectors The exterior product of two vectors is a bivector, but not all bivectors are exterior products of two vectors. For example, in four dimensions the bivector

B = e 1 ∧ e 2 + e 3 ∧ e 4 = e 1 e 2 + e 3 e 4 = e 12 + e 34 {\displaystyle \mathbf {B} =\mathbf {e} _{1}\wedge \mathbf {e} _{2}+\mathbf {e} _{3}\wedge \mathbf {e} _{4}=\mathbf {e} _{1}\mathbf {e} _{2}+\mathbf {e} _{3}\mathbf {e} _{4}=\mathbf {e} _{12}+\mathbf {e} _{34}}

cannot be written as the exterior product of two vectors. A bivector that can be written as the exterior product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions; in four dimensions every bivector is the sum of at most two exterior products. A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by a directed plane area.

Product of two bivectors The geometric product of two bivectors, A and B, is

A B = A ⋅ B + A × B + A ∧ B . {\displaystyle \mathbf {A} \mathbf {B} =\mathbf {A} \cdot \mathbf {B} +\mathbf {A} \times \mathbf {B} +\mathbf {A} \wedge \mathbf {B} .}

The quantity A · B is the scalar-valued scalar product, while A ∧ B is the grade 4 exterior product that arises in four or more dimensions. The quantity A × B is the bivector-valued commutator product, given by

A × B = 1 2 ( A B − B A ) , {\displaystyle \mathbf {A} \times \mathbf {B} ={\tfrac {1}{2}}(\mathbf {AB} -\mathbf {BA} ),}

The space of bivectors ⋀2Rn is a Lie algebra over R, with the commutator product as the Lie bracket. The full geometric product of bivectors generates the even subalgebra. Of particular interest is the product of a bivector with itself. As the commutator product is antisymmetric the product simplifies to

A A = A ⋅ A + A ∧ A . {\displaystyle \mathbf {A} \mathbf {A} =\mathbf {A} \cdot \mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}

If the bivector is simple the last term is zero and the product is the scalar-valued A · A, which can be used as a check for simplicity. In particular the exterior product of bivectors only exists in four or more dimensions, so all bivectors in two and three dimensions are simple.

General bivectors and matrices Bivectors are isomorphic to skew-symmetric matrices in any number of dimensions. For example, the general bivector B23e23 + B31e31 + B12e12 in three dimensions maps to the matrix

M B = ( 0 B 12 − B 31 − B 12 0 B 23 B 31 − B 23 0 ) . {\displaystyle M_{B}={\begin{pmatrix}0&B_{12}&-B_{31}\\-B_{12}&0&B_{23}\\B_{31}&-B_{23}&0\end{pmatrix}}.}

This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector minus the exterior product; an example is the angular velocity tensor. Skew symmetric matrices generate orthogonal matrices with determinant 1 through the exponential map. In particular, applying the exponential map to a bivector that is associated with a rotation yields a rotation matrix. The rotation matrix MR given by the skew-symmetric matrix above is

M R = exp ⁡ M B . {\displaystyle M_{R}=\exp {M_{B}}.}

The rotation described by MR is the same as that described by the rotor R given by

R = exp ⁡ 1 2 B , {\displaystyle R=\exp {{\tfrac {1}{2}}B},}

and the matrix MR can be also calculated directly from rotor R. In three dimensions, this is given by

M R = ( ( R e 1 R − 1 ) ⋅ e 1 ( R e 2 R − 1 ) ⋅ e 1 ( R e 3 R − 1 ) ⋅ e 1 ( R e 1 R − 1 ) ⋅ e 2 ( R e 2 R − 1 ) ⋅ e 2 ( R e 3 R − 1 ) ⋅ e 2 ( R e 1 R − 1 ) ⋅ e 3 ( R e 2 R − 1 ) ⋅ e 3 ( R e 3 R − 1 ) ⋅ e 3 ) . {\displaystyle M_{R}={\begin{pmatrix}(R\mathbf {e} _{1}R^{-1})\cdot \mathbf {e} _{1}&(R\mathbf {e} _{2}R^{-1})\cdot \mathbf {e} _{1}&(R\mathbf {e} _{3}R^{-1})\cdot \mathbf {e} _{1}\\(R\mathbf {e} _{1}R^{-1})\cdot \mathbf {e} _{2}&(R\mathbf {e} _{2}R^{-1})\cdot \mathbf {e} _{2}&(R\mathbf {e} _{3}R^{-1})\cdot \mathbf {e} _{2}\\(R\mathbf {e} _{1}R^{-1})\cdot \mathbf {e} _{3}&(R\mathbf {e} _{2}R^{-1})\cdot \mathbf {e} _{3}&(R\mathbf {e} _{3}R^{-1})\cdot \mathbf {e} _{3}\end{pmatrix}}.}

Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix M the eigenvalues can be calculated by solving the characteristic equation for that matrix 0 = det(M − λI). By the fundamental theorem of algebra this has three roots (only one of which is real as there is only one eigenvector, i.e., the axis of rotation). The other roots must be a complex conjugate pair. They have unit magnitude so purely imaginary logarithms, equal to the magnitude of the bivector associated with the rotation, which is also the angle of rotation. The eigenvectors associated with the complex eigenvalues are in the plane of the bivector, so the exterior product of two non-parallel eigenvectors results in the bivector (or a multiple thereof).

Two dimensions When working with coordinates in geometric algebra it is usual to write the basis vectors as (e1, e2, ...), a convention that will be used here. A vector in real two-dimensional space R2 can be written a = a1e1 + a2e2, where a1 and a2 are real numbers, e1 and e2 are orthonormal basis vectors. The geometric product of two such vectors is

a b = ( a 1 e 1 + a 2 e 2 ) ( b 1 e 1 + b 2 e 2 ) = a 1 b 1 e 1 e 1 + a 1 b 2 e 1 e 2 + a 2 b 1 e 2 e 1 + a 2 b 2 e 2 e 2 = a 1 b 1 + a 2 b 2 + ( a 1 b 2 − a 2 b 1 ) e 1 e 2 . {\displaystyle {\begin{aligned}\mathbf {a} \mathbf {b} &=(a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2})(b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2})\\&=a_{1}b_{1}\mathbf {e} _{1}\mathbf {e} _{1}+a_{1}b_{2}\mathbf {e} _{1}\mathbf {e} _{2}+a_{2}b_{1}\mathbf {e} _{2}\mathbf {e} _{1}+a_{2}b_{2}\mathbf {e} _{2}\mathbf {e} _{2}\\&=a_{1}b_{1}+a_{2}b_{2}+(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{1}\mathbf {e} _{2}.\end{aligned}}}

This can be split into the symmetric, scalar-valued, scalar product and an antisymmetric, bivector-valued exterior product:

a ⋅ b = a 1 b 1 + a 2 b 2 , a ∧ b = ( a 1 b 2 − a 2 b 1 ) e 1 e 2 = ( a 1 b 2 − a 2 b 1 ) e 12 . {\displaystyle {\begin{aligned}\mathbf {a} \cdot \mathbf {b} &=a_{1}b_{1}+a_{2}b_{2},\\\mathbf {a} \wedge \mathbf {b} &=(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{1}\mathbf {e} _{2}=(a_{1}b_{2}-a_{2