Bivector

thumb|170px|Parallel plane segments with the same orientation and area corresponding to the same 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 has an attitude (or direction) of the plane spanned by and , 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 and ), and has an orientation being the side of on which lies within the plane spanned by and .

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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 and , their exterior product 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.

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 , and , the geometric product satisfies the following properties:

; Associativity : <math> (\mathbf{ab})\mathbf{c} = \mathbf{a}(\mathbf{bc}) </math>

; Left and right distributivity : <math>\begin{align}

\mathbf{a}(\mathbf{b} + \mathbf{c}) &= \mathbf{ab} + \mathbf{ac} \\

(\mathbf{b} + \mathbf{c})\mathbf{a} &= \mathbf{ba} + \mathbf{ca}

\end{align}</math>

; Scalar square : , where is the quadratic form, which need not be positive-definite.

Scalar product

From associativity, , is a scalar times . When is not parallel to and hence not a scalar multiple of , cannot be a scalar. But

: <math>\tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}) = \tfrac{1}{2} \left((\mathbf{a} + \mathbf{b})^2 - \mathbf{a}^2 - \mathbf{b}^2\right)</math>

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

: <math>\mathbf{a} \cdot \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}).</math>

It is symmetric, scalar-valued, and can be used to determine the angle between two vectors: in particular if and 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:

: <math>\mathbf{a} \wedge \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} - \mathbf{ba})</math>

It is antisymmetric in and

: <math>\mathbf{b} \wedge \mathbf{a} = \tfrac{1}{2}(\mathbf{ba} - \mathbf{ab}) = -\tfrac{1}{2}(\mathbf{ab} - \mathbf{ba}) = -\mathbf{a} \wedge \mathbf{b}</math>

and by addition:

: <math>\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}</math>

That is, the geometric product is the sum of the symmetric scalar product and alternating exterior product.

To examine the nature of , consider the formula

: <math>(\mathbf{a} \cdot \mathbf{b})^2 - (\mathbf{a} \wedge \mathbf{b})^2 = \mathbf{a}^2\mathbf{b}^2,</math>

which using the Pythagorean trigonometric identity gives the value of

: <math>(\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</math>

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 , where 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:

: <math>\mathbf{A} = \mathbf{a} \wedge \mathbf{b} = -\mathbf{b} \wedge \mathbf{a} \,,</math>

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 <math>\mathcal{G}_n</math> or , where is the dimension of the vector space . is both a vector space and an algebra, generated by all the products between vectors in , 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 and is written , 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 . This algebra results from considering all repeated sums and geometric products of scalars and bivectors. It has dimension , and contains 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, , while in three it is isomorphic to the quaternions, . 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 and , is , where is the angle between the vectors. It is written , where 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 . 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

: <math>\frac{\mathbf{B{\left\vert\mathbf{B}\right\vert}.</math>

Of particular utility are the unit bivectors formed from the products of the standard basis of the vector space. If and are distinct basis vectors then the product is a bivector. As and are orthogonal, , written , 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 . For instance, in four dimensions the basis for is (, , , , , ) or (, , , , , ).

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

: <math>\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}</math>

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, and , is

: <math>\mathbf{A}\mathbf{B} = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B} + \mathbf{A} \wedge \mathbf{B}.</math>

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

: <math>\mathbf{A} \times \mathbf{B} = \tfrac{1}{2}(\mathbf{AB} - \mathbf{BA}),</math>

The space of bivectors is a Lie algebra over , 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

: <math>\mathbf{A}\mathbf{A} = \mathbf{A} \cdot \mathbf{A} + \mathbf{A} \wedge \mathbf{A}.</math>

If the bivector is simple the last term is zero and the product is the scalar-valued , 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.

: <math>\mathbf{v}' = R\mathbf{v}R^{-1}.</math>

Three dimensions

In three dimensions the geometric product of two vectors is

: <math>\begin{align} \mathbf{ab} &= (a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3)(b_1\mathbf{e}_1 + b_2\mathbf{e}_2 + b_3\mathbf{e}_3) \\ &= a_1 b_1{\mathbf{e}_1}^2 + a_2 b_2{\mathbf{e}_2}^2 + a_3 b_3{\mathbf{e}_3}^2 + (a_2 b_3 - a_3 b_2)\mathbf{e}_2\mathbf{e}_3 + (a_3 b_1 - a_1 b_3)\mathbf{e}_3\mathbf{e}_1 + (a_1 b_2 - a_2 b_1)\mathbf{e}_1\mathbf{e}_2. \end{align}</math>

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

: <math>\begin{align} \mathbf{a} \cdot \mathbf{b} &= a_1b_1 + a_2b_2 + a_3b_3 \\ \mathbf{a} \wedge \mathbf{b} &= (a_2 b_3 - a_3 b_2)\mathbf{e}_{23} + (a_3 b_1 - a_1 b_3)\mathbf{e}_{31} + (a_1 b_2 - a_2 b_1)\mathbf{e}_{12}. \end{align}</math>

In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectors , and form a basis for the space of bivectors , which is itself a three-dimensional linear space. So if a general bivector is:

: <math>\mathbf{A} = A_{23}\mathbf{e}_{23} + A_{31}\mathbf{e}_{31} + A_{12}\mathbf{e}_{12}, </math>

they can be added like vectors

: <math>\mathbf{A} + \mathbf{B} = (A_{23} + B_{23})\mathbf{e}_{23} + (A_{31} + B_{31})\mathbf{e}_{31} + (A_{12} + B_{12})\mathbf{e}_{12}.</math>

while when multiplied they produce the following

: <math>\mathbf{A} \mathbf{B} = -A_{23}B_{23} - A_{31}B_{31} - A_{12}B_{12} + (A_{12}B_{31} - A_{31}B_{12})\mathbf{e}_{23} + (A_{23}B_{12} - A_{12}B_{23})\mathbf{e}_{31} + (A_{31}B_{23} - A_{23}B_{31})\mathbf{e}_{12}</math>

which can be split into symmetric scalar and antisymmetric bivector parts as follows

: <math>\begin{align} \mathbf{A} \cdot \mathbf{B} &= -A_{12}B_{12} - A_{31}B_{31} - A_{23}B_{23} \\ \mathbf{A} \times \mathbf{B} &= (A_{23}B_{31} - A_{31}B_{23})\mathbf{e}_{12} + (A_{12}B_{23} - A_{23}B_{12})\mathbf{e}_{13} + (A_{31}B_{12} - A_{12}B_{31})\mathbf{e}_{23}. \end{align}</math>

The exterior product of two bivectors in three dimensions is zero.

A bivector can be written as the product of its magnitude and a unit bivector, so writing for and using the Taylor series for the exponential map it can be shown that

: <math>\exp{\mathbf{B = \exp({\beta\frac{\mathbf{B{\beta) = \cos{\beta} + \frac{\mathbf{B{\beta}\sin{\beta}.</math>

This is another version of Euler's formula, but with a general bivector in three dimensions. Unlike in two dimensions bivectors are not commutative so properties that depend on commutativity do not apply in three dimensions. For example, in general in three (or more) dimensions.

The full geometric algebra in three dimensions, , has basis (, , , , , , , ). The element is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors to which they are related, as discussed below.

Quaternions

Bivectors are not closed under the geometric product, but the even subalgebra is. In three dimensions it consists of all scalar and bivector elements of the geometric algebra, so a general element can be written for example , where is the scalar part and is the bivector part. It is written and has basis . The product of two general elements of the even subalgebra is

: <math>(a + \mathbf{A})(b + \mathbf{B}) = ab + a\mathbf{B} + b\mathbf{A} + \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B}.</math>

The even subalgebra, that is the algebra consisting of scalars and bivectors, is isomorphic to the quaternions, . This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates to the negative products in the bivector scalar product . Other quaternion properties can be similarly related to or derived from geometric algebra.

This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done the quaternion product is merely the geometric product. It also relates quaternions in three dimensions to complex numbers in two, as each is isomorphic to the even subalgebra for the dimension, a relationship that generalises to higher dimensions.

Rotation vector

The rotation vector, from the axis–angle representation of rotations, is a compact way of representing rotations in three dimensions. In its most compact form, it consists of a vector, the product of a unit vector that is the axis of rotation with the (signed) angle of rotation , so that the magnitude of the overall rotation vector equals the (unsigned) rotation angle.

The quaternion associated with the rotation is

: <math>q = \left(\cos{\tfrac{2} \theta}, \omega \sin{\tfrac{1}{2} \theta}\right)</math>

In geometric algebra the rotation is represented by a bivector. This can be seen in its relation to quaternions. Let be a unit bivector in the plane of rotation, and let be the angle of rotation. Then the rotation bivector is . The quaternion closely corresponds to the exponential of half of the bivector . That is, the components of the quaternion correspond to the scalar and bivector parts of the following expression:

<math display="block">\exp{\tfrac{1}{2} \boldsymbol{\Omega} \theta} = \cos{\tfrac{1}{2} \theta} + \boldsymbol{\Omega}\sin{\tfrac{1}{2} \theta} </math>

The exponential can be defined in terms of its power series, and easily evaluated using the fact that squared is .

So rotations can be represented by bivectors. Just as quaternions are elements of the geometric algebra, they are related by the exponential map in that algebra.

Rotors

The bivector generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions:

<math display="block">\mathbf{v}' = \exp(-\tfrac{1}{2} \boldsymbol{\Omega} \theta)\,\mathbf{v} \exp(\tfrac{1}{2} \boldsymbol{\Omega} \theta).</math>

As in two dimensions, the quantity is called a rotor and written . The quantity is then , and they generate rotations as

<math display = "block">\mathbf{v}' = R\mathbf{v}R^{-1}.</math>

This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions. This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors and represent the same rotation.

Axial vectors

275px|thumb|The 3-angular momentum as a bivector (plane element) and [[axial vector, of a particle of mass with instantaneous 3-position and 3-momentum .]]

The rotation vector is an example of an axial vector. Axial vectors, or pseudovectors, are vectors with the special feature that their coordinates undergo a sign change relative to the usual vectors (also called "polar vectors") under inversion through the origin, reflection in a plane, or other orientation-reversing linear transformation. Examples include quantities like torque, angular momentum and vector magnetic fields. Quantities that would use axial vectors in vector algebra are properly represented by bivectors in geometric algebra. More precisely, if an underlying orientation is chosen, the axial vectors are naturally identified with the usual vectors; the Hodge dual then gives the isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa; that is

: <math>\mathbf{A} = {\star} \mathbf{a} \,,\quad \mathbf{a} = {\star} \mathbf{A}</math>

where is the Hodge star. Note that if the underlying orientation is reversed by inversion through the origin, both the identification of the axial vectors with the usual vectors and the Hodge dual change sign, but the bivectors don't budge. Alternately, using the unit pseudoscalar in , gives

: <math>\mathbf{A} = \mathbf{a}i \,,\quad \mathbf{a} = - \mathbf{A} i.</math>

This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalar squares to , so a negative is needed in one of the products.

This relationship extends to operations like the vector-valued cross product and bivector-valued exterior product, as when written as determinants they are calculated in the same way:

: <math>\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix} \,,\quad \mathbf{a} \wedge \mathbf{b} = \begin{vmatrix} \mathbf{e}_{23} & \mathbf{e}_{31} & \mathbf{e}_{12}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix}\,,</math>

so are related by the Hodge dual:

: <math>{\star} (\mathbf a \wedge \mathbf b ) = \mathbf {a \times b} \,,\quad {\star} (\mathbf {a \times b} ) = \mathbf a \wedge \mathbf b\,.</math>

Bivectors have a number of advantages over axial vectors. They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are. For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalise to other dimensions; in particular bivectors can be used to describe quantities like torque and angular momentum in two as well as three dimensions. Also, they closely match geometric intuition in a number of ways, as seen in the next section.

Geometric interpretation

[[File:Wedge product.JPG|thumb|170px|Parallel plane segments with the same orientation and area corresponding to the same bivector .

Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector , where and are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows:

: <math>R = \exp(\tfrac{1}{2} (\mathbf{B}_1 + \mathbf{B}_2)) = \exp(\tfrac{1}{2} \mathbf{B}_1)\,\exp(\tfrac{1}{2} \mathbf{B}_2) = \exp(\tfrac{1}{2} \mathbf{B}_2)\,\exp(\tfrac{1}{2} \mathbf{B}_1)</math>

It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors.

Spacetime rotations

Spacetime is a mathematical model for our universe used in special relativity. It consists of three space dimensions and one time dimension combined into a single four-dimensional space. It is naturally described using geometric algebra and bivectors, with the Euclidean metric replaced by a Minkowski metric. That algebra is identical to that of Euclidean space, except the signature is changed, so

: <math>\mathbf{e}_i^2 = \begin{cases} 1, & i = 1, 2, 3 \\ -1, & i = 4 \end{cases}</math>

(Note the order and indices above are not universal – here is the time-like dimension). The geometric algebra is , and the subspace of bivectors is .

The simple bivectors are of two types. The simple bivectors , and have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, . These bivectors generate ordinary rotations in .

The simple bivectors , and have positive squares and as planes span a space dimension and the time dimension. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows:

: <math>\exp{\tfrac{1}{2}{\boldsymbol{\Omega}\theta = \cosh{\tfrac{1}{2} \theta} + \boldsymbol{\Omega}\sinh{\tfrac{1}{2} \theta},</math>

where is the bivector (, etc.), identified via the metric with an antisymmetric linear transformation of . These are Lorentz boosts, expressed in a particularly compact way, using the same kind of algebra as in and .

In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector is of the form

: <math>R = \exp{\tfrac{1}{2} \mathbf{A.</math>

The set of all rotations in spacetime form the Lorentz group, and from them most of the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same kind of algebra.

Maxwell's equations

(Note: in this section traditional 3-vectors are indicated by lines over the symbols and spacetime vector and bivectors by bold symbols, with the vectors and exceptionally in uppercase)

Maxwell's equations are used in physics to describe the relationship between electric and magnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from . If the electric and magnetic fields in are and then the electromagnetic bivector is

: <math>\mathbf{F} = \frac{1}{c}\overline{E}\mathbf{e}_4 + \overline{B}\mathbf{e}_{123},</math>

where is again the basis vector for the time-like dimension and is the speed of light. The product yields the bivector that is Hodge dual to in three dimensions, as discussed above, while as a product of orthogonal vectors is also bivector-valued. As a whole it is the electromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the 4-current , a vector quantity given by

: <math>\mathbf{J} = \overline{j} + c\rho\mathbf{e}_4,</math>

where is current density and is charge density. They are related by a differential operator ∂, which is

: <math>\partial = \nabla - \mathbf{e}_4\frac{1}{c}\frac{\partial}{\partial t}.</math>

The operator ∇ is a differential operator in geometric algebra, acting on the space dimensions and given by . When applied to vectors is the divergence and is the curl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity they act as grade lowering and raising differential operators. In particular if is a scalar then this operator is just the gradient, and it can be thought of as a geometric algebraic del operator.

Together these can be used to give a particularly compact form for Maxwell's equations with sources:

: <math>\partial\mathbf{F} = \mathbf{J}.</math>

This equation, when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell's four equations. It is also related to the electromagnetic four-potential, a vector given by

: <math>\mathbf{A} = \overline{A} + \frac{1}{c}V\mathbf{e}_4,</math>

where is the vector magnetic potential and is the electric potential. It is related to the electromagnetic bivector as follows

: <math>\partial\mathbf{A} = -\mathbf{F},</math>

using the same differential operator .

Higher dimensions

As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The geometric algebra for the real space is , and the subspace of bivectors is .

The number of simple bivectors needed to form a general bivector rises with the dimension, so for odd it is , for even it is . So for four and five dimensions only two simple bivectors are needed but three are required for six and seven dimensions. For example, in six dimensions with standard basis (, , , , , ) the bivector

: <math>\mathbf{e}_{12} + \mathbf{e}_{34} + \mathbf{e}_{56}</math>

is the sum of three simple bivectors but no less. As in four dimensions it is always possible to find orthogonal simple bivectors for this sum.

Rotations in higher dimensions

As in three and four dimensions rotors are generated by the exponential map, so

: <math>\exp{\tfrac{1}{2} \mathbf{B</math>

is the rotor generated by bivector . Simple rotations, that take place in a plane of rotation around a fixed blade of dimension are generated by simple bivectors, while other bivectors generate more complex rotations which can be described in terms of the simple bivectors they are sums of, each related to a plane of rotation. All bivectors can be expressed as the sum of orthogonal and commutative simple bivectors, so rotations can always be decomposed into a set of commutative rotations about the planes associated with these bivectors. The group of the rotors in dimensions is the spin group, .

One notable feature, related to the number of simple bivectors and so rotation planes, is that in odd dimensions every rotation has a fixed axis – it is misleading to call it an axis of rotation as in higher dimensions rotations are taking place in multiple planes orthogonal to it. This is related to bivectors, as bivectors in odd dimensions decompose into the same number of bivectors as the even dimension below, so have the same number of planes, but one extra dimension. As each plane generates rotations in two dimensions in odd dimensions there must be one dimension, that is an axis, that is not being rotated.

Bivectors are also related to the rotation matrix in dimensions. As in three dimensions the characteristic equation of the matrix can be solved to find the eigenvalues. In odd dimensions this has one real root, with eigenvector the fixed axis, and in even dimensions it has no real roots, so either all or all but one of the roots are complex conjugate pairs. Each pair is associated with a simple component of the bivector associated with the rotation. In particular, the log of each pair is the magnitude up to a sign, while eigenvectors generated from the roots are parallel to and so can be used to generate the bivector. In general the eigenvalues and bivectors are unique, and the set of eigenvalues gives the full decomposition into simple bivectors; if roots are repeated then the decomposition of the bivector into simple bivectors is not unique.

Projective geometry

Geometric algebra can be applied to projective geometry in a straightforward way. The geometric algebra used is , the algebra of the real vector space . This is used to describe objects in the real projective space . The non-zero vectors in or are associated with points in the projective space so vectors that differ only by a scale factor, so their exterior product is zero, map to the same point. Non-zero simple bivectors in represent lines in , with bivectors differing only by a (positive or negative) scale factor representing the same line.

A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example, given two distinct points in represented by vectors and the line containing them is given by (or ). Two lines intersect in a point if for their bivectors and . This point is given by the vector

: <math>\mathbf{p} = \mathbf{A} \lor \mathbf{B} = (\mathbf{A} \times \mathbf{B}) J^{-1}.</math>

The operation "" is the meet, which can be defined as above in terms of the join, for non-zero . Using these operations projective geometry can be formulated in terms of geometric algebra. For example, given a third (non-zero) bivector the point lies on the line given by if and only if

: <math>\mathbf{p} \land \mathbf{C} = 0.</math>

So the condition for the lines given by , and to be collinear is

: <math>(\mathbf{A} \lor \mathbf{B}) \land \mathbf{C} = 0,</math>

which in and simplifies to

: <math>\langle \mathbf{ABC} \rangle = 0,</math>

where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra.

More generally, every real geometric algebra is isomorphic to a matrix algebra. These contain bivectors as a subspace, though often in a way which is not especially useful. These matrices are mainly of interest as a way of classifying Clifford algebras.

See also

• Dyadics
• Multivector
• Multilinear algebra

Notes

General references

• Reprinted, Dover Publications, 2005, .