Inmathematics, abivector or2-vector is a quantity inexterior algebra orgeometric algebra that extends the idea ofscalars andvectors. 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 tocomplex numbers intwo dimensions and to bothpseudovectors andvector quaternions in three dimensions. They can be used to generaterotations 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 adirected plane segment (ororiented plane segment), much asvectors can be thought of as characterizingdirected line segments.^[2] The bivectora ∧b has anattitude (ordirection) of theplane spanned bya andb, has an area that is a scalar multiple of any referenceplane segment with the same attitude (and in geometric algebra, it has amagnitude equal to the area of theparallelogram with edgesa andb), and has anorientation being the side ofa on whichb lies within the plane spanned bya andb.^[2][3] 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 theexterior product on vectors: given two vectorsa andb, their exterior producta ∧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 calledsimple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case.^[4] The exterior product of two vectors isalternating, soa ∧a is the zero bivector, andb ∧a is the negative of the bivectora ∧b, producing the opposite orientation. Concepts directly related to bivector are rank-2antisymmetric tensor andskew-symmetric matrix.

The bivector was first defined in 1844 by German mathematicianHermann Grassmann inexterior algebra as the result of theexterior product of two vectors. Just the previous year, in Ireland,William Rowan Hamilton had discoveredquaternions. Hamilton coined bothvector andbivector, the latter in hisLectures on Quaternions (1853) as he introducedbiquaternions, which havebivectors for their vector parts. It was not until English mathematicianWilliam Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and foundedClifford algebra, that the bivector of this article arose.Henry Forder used the termbivector to develop exterior algebra in 1941.^[5]

In the 1890sJosiah Willard Gibbs andOliver Heaviside developedvector calculus, which included separatecross product anddot products that were derived from quaternion multiplication.^[6][7][8] The success of vector calculus, and of the bookVector Analysis by Gibbs andWilson, 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 usedbivector in Hamilton's sense, a use that has sometimes been copied.^[9][10][11]Today the bivector is largely studied as a topic ingeometric algebra, a Clifford algebra overreal orcomplexvector spaces with aquadratic form. Its resurgence was led byDavid Hestenes who, along with others, applied geometric algebra to a range of new applications inphysics.^[12]

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 aEuclidean metric and so apositive-definitequadratic form.

The bivector arises from the definition of thegeometric product over a vector space with an associated quadratic form sometimes called themetric. For vectorsa,b andc, the geometric product satisfies the following properties:

Associativity

${\displaystyle (\mathbf{a}\mathbf{b})\mathbf{c}=\mathbf{a}(\mathbf{b}\mathbf{c})}$

Left and rightdistributivity

Scalar square

⁠${\displaystyle {\mathbf{a}}^2=Q(\mathbf{a})}$⁠, whereQ is the quadratic form, which need not bepositive-definite.

From associativity,a(ab) =a²b, is a scalar timesb. Whenb is not parallel to and hence not a scalar multiple ofa,ab cannot be a scalar. But

${\displaystyle \frac{1}{2}(\mathbf{a}\mathbf{b}+\mathbf{b}\mathbf{a})=\frac{1}{2}\left((\mathbf{a}+\mathbf{b}{)}^2-{\mathbf{a}}^2-{\mathbf{b}}^2\right)}$

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

${\displaystyle \mathbf{a}\cdot \mathbf{b}=\frac{1}{2}(\mathbf{a}\mathbf{b}+\mathbf{b}\mathbf{a}).}$

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

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 itsantisymmetric part:

${\displaystyle \mathbf{a}\wedge \mathbf{b}=\frac{1}{2}(\mathbf{a}\mathbf{b}-\mathbf{b}\mathbf{a})}$

It is antisymmetric ina andb

${\displaystyle \mathbf{b}\wedge \mathbf{a}=\frac{1}{2}(\mathbf{b}\mathbf{a}-\mathbf{a}\mathbf{b})=-\frac{1}{2}(\mathbf{a}\mathbf{b}-\mathbf{b}\mathbf{a})=-\mathbf{a}\wedge \mathbf{b}}$

and by addition:

${\displaystyle \mathbf{a}\cdot \mathbf{b}+\mathbf{a}\wedge \mathbf{b}=\frac{1}{2}(\mathbf{a}\mathbf{b}+\mathbf{b}\mathbf{a})+\frac{1}{2}(\mathbf{a}\mathbf{b}-\mathbf{b}\mathbf{a})=\mathbf{a}\mathbf{b}}$

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

To examine the nature ofa ∧b, consider the formula

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

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

${\displaystyle (\mathbf{a}\wedge \mathbf{b}{)}^2=(\mathbf{a}\cdot \mathbf{b}{)}^2-{\mathbf{a}}^2{\mathbf{b}}^2={\left|\mathbf{a}\right|}^2{\left|\mathbf{b}\right|}^2({\cos }^2\theta -1)=-{\left|\mathbf{a}\right|}^2{\left|\mathbf{b}\right|}^2{\sin }^2\theta }$

With a negative square, it cannot be a scalar or vector quantity, so it is a new sort of object, abivector. It hasmagnitude|a| |b| |sinθ|, whereθ 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:

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

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

The algebra generated by the geometric product (that is, all objects formed by taking repeated sums and geometric products of scalars and vectors) is thegeometric algebra over the vector space. For an Euclidean vector space, this algebra is written${\displaystyle {\mathcal{G}}_n}$ orCl_n(R), wheren is the dimension of the vector spaceRⁿ.Cl_n(R) is both a vector space and an algebra, generated by all the products between vectors inRⁿ, so it contains all vectors and bivectors. More precisely, as a vector space it contains the vectors and bivectors aslinear subspaces, though not assubalgebras (since the geometric product of two vectors is not generally another vector).

The space of all bivectors has dimension⁠1/2⁠n(n − 1) and is written⋀²Rⁿ,^[13] and is the secondexterior power of the original vector space.

The subalgebra generated by the bivectors is theeven subalgebra of the geometric algebra, writtenCl^[0]
_n(R). This algebra results from considering all repeated sums and geometric products of scalars and bivectors. It has dimension2ⁿ⁻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 isisomorphic to thecomplex numbers,C, while in three it is isomorphic to thequaternions,H. The even subalgebra contains therotations in any dimension.

As noted in the previous section the magnitude of a simple bivector, that is one that is the exterior product of two vectorsa andb, is|a| |b| sinθ, whereθ is the angle between the vectors. It is written|B|, whereB is the bivector.

For general bivectors, the magnitude can be calculated by taking thenorm 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.

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

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

Of particular utility are the unit bivectors formed from the products of thestandard basis of the vector space. Ife_i ande_j are distinct basis vectors then the producte_i ∧e_j is a bivector. Ase_i ande_j are orthogonal,e_i ∧e_j =e_ie_j, writtene_ij, and has unit magnitude as the vectors areunit 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).^[14]

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

${\displaystyle \mathbf{B}={\mathbf{e}}_1\wedge {\mathbf{e}}_2+{\mathbf{e}}_3\wedge {\mathbf{e}}_4={\mathbf{e}}_1{\mathbf{e}}_2+{\mathbf{e}}_3{\mathbf{e}}_4={\mathbf{e}}_{12}+{\mathbf{e}}_{34}}$

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.^[4]

The geometric product of two bivectors,A andB, is

${\displaystyle \mathbf{A}\mathbf{B}=\mathbf{A}\cdot \mathbf{B}+\mathbf{A}\times \mathbf{B}+\mathbf{A}\wedge \mathbf{B}.}$

The quantityA ·B is the scalar-valued scalar product, whileA ∧B is the grade 4 exterior product that arises in four or more dimensions. The quantityA ×B is the bivector-valuedcommutator product, given by

${\displaystyle \mathbf{A}\times \mathbf{B}=\frac{1}{2}(\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}),}$^[15]

The space of bivectors⋀²Rⁿ is aLie algebra overR, 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

${\displaystyle \mathbf{A}\mathbf{A}=\mathbf{A}\cdot \mathbf{A}+\mathbf{A}\wedge \mathbf{A}.}$

If the bivector issimple the last term is zero and the product is the scalar-valuedA ·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.^[4]

Bivectors are isomorphic toskew-symmetric matrices in any number of dimensions. For example, the general bivectorB₂₃e₂₃ +B₃₁e₃₁ +B₁₂e₁₂ in three dimensions maps to the matrix

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 theangular velocity tensor.

Skew symmetric matrices generateorthogonal matrices withdeterminant1 through the exponential map. In particular, applying the exponential map to a bivector that is associated with a rotation yields arotation matrix. The rotation matrixM_R given by the skew-symmetric matrix above is

${\displaystyle M_R=\exp M_B.}$

The rotation described byM_R is the same as that described by the rotorR given by

${\displaystyle R=\exp \frac{1}{2}B,}$

and the matrixM_R can be also calculated directly from rotorR. In three dimensions, this is given by

Bivectors are related to theeigenvalues of a rotation matrix. Given a rotation matrixM the eigenvalues can be calculated by solving thecharacteristic equation for that matrix0 = det(M −λI). By thefundamental 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).

When working with coordinates in geometric algebra it is usual to write thebasis vectors as (e₁,e₂, ...), a convention that will be used here.

Avector in real two-dimensional spaceR² can be writtena =a₁e₁ +a₂e₂, wherea₁ anda₂ are real numbers,e₁ ande₂ areorthonormal basis vectors. The geometric product of two such vectors is

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

All bivectors in two dimensions are of this form, that is multiples of the bivectore₁e₂, writtene₁₂ to emphasise it is a bivector rather than a vector. The magnitude ofe₁₂ is1, with

${\displaystyle {\mathbf{e}}_{12}^2=-1,}$

so it is called theunit bivector. The term unit bivector can be used in other dimensions but it is only uniquely defined (up to a sign) in two dimensions and all bivectors are multiples ofe₁₂. As the highest grade element of the algebrae₁₂ is also thepseudoscalar which is given the symboli.

With the properties of negative square and unit magnitude, the unit bivector can be identified with theimaginary unit fromcomplex numbers. The bivectors and scalars together form the even subalgebra of the geometric algebra, which isisomorphic to the complex numbersC. The even subalgebra has basis(1,e₁₂), the whole algebra has basis(1,e₁,e₂,e₁₂).

The complex numbers are usually identified with thecoordinate axes and two-dimensional vectors, which would mean associating them with the vector elements of the geometric algebra. There is no contradiction in this, as to get from a general vector to a complex number an axis needs to be identified as the real axis,e₁ say. This multiplies by all vectors to generate the elements of even subalgebra.

All the properties of complex numbers can be derived from bivectors, but two are of particular interest. First as with complex numbers products of bivectors and so the even subalgebra arecommutative. This is only true in two dimensions, so properties of the bivector in two dimensions that depend on commutativity do not usually generalise to higher dimensions.

Second a general bivector can be written

${\displaystyle \theta {\mathbf{e}}_{12}=i\theta ,}$

whereθ is a real number. Putting this into theTaylor series for theexponential map and using the propertye₁₂² = −1 results in a bivector version ofEuler's formula,

${\displaystyle \exp \theta {\mathbf{e}}_{12}=\exp i\theta =\cos \theta +i\sin \theta ,}$

which when multiplied by any vector rotates it through an angleθ about the origin:

${\displaystyle (x^{\prime }{\mathbf{e}}_1+y^{\prime }{\mathbf{e}}_2)=(x{\mathbf{e}}_1+y{\mathbf{e}}_2)\exp i\theta .}$

The product of a vector with a bivector in two dimensions isanticommutative, so the following products all generate the same rotation

${\displaystyle {\mathbf{v}}^{\prime }=\mathbf{v}\exp i\theta =\exp (-i\theta )\mathbf{v}=\exp (-i\theta /2)\mathbf{v}\exp (i\theta /2).}$

Of these the last product is the one that generalises into higher dimensions. The quantity needed is called arotor and is given the symbolR, so in two dimensions a rotor that rotates through angleθ can be written

${\displaystyle R=\exp (-\frac{1}{2}i\theta )=\exp (-\frac{1}{2}\theta {\mathbf{e}}_{12}),}$

and the rotation it generates is^[16]

${\displaystyle {\mathbf{v}}^{\prime }=R\mathbf{v}{R}^{-1}.}$

Inthree dimensions the geometric product of two vectors is

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

In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectorse₂₃,e₃₁ ande₁₂ form a basis for the space of bivectors⋀²R³, which is itself a three-dimensional linear space. So if a general bivector is:

${\displaystyle \mathbf{A}=A_{23}{\mathbf{e}}_{23}+A_{31}{\mathbf{e}}_{31}+A_{12}{\mathbf{e}}_{12},}$

they can be added like vectors

${\displaystyle \mathbf{A}+\mathbf{B}=(A_{23}+B_{23}){\mathbf{e}}_{23}+(A_{31}+B_{31}){\mathbf{e}}_{31}+(A_{12}+B_{12}){\mathbf{e}}_{12}.}$

while when multiplied they produce the following

${\displaystyle \mathbf{A}\mathbf{B}=-A_{23}{B}_{23}-A_{31}{B}_{31}-A_{12}{B}_{12}+(A_{12}{B}_{31}-A_{31}{B}_{12}){\mathbf{e}}_{23}+(A_{23}{B}_{12}-A_{12}{B}_{23}){\mathbf{e}}_{31}+(A_{31}{B}_{23}-A_{23}{B}_{31}){\mathbf{e}}_{12}}$

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

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

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

${\displaystyle \exp \mathbf{B}=\exp (\beta \frac{\mathbf{B}}{\beta })=\cos \beta +\frac{\mathbf{B}}{\beta }\sin \beta .}$

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 generalexp(A +B) ≠ exp(A) exp(B) in three (or more) dimensions.

The full geometric algebra in three dimensions,Cl₃(R), has basis (1,e₁,e₂,e₃,e₂₃,e₃₁,e₁₂,e₁₂₃). The elemente₁₂₃ is a trivector and thepseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified withpseudovectors^[17] to which they are related, asdiscussed below.

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 examplea +A, wherea is the scalar part andA is the bivector part. It is writtenCl^[0]
₃ and has basis(1,e₂₃,e₃₁,e₁₂). The product of two general elements of the even subalgebra is

${\displaystyle (a+\mathbf{A})(b+\mathbf{B})=ab+a\mathbf{B}+b\mathbf{A}+\mathbf{A}\cdot \mathbf{B}+\mathbf{A}\times \mathbf{B}.}$

The even subalgebra, that is the algebra consisting of scalars and bivectors, isisomorphic to thequaternions,H. 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 productA ·B. 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.

The rotation vector, from theaxis–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 aunit vectorω that is theaxis of rotation with the (signed)angle of rotationθ, so that the magnitude of the overall rotation vectorθω equals the (unsigned) rotation angle.

The quaternion associated with the rotation is

${\displaystyle q=\left(\cos \frac{1}{2}\theta ,\omega \sin \frac{1}{2}\theta \right)}$

In geometric algebra the rotation is represented by a bivector. This can be seen in its relation to quaternions. LetΩ be a unit bivector in the plane of rotation, and letθ be theangle of rotation. Then the rotation bivector isΩθ. The quaternion closely corresponds to the exponential of half of the bivectorΩθ. That is, the components of the quaternion correspond to the scalar and bivector parts of the following expression:$${\displaystyle \exp \frac{1}{2}\mathbf{\Omega }\theta =\cos \frac{1}{2}\theta +\mathbf{\Omega }\sin \frac{1}{2}\theta }$$

The exponential can be defined in terms of itspower series, and easily evaluated using the fact thatΩ squared is−1.

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.

The bivectorΩθ generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions:$${\displaystyle {\mathbf{v}}^{\prime }=\exp (-\frac{1}{2}\mathbf{\Omega }\theta )\mathbf{v}\exp (\frac{1}{2}\mathbf{\Omega }\theta ).}$$

As in two dimensions, the quantityexp(−⁠1/2⁠Ωθ) is called arotor and writtenR. The quantityexp(⁠1/2⁠Ωθ) is thenR⁻¹, and they generate rotations as$${\displaystyle {\mathbf{v}}^{\prime }=R\mathbf{v}{R}^{-1}.}$$

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 adouble cover over the rotation group, so the rotorsR and−R represent the same rotation.

The rotation vector is an example of anaxial 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.^[18] Examples include quantities liketorque,angular momentum and vectormagnetic fields. Quantities that would use axial vectors invector algebra are properly represented by bivectors in geometric algebra.^[19] More precisely, if an underlying orientation is chosen, the axial vectors are naturally identified with the usual vectors; theHodge dual then gives the isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa; that is

${\displaystyle \mathbf{A}=\star \mathbf{a},\mathbf{a}=\star \mathbf{A}}$

where⁠${\displaystyle \star }$⁠ 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 theunit pseudoscalar inCl₃(R),i =e₁e₂e₃ gives

${\displaystyle \mathbf{A}=\mathbf{a}i,\mathbf{a}=-\mathbf{A}i.}$

This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalari squares to−1, so a negative is needed in one of the products.

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

so are related by the Hodge dual:

${\displaystyle \star (\mathbf{a}\wedge \mathbf{b})=\mathbf{a}\times \mathbf{b},\star (\mathbf{a}\times \mathbf{b})=\mathbf{a}\wedge \mathbf{b}.}$

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 thetriple product should result in apseudoscalar, 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.^[20]

As suggested by their name and that of the algebra, one of the attractions of bivectors is that they have a natural geometric interpretation. This can be described in any dimension but is best done in three where parallels can be drawn with more familiar objects, before being applied to higher dimensions. In two dimensions the geometric interpretation is trivial, as the space is two-dimensional so has only one plane, and all bivectors are associated with it differing only by a scale factor.

All bivectors can be interpreted asplanes, or more precisely as directed plane segments. In three dimensions there are three properties of a bivector that can be interpreted geometrically:

• The arrangement of the plane in space, precisely theattitude of the plane (or alternately therotation,geometric orientation orgradient of the plane), is associated with the ratio of the bivector components. In particular the three basis bivectors,e₂₃,e₃₁ ande₁₂, or scalar multiples of them, are associated with theyz-plane,zx-plane andxy-plane respectively.
• Themagnitude of the bivector is associated with thearea of the plane segment. The area does not have a particular shape so any shape can be used. It can even be represented in other ways, such as by an angular measure. But if the vectors are interpreted as lengths the bivector is usually interpreted as an area with the same units, as follows.
• Like the direction of avector a plane associated with a bivector has a direction, a circulation or a sense of rotation in the plane, which takes two values seen asclockwise and counterclockwise when viewed from viewpoint not in the plane. This is associated with a change of sign in the bivector, that is if the direction is reversed the bivector is negated. Alternately if two bivectors have the same attitude and magnitude but opposite directions then one is the negative of the other.
• If imagined as a parallelogram, with the origin for the vectors at 0, then signed area is thedeterminant of the vectors' Cartesian coordinates (a_xb_y −b_xa_y).^[21]

In three dimensions all bivectors can be generated by the exterior product of two vectors. If the bivectorB =a ∧b then the magnitude ofB is

${\displaystyle |\mathbf{B}|=|\mathbf{a}||\mathbf{b}|\sin \theta ,}$

whereθ is the angle between the vectors. This is the area of theparallelogram with edgesa andb, as shown in the diagram. One interpretation is that the area is swept out byb as it moves alonga. The exterior product is antisymmetric, so reversing the order ofa andb to makea move alongb results in a bivector with the opposite direction that is the negative of the first. The plane of bivectora ∧b contains botha andb so they are both parallel to the plane.

Bivectors and axial vectors are related byHodge dual. In a real vector space the Hodge dual relates a subspace to itsorthogonal complement, so if a bivector is represented by a plane then the axial vector associated with it is simply the plane'ssurface normal. The plane has two normals, one on each side, giving the two possibleorientations for the plane and bivector.

This relates thecross product to theexterior product. It can also be used to represent physical quantities, liketorque andangular momentum. In vector algebra they are usually represented by vectors, perpendicular to the plane of theforce,linear momentum or displacement that they are calculated from. But if a bivector is used instead the plane is the plane of the bivector, so is a more natural way to represent the quantities and the way they act. It also unlike the vector representation generalises into other dimensions.

The product of two bivectors has a geometric interpretation. For non-zero bivectorsA andB the product can be split into symmetric and antisymmetric parts as follows:

${\displaystyle \mathbf{A}\mathbf{B}=\mathbf{A}\cdot \mathbf{B}+\mathbf{A}\times \mathbf{B}.}$

Like vectors these have magnitudes|A ·B| = |A| |B| cosθ and|A ×B| = |A| |B| sinθ, whereθ is the angle between the planes. In three dimensions it is the same as the angle between the normal vectors dual to the planes, and it generalises to some extent in higher dimensions.

Bivectors can be added together as areas. Given two non-zero bivectorsB andC in three dimensions it is always possible to find a vector that is contained in both,a say, so the bivectors can be written as exterior products involvinga:

This can be interpreted geometrically as seen in the diagram: the two areas sum to give a third, with the three areas forming faces of aprism witha,b,c andb +c as edges. This corresponds to the two ways of calculating the area using thedistributivity of the exterior product:

This only works in three dimensions as it is the only dimension where a vector parallel to both bivectors must exist. In higher dimensions bivectors generally are not associated with a single plane, or if they are (simple bivectors) two bivectors may have no vector in common, and so sum to a non-simple bivector.

In four dimensions, the basis elements for the space⋀²R⁴ of bivectors are (e₁₂,e₁₃,e₁₄,e₂₃,e₂₄,e₃₄), so a general bivector is of the form

${\displaystyle \mathbf{A}=a_{12}{\mathbf{e}}_{12}+a_{13}{\mathbf{e}}_{13}+a_{14}{\mathbf{e}}_{14}+a_{23}{\mathbf{e}}_{23}+a_{24}{\mathbf{e}}_{24}+a_{34}{\mathbf{e}}_{34}.}$

In four dimensions, the Hodge dual of a bivector is a bivector, and the space⋀²R⁴ is dual to itself. Normal vectors are not unique, instead every plane is orthogonal to all the vectors in its Hodge dual space. This can be used to partition the bivectors into two 'halves', in the following way. We have three pairs of orthogonal bivectors:(e₁₂,e₃₄),(e₁₃,e₂₄) and(e₁₄,e₂₃). There are four distinct ways of picking one bivector from each of the first two pairs, and once these first two are picked their sum yields the third bivector from the other pair. For example,(e₁₂,e₁₃,e₁₄) and(e₂₃,e₂₄,e₃₄).

In four dimensions bivectors are generated by the exterior product of vectors inR⁴, but with one important difference fromR³ andR². In four dimensions not all bivectors are simple. There are bivectors such ase₁₂ +e₃₄ that cannot be generated by the exterior product of two vectors. This also means they do not have a real, that is scalar, square. In this case

${\displaystyle ({\mathbf{e}}_{12}+{\mathbf{e}}_{34}{)}^2={\mathbf{e}}_{12}{\mathbf{e}}_{12}+{\mathbf{e}}_{12}{\mathbf{e}}_{34}+{\mathbf{e}}_{34}{\mathbf{e}}_{12}+{\mathbf{e}}_{34}{\mathbf{e}}_{34}=-2+2{\mathbf{e}}_{1234}.}$

The elemente₁₂₃₄ is the pseudoscalar inCl₄, distinct from the scalar, so the square is non-scalar.

All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as

${\displaystyle {\mathbf{e}}_{12}+{\mathbf{e}}_{34}={\mathbf{e}}_1\wedge {\mathbf{e}}_2+{\mathbf{e}}_3\wedge {\mathbf{e}}_4.}$

Similarly, every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover, for a generic bivector the choice of simple bivectors is unique, that is, there is only one way to decompose into orthogonal bivectors; the only exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique.^[4] The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero.

As in three dimensions bivectors in four dimension generate rotations through the exponential map, and all rotations can be generated this way. As in three dimensions ifB is a bivector then the rotorR isexp⁠1/2⁠B and rotations are generated in the same way:

${\displaystyle v^{\prime }=Rv{R}^{-1}.}$

The rotations generated are more complex though. They can be categorised as follows:

simple rotations are those that fix a plane in 4D, and rotate by an angle "about" this plane.

double rotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified

isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique.

These are generated by bivectors in a straightforward way. Simple rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors that the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique.^[22]

Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivectorB =B₁ +B₂, whereB₁ andB₂ are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows:

${\displaystyle R=\exp (\frac{1}{2}({\mathbf{B}}_1+{\mathbf{B}}_2))=\exp (\frac{1}{2}{\mathbf{B}}_1)\exp (\frac{1}{2}{\mathbf{B}}_2)=\exp (\frac{1}{2}{\mathbf{B}}_2)\exp (\frac{1}{2}{\mathbf{B}}_1)}$

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

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

(Note the order and indices above are not universal – heree₄ is the time-like dimension). The geometric algebra isCl_3,1(R), and the subspace of bivectors is⋀²R^3,1.

The simple bivectors are of two types. The simple bivectorse₂₃,e₃₁ ande₁₂ have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space,R³. These bivectors generate ordinary rotations inR³.

The simple bivectorse₁₄,e₂₄ ande₃₄ 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:

${\displaystyle \exp \frac{1}{2}\mathbf{\Omega }\theta =\cosh \frac{1}{2}\theta +\mathbf{\Omega }\sinh \frac{1}{2}\theta ,}$

whereΩ is the bivector (e₁₄, etc.), identified via the metric with an antisymmetric linear transformation ofR^3,1. These areLorentz boosts, expressed in a particularly compact way, using the same kind of algebra as inR³ andR⁴.

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

${\displaystyle R=\exp \frac{1}{2}\mathbf{A}.}$

The set of all rotations in spacetime form theLorentz 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.

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

Maxwell's equations are used in physics to describe the relationship betweenelectric andmagnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from⋀²R^3,1. If the electric and magnetic fields inR³ areE andB then theelectromagnetic bivector is

${\displaystyle \mathbf{F}=\frac{1}{c}\overline{E}{\mathbf{e}}_4+\overline{B}{\mathbf{e}}_{123},}$

wheree₄ is again the basis vector for the time-like dimension andc is thespeed of light. The productBe₁₂₃ yields the bivector that is Hodge dual toB in three dimensions, asdiscussed above, whileEe₄ as a product of orthogonal vectors is also bivector-valued. As a whole it is theelectromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the4-currentJ, a vector quantity given by

${\displaystyle \mathbf{J}=\overline{j}+c\rho {\mathbf{e}}_4,}$

wherej iscurrent density andρ ischarge density. They are related by a differential operator ∂, which is

${\displaystyle \mathrm{\partial }=\mathrm{\nabla }-{\mathbf{e}}_4\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}.}$

The operator ∇ is adifferential operator in geometric algebra, acting on the space dimensions and given by∇M = ∇·M + ∇∧M. When applied to vectors∇·M is thedivergence and∇∧M is thecurl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantityM they act as grade lowering and raising differential operators. In particular ifM is a scalar then this operator is just thegradient, and it can be thought of as a geometric algebraicdel operator.

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

${\displaystyle \mathrm{\partial }\mathbf{F}=\mathbf{J}.}$

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 theelectromagnetic four-potential, a vectorA given by

${\displaystyle \mathbf{A}=\overline{A}+\frac{1}{c}V{\mathbf{e}}_4,}$

whereA is the vector magnetic potential andV is the electric potential. It is related to the electromagnetic bivector as follows

${\displaystyle \mathrm{\partial }\mathbf{A}=-\mathbf{F},}$

using the same differential operator∂.^[23]

As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The geometric algebra for the real spaceRⁿ isCl_n(R), and the subspace of bivectors is⋀²Rⁿ.

The number of simple bivectors needed to form a general bivector rises with the dimension, so forn odd it is(n − 1) / 2, forn even it isn / 2. So for four andfive dimensions only two simple bivectors are needed but three are required forsix andseven dimensions. For example, in six dimensions with standard basis (e₁,e₂,e₃,e₄,e₅,e₆) the bivector

${\displaystyle {\mathbf{e}}_{12}+{\mathbf{e}}_{34}+{\mathbf{e}}_{56}}$

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.

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

${\displaystyle \exp \frac{1}{2}\mathbf{B}}$

is the rotor generated by bivectorB. Simple rotations, that take place in aplane of rotation around a fixedblade of dimension(n − 2) 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 inn dimensions is thespin group,Spin(n).

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.^[24]

Bivectors are also related to the rotation matrix inn dimensions. As in three dimensions thecharacteristic equation of the matrix can be solved to find theeigenvalues. 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.

Geometric algebra can be applied toprojective geometry in a straightforward way. The geometric algebra used isCl_n(R),n ≥ 3, the algebra of the real vector spaceRⁿ. This is used to describe objects in thereal projective spaceRPⁿ⁻¹. The non-zero vectors inCl_n(R) orRⁿ 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⋀²Rⁿ represent lines inRPⁿ⁻¹, 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 inRPⁿ⁻¹ represented by vectorsa andb the line containing them is given bya ∧b (orb ∧a). Two lines intersect in a point ifA ∧B = 0 for their bivectorsA andB. This point is given by the vector

${\displaystyle \mathbf{p}=\mathbf{A}\vee \mathbf{B}=(\mathbf{A}\times \mathbf{B})J^{-1}.}$

The operation "∨" is the meet, which can be defined as above in terms of the join,J =A ∧B^{[clarification needed]} for non-zeroA ∧B. Using these operations projective geometry can be formulated in terms of geometric algebra. For example, given a third (non-zero) bivectorC the pointp lies on the line given byC if and only if

${\displaystyle \mathbf{p}\wedge \mathbf{C}=0.}$

So the condition for the lines given byA,B andC to be collinear is

${\displaystyle (\mathbf{A}\vee \mathbf{B})\wedge \mathbf{C}=0,}$

which inCl₃(R) andRP² simplifies to

${\displaystyle ⟨\mathbf{A}\mathbf{B}\mathbf{C}⟩=0,}$

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.^[15]

Asnoted above a bivector can be written as a skew-symmetric matrix, which through the exponential map generates a rotation matrix that describes the same rotation as the rotor, also generated by the exponential map but applied to the vector. But it is also used with other bivectors such as theangular velocity tensor and theelectromagnetic tensor, respectively a 3×3 and 4×4 skew-symmetric matrix or tensor.

Real bivectors in⋀²Rⁿ are isomorphic ton ×n skew-symmetric matrices, or alternately to antisymmetrictensors of degree 2 onRⁿ. While bivectors are isomorphic to vectors (via the dual) in three dimensions they can be represented by skew-symmetric matrices in any dimension. This is useful for relating bivectors to problems described by matrices, so they can be re-cast in terms of bivectors, given a geometric interpretation, then often solved more easily or related geometrically to other bivector problems.^[25]

More generally, every real geometric algebra isisomorphic 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.^[26]

Look upbivector in Wiktionary, the free dictionary.

• Dyadics
• Multivector
• Multilinear algebra

• Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2009)."§ 2.3.3 Visualizing bivectors".Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 31ff.ISBN 978-0-12-374942-0.
• Whitney, Hassler (1957).Geometric Integration Theory. Princeton:Princeton University Press.ISBN 978-0-486-44583-0.
• Lounesto, Pertti (2001).Clifford algebras and spinors. Cambridge:Cambridge University Press.ISBN 978-0-521-00551-7.^{[permanent dead link‍]}
• Chris Doran & Anthony Lasenby (2003)."§ 1.6 The outer product".Geometric Algebra for Physicists. Cambridge:Cambridge University Press. p. 11et seq.ISBN 978-0-521-71595-9.

• Clifford algebras
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• Multilinear algebra
• Vector calculus
• Tensors
• Planes (geometry)

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