Bivector

[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).

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.

Its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics.

[12] For this article, the bivector will be considered only in real geometric algebras, which may be applied in most areas of physics.

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: From associativity, a(ab) = a2b, is a scalar times b.

To examine the nature of a ∧ b, consider the formula which using the Pythagorean trigonometric identity gives the value of (a ∧ b)2 With a negative square, it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector.

This algebra results from considering all repeated sums and geometric products of scalars and bivectors.

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.

The exponential can be defined in terms of its power series, and easily evaluated using the fact that Ω squared is −1.

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.

[18] Examples include quantities like torque, angular momentum and vector magnetic fields.

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: so are related by the Hodge dual: 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.

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.

The exterior product is antisymmetric, so reversing the order of a and b to make a move along b results in a bivector with the opposite direction that is the negative of the first.

Given two non-zero bivectors B and C 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 involving a: 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 a prism with a, b, c and b + c as edges.

[4] The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero.

Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique.

That algebra is identical to that of Euclidean space, except the signature is changed, so (Note the order and indices above are not universal – here e4 is the time-like dimension).

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

(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 J and A 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 ⋀2R3,1.

If the electric and magnetic fields in R3 are E and B then the electromagnetic bivector is where e4 is again the basis vector for the time-like dimension and c is the speed of light.

The geometric algebra for the real space Rn is Cln(R), and the subspace of bivectors is ⋀2Rn.

A description of the projective geometry can be constructed in the geometric algebra using basic operations.

This point is given by the vector The operation "∨" is the meet, which can be defined as above in terms of the join, J = A ∧ B [clarification needed] for non-zero A ∧ B.

[15] As noted 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.

Real bivectors in ⋀2Rn are isomorphic to n × n skew-symmetric matrices, or alternately to antisymmetric tensors of degree 2 on Rn.