Bivectors

A bivector is an element of the exterior square Λ^2 V of a real vector space V, equivalently an antisymmetric rank-2 tensor. It can be written as a wedge product a ∧ b of two vectors, and any bivector is a sum of such wedge products. If V has dimension n, the space of bivectors has dimension n(n−1)/2, with components B_{ij} = −B_{ji}.

Geometrically, a bivector encodes oriented area: it represents the oriented parallelogram spanned by two vectors, with its magnitude equal to the parallelogram’s area and its orientation determined by the order of the spanning vectors. In three-dimensional Euclidean space there is a natural duality between bivectors and axial vectors: the dual of a bivector corresponds to a vector perpendicular to its plane, and the cross product a × b can be identified with the dual of the bivector a ∧ b. In higher dimensions the duality depends on the metric and orientation.

Algebraically, the geometric (Clifford) product of two vectors a and b decomposes into a scalar plus a bivector: ab = a·b + a∧b. Bivectors form a subspace on which rotations are generated: a rotor R = exp(−Bθ/2) rotates vectors via a' = R a R^{-1}, where B is a unit bivector representing the rotation plane and θ is the angle.

Bivectors are central in exterior algebra and geometric algebra as coordinate-free descriptions of oriented area, with applications in physics and geometry, including computer graphics and electromagnetism in four dimensions.