Bivector - Three Dimensions

Three Dimensions

In three dimensions the geometric product of two vectors is

This can be split into the symmetric, scalar valued, interior 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 bivectors e₂₃, e₃₁ and e₁₂ form a basis for the space of bivectors Λ2ℝ3, which itself a three dimensional linear space. So if a general bivector is:

they can be added like vectors

while when multiplied they produce the following

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

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 eA + B ≠ eAeB 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 element e₁₂₃ 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.