Two Dimensions
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.
A vector in real two dimensional space ℝ2 can be written a = a1e1 + a2e2, where a1 and a2 are real numbers, e1 and e2 are orthonormal basis vectors. The geometric product of two such vectors is
This can be split into the symmetric, scalar valued, interior product and an antisymmetric, bivector valued exterior product:
All bivectors in two dimensions are of this form, that is multiples of the bivector e1e2, written e12 to emphasise it is a bivector rather than a vector. The magnitude of e12 is 1, with
so it is called the unit bivector. The term unit bivector can be used in other dimensions but it is only uniquely defined in two dimensions and all bivectors are multiples of e12. As the highest grade element of the algebra e12 is also the pseudoscalar which is given the symbol i.
Read more about this topic: Bivector
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