What's the meaning of the unit bivector i? I'm reading the Oersted Medal Lecture by David Hestenes to improve my understanding of Geometric Algebra and its applications in Physics. I understand he does not start from a mathematical "clean slate", but I don't care for that. I want to understand what he's saying and what I can do with this geometric algebra.
On page 10 he introduces the unit bivector i. I understand (I think) what unit vectors are: multiply by a scalar and get a scaled directional line. But a bivector is a(n oriented) parallellogram (plane). So if I multiply the unit bivector i with a scalar, I get a scaled parallellogram?

for orthogonal vectors a, b, c, d, etc (equivalent to column vectors in Hilbert Spaces)
where ^ means outer product and . means inner product)
and where * means geometric product (sum of inner and outer product)
and a*a = 1, etc (due to inner product term)
and a * b = a b = a^b  (when a and b are orthogonal)
and a b = - b a due to anti-commutative nature of vectors (right hand rule)
so product a b * a b = a b a b = - a a b b = -1*1 = -1
so finally a^b = sqrt(-1) = i so each bivector represents a unique spinor for that pair of dimensions.
The understanding of graded components: grade-1 vectors, grade-2 bivectors, grade-3 trivectors etc are new concepts compared to Hilbert space notations. The relationships between geometric, inner and outer products are tricky, so a longer explanation is really required to get this relationship. 
So for more details see: http://www.matzkefamily.net/doug/papers/PHD/phd_matzke_final.pdf and other papers and videos related to my work in geometric algebra (related to quantum computing)
