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?
 A: The bivector "i" is the Hestenes thing which corresponds to what is normally called dx wedge dy. This is an antisymmetric 2-tensor with components 1 and -1 at the x,y and y,x positions, and it is represented by a little area square in the x-y plane. This is a differential form.
What Hestenes does to produce geometric algebra is to multiply every vector index by gamma matrices, and these gamma matrices make an antisymmetric algebra with the following law:
$$ \gamma_i \gamma_j + \gamma_j \gamma_i = 2g_{ij}$$
Where g is the metric tensor, usually $g_{ij}=\delta_{ij}$ for 3 dimensional (or Euclidean) geometric algebra. In 3d, the $\gamma$ matrices have a standard representation with the Pauli matrices.
This means that when Hestenes is talking about the unit vector, he isn't thinking of it like a unit vector, but as what other people would call the slash of the unit vector, which is the $gamma$ matrices dotted with this unit vector, or in this case, just $\gamma_x$. The $\gamma$ matrices square to 1, and anticommute. So when he multiplies two unit vectors in geometric algebra, he gets
$$ \gamma_x \gamma_y = {1\over 2} (\gamma_x \gamma_y - \gamma_y\gamma_x)=\sigma_{xy}$$
Where the last equality is a definition: $\sigma_{ij}$ is defined as the antisymmetric product of $\gamma$ matrices:
$$ 2\sigma_{ij} = \gamma_i \gamma_j - \gamma_j \gamma_i $$
(most authors omit the factor of 2 on the left hand side). This definition of $\sigma$ is redundant off the diagonal, by antisymmetry of $\gamma$ multiplication. It's just defined this way to make sure that $\sigma_{ii}$ is zero, not 1.
So for Hestense, the unit two-form $dx\wedge dy$ is contracted with sigma, so it is just $\sigma_{xy}$. This means that it has the property that it squares to -1. You can see this by squaring and anticommuting the $\gamma$ matrices.
Hestene's approach hides the gamma matrices. The upside is that this gives you a quick intuition for $\gamma$ algebra without cumbersome notation for the $\gamma$ matrices or explicit matrix representations. The downside is that you don't learn symmetric tensors, and the notation is wildly different from what everyone else uses, with insufficient payoff (as far as I can see) to make up for it. Keeping the gammas explicit makes GA readable to me, but Hestenes prefers to hide them. It's no big deal to translate.
A: This link should answer your question, I think.
Don't be put off by some comments you will find on the internet, it is difficult for people who have spent years learning many formalisms to think of abandoning them; in your case you can acquire in a relatively short time the same effect of all those years (for not just physics but maths, computing, graphics, robotics, it is universal).
See this for GA baked in to a university course at your level.
(Btw, it is better to think of a bivector as a patch of area with a value equal to...)
A: The unit bivector represents the 2D subspaces.  In 2D Euclidean GA, there are 4 coordinates:


*

*1 scalar coordinate

*2 vector coordinates

*1 bivector coordinate


A "vector" is then (s, e1, e2, e1^e2).
The unit bivector is frequently drawn as a parallelogram, but that's just a visualization aid.  It conceptually more closely resembles a directed area where the sign indicates direction.
A: A bivector operating on a vector rotates the vector. 
A: 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)
