Reasoning behind the cross products used to find area Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross product is used instead of the dot product. 
I was hoping math.se could explain this, it has been irking me too much. I understand the role of the cross product in torque and magnetism, but this one escapes me. 

I think the signed part of area is the most difficult to assign some intuitive meaning.
Consider two vectors in $\mathbb{R^2}$, and let $A : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ be the signed area. Then $A$ should be linear in each variable separately, since we should have $A(\lambda x,y) = \lambda A(x,y)$, and $A(x_1+x_2,y) = A(x_1,y)+A(x_2,y)$ (see the following diagram to convince yourself why this should be the case):

The area between a vector and itself should be zero (that is $A(z,z) = 0$ for all $z$), so the linearity requirement gives $A(x+y,x+y) = A(x,x)+A(x,y)+A(y,x) + A(y,y) = A(x,y)+A(y,x) = 0$, and so $A(x,y) = -A(y,x)$.
Then $A(x,y) = A(\sum_i x_i e_i, \sum_j y_j e_j) = \sum_i \sum_j x_i y_i A(e_i,e_j) = (x_1 y_1 -x_2y_2 )A(e_1,e_2)$.
It seems reasonable to assign an area of one to the area spanned by $e_1,e_2$, hence $A(e_1,e_2) = 1$, which gives $A(x,y) = x_1 y_1 -x_2y_2 $ (which, of course, equals $\det \begin{bmatrix} x & y \end{bmatrix}$).
