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 one way to approach this is to generalize to higher dimensions.  Given $k$ vectors in $n$ dimensional space, you can form an $n \times k$ matrix who columns are the $k$ vectors.  Then you can pick $\left( n\atop k \right)$ submatrices of size $k\times k$.  Calculate the determinant of each of these $k \times k$ matrices.  Then take the sum of the squares of these numbers.  This will be the square of the $k$-dimensional volume of the parallelepiped generated by the $k$-vectors in $n$-dimensional space.
Why is this true?  First you want to learn about alternating forms, also called exterior algebras: http://en.wikipedia.org/wiki/Exterior_algebra.  Then you will see that if you perform an orthogonal transformation on $\mathbb R^n$ to place the $k$ vectors so that only their first $k$-coordinates are non-zero, that this will also induce an orthogonal transformation on the exterior algebra $\Lambda^k(\mathbb R^n)$.  Then only one of the coordinates of the $k$-form will be non-zero, and its value will the the determinant of the $k\times k$ matrix of the transformed $k$-vectors,
This is a very brief, and somewhat sophisticated description.  But I have to say it took me years to understand this.  I don't know if I can easily reduce it to a short answer here.  Not at least without several hours of thinking of how to write it.
