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. 

What I'm missing in the other answers is that a surface area, in physics, separates the inside of a body from its outside, most of the time. Therefore it is natural to have a vector at the surface area which points to (the inside or) the outside. Which is precisely the outer product (called the normal on the surface if the area is taken apart as a scalar). It would be hard to describe e.g. conservation of mass in Fluid Flow without such an "area with a direction". A sample - and simple - application is found here in MSE.
