The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of computing this resultant vector. One way in particular is known from the symbolic determinant involving i j k and the entries of a and b.
I recall a professor awhile back trying to explain why a $\times$ b can be seen from this by appealing to the notion of finding the projection of the parallelogram with sides a and b onto the xy, xz and yz planes (imagine a and b floating in 3-space, so we're finding the projection of a and b onto these planes and finding the new area of the parallelogram after this projection). The goal was to show that the factor scaling the i component of a $\times$ b was the area of the parallelogram formed by a and b after being projected onto the yz plane, and similarly for the j component being scaled by the area of the projected parallelogram of a and b in the xz plane and the k component from the xy plane.
The explanation seemed straightforward at the time, but in trying to remember it I'm a little lost. Is anyone familiar with this notion of finding the areas under projection and these areas being the magnitudes of the i j and k components of a $\times$ b?
I hope that isn't too vague. I know that we had an explanation for the sign change of the j component as well being connected to these smashed projections but I don't recall how that worked either.