I have two planes in an $n$-dimensional space, and a parallelogram in one of the planes, defined by two vectors ($v_1$, $v_2$), with area $S$. I want to find the area of the parallelogram defined by the projections of the vectors onto the other plane ($v'_1$, $v'_2$).
I can obviously just obtain the projections and compute the area, but I'm after a "simpler" result. For example, in 3-dimensional space a single angle $\theta$ can be defined between the two planes, and the projected area is then $S'=S\cos\theta$. But in an $n$-dimensional space there are, in general, two principal angles between the planes, is there a similarly simple relation in this case? Intuitively, I believe that if the principal angles are $\theta_1$ and $\theta_2$, maybe the area is $S'=S\cos\theta_1\cos\theta_2$ (which would still be applicable to 3 dimensions, since then $\theta_2=0$). Am I right? Where can I find a reference for this (or the correct) result?