I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and Grassmanian of $k$-planes, which I don't understand. I know how to check the intersection a plane and a line, but I don't know how to check the intersection Grassmanian of $n-k$-planes and Grassmanian of $k$-planes.

I would appreciate it if one could either explain it or introduce some reference.