Jordan Plane and Numerical Range of Operators Involving Two Projections

We use principal angles between two subspaces to define Jordan planes. Jordan planes provide an optimal way to decompose $\mathbb{C}^n$ in relation to given two subspaces. We apply Jordan planes to show that two pairs of of subspaces $(M,N)$ and $(M^{\perp},N^{\perp})$ are unitarily equivalent if $M$ and $N$ are subspaces of $\mathbb{C}^n$ in generic position. We compute numerical ranges of sum and product of two orthogonal projections by using Jordan planes.