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We use principal angles between two subspaces to define Jordan planes. Jordanplanes provide an optimal way to decompose $\mathbb{C}^n$ in relation to giventwo 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 $N$ aresubspaces of $\mathbb{C}^n$ in generic position. We compute numerical ranges ofsum and product of two orthogonal projections by using Jordan planes.

Jordan Plane and Numerical Range of Operators Involving Two Projections