1. Choose the angle between planes
The planes share the $e_1$ direction, so the principal angles should be $(0,\alpha)$.
2. Visual picture
Blue: plane $U$. Orange: plane $W$. The drawing is a 2D projection of the 3D geometry.
3. Matrices
4. Distances
5. Independent-study practice tasks with answers
Task A. Explain why $\min_{u\in U,w\in W}\|u-w\|=0$ for any two linear subspaces.
Answer: Both subspaces contain $0$, so choose $u=w=0$.
Answer: Both subspaces contain $0$, so choose $u=w=0$.
Task B. If the principal angles are $0$ and $\pi/3$, compute the geodesic distance.
Answer: $d_{\operatorname{Gr}}=\sqrt{0^2+(\pi/3)^2}=\pi/3$.
Answer: $d_{\operatorname{Gr}}=\sqrt{0^2+(\pi/3)^2}=\pi/3$.
Task C. If the principal angles are $0.2$ and $0.6$, compute the chordal distance.
Answer: $d_{\mathrm{chordal}}=\sqrt{\sin^2(0.2)+\sin^2(0.6)}\approx 0.5985$.
Answer: $d_{\mathrm{chordal}}=\sqrt{\sin^2(0.2)+\sin^2(0.6)}\approx 0.5985$.
Similar practice. Try angles $0.1,0.4,0.7$. Compute geodesic and chordal distances.
Answer: $d_{\operatorname{Gr}}\approx0.8124$ and $d_{\mathrm{chordal}}\approx0.7580$.
Answer: $d_{\operatorname{Gr}}\approx0.8124$ and $d_{\mathrm{chordal}}\approx0.7580$.