Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 15E / d / Solution

The operator $R_g$ reverses the $|{\psi }_g⟩$ component and fixes $|{\psi }_b⟩$, so it is reflection in the horizontal $|{\psi }_b⟩$ axis. The operator $R_{|\Omega ⟩}$ reverses the component parallel to $|\Omega ⟩$ and fixes its perpendicular line; multiplication by $-1$ reverses those roles. Hence $-R_{|\Omega ⟩}$ is reflection in the line spanned by $|\Omega ⟩$.