Consider two 4D vectors: $v_1=(\cos\varphi_1\sin\theta_1\sin\psi_1,\sin\varphi_1\sin\theta_1\sin\psi_1,\cos\theta_1\sin\psi_1,\cos\psi_1)$ and $v_2$, this vectors are orthogonal $v_1 \cdot v_2=0$, I guess that I can express $v_2(\varphi_1,\theta_1,\psi_1,\alpha,\beta)$, where $\alpha, \; \beta$ are angles, which define rotations around $v_1$. How can I do this?
For 3D it's a simple task: I find some orthogonal to $v_1$ vector and rotate it around $v_1$ by $R(\alpha)$ matrix. But in 4D, rotaions around vector don't define and I don't understand how can I costruct this two-parametric ($\alpha, \beta$) manifold.