Let $\mathbf{R}=[r_1 \ r_2 \ r_3]$, $\mathbf{Q}=[q_1 \ q_2 \ q_3]$
be two different 3-dimensional rotation matrices built from orthonormal column vectors $r_i$ and $q_i$ appropriately.
Question:
- Could be obtained general form for a matrix $\mathbf{ Q}$ given the matrix $\mathbf{R}$ that it would be satisfied $${\sum_{i=1}^3}q_i + {\sum_{i=1}^3}r_i =0$$ Obvious solution is $q_i=-r_j$ and its "orthogonal" permutations.
- Are other solutions possible also? If not how to prove it ?