There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases.

$X$ is $2\times 2$ or $3\times 3$ rotation matrix with an unusual domain specific constriant:

• $X^TX = XX^T = I$
• $Xa + X^Tb = c$

Is there a solution for $X$ in terms of $a,b,c$? Based on where the problem came from, I know there isn't always a solution, but I have stumped myself trying to figure out how to solve it when there is one.

I have tried working out the $2\times 2$ case element-wise, and arrived at the following, equally(?) difficult problem:

$X = \begin{bmatrix}x_{11} & x_{12} \\ -x_{12} & x_{11}\end{bmatrix}$

$\begin{bmatrix}a_1+b_1 & b_2-a_2 \\ a_2+b_2 & a_1-b_1\end{bmatrix}\begin{bmatrix}x_{11}\\x_{12}\end{bmatrix}=c$

$Ax = c$ where $x^Tx=1$