My question concerns whether it is possible to determine a unique vector given two of the vectors rotations.
Assuming we have three known vectors in 3D; $a$, $b$ and $c$, and two angles $\phi$ and $\psi$.
Let b be the rotation of a by angle $\phi$ around axis $a\times b$, and $c$ is the rotation of $a$ by angle $\psi$ around $a \times c$.
Firstly, can we determine a vector $d$, s.t. $d$ is the rotation of $c$ around $c \times d$ with angle $\psi$ and furthermore the rotation of $b$ around $b \times d$ with angle $\phi$.
And if yes, is this vector unique?