# Uniqueness of a vector given two rotations

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?