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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?

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