Eigenvector Slew - Problem

Problem

Let

be a body fixed reference system for a 3 axis stabilized spacecraft. The initial attitude is given by

One wants to find an axis relative the spacecraft body

and a rotation angle such that after the rotation with the angle one has that

where

are the new target directions.

In vector form this means that

\hat{d}=r_a \cdot \hat{r} + \cos \alpha \cdot ( \hat{a} - r_a \cdot \hat{r})+ \sin \alpha \cdot \hat{r} \times \hat{a}
\hat{e}=r_b \cdot \hat{r} + \cos \alpha \cdot ( \hat{b} - r_b \cdot \hat{r})+ \sin \alpha \cdot \hat{r} \times \hat{b}
\hat{f}=r_c \cdot \hat{r} + \cos \alpha \cdot ( \hat{c} - r_c \cdot \hat{r})+ \sin \alpha \cdot \hat{r} \times \hat{c}.

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