I'm working on a problem where I seem to have run into a system of non-linear equations. I have ten equations and ten unknowns. In the equations below, all of the $\phi\_{ij}$'s are known, but all of the $a\_{1},...,e\_{2}$ are unknown.

$$
a\_{1}b\_{1}+a\_{2}b\_{2}=\phi\_{ab}\\
a\_{1}c\_{1}+a\_{2}c\_{2}=\phi\_{ac}\\
a\_{1}d\_{1}+a\_{2}d\_{2}=\phi\_{ad}\\
a\_{1}e\_{1}+a\_{2}e\_{2}=\phi\_{ae}\\
b\_{1}c\_{1}+b\_{2}c\_{2}=\phi\_{bc}\\
b\_{1}d\_{1}+b\_{2}d\_{2}=\phi\_{bd}\\
b\_{1}e\_{1}+b\_{2}e\_{2}=\phi\_{be}\\
c\_{1}d\_{1}+c\_{2}d\_{2}=\phi\_{cd}\\
c\_{1}e\_{1}+c\_{2}e\_{2}=\phi\_{ce}\\
d\_{1}e\_{1}+d\_{2}e\_{2}=\phi\_{de}\\
$$

Would anyone have any strategies for solving this system of non-linear equations for the $a\_{1},...,e\_{2}$? Is it possible to convert this into a system of linear equations somehow?

Additional info (based on comments):

The dot products of the vectors with themselves are all known quantities, and in fact equal to one:
$$
\phi\_{aa}=1\\
\phi\_{bb}=1\\
\phi\_{cc}=1\\
\phi\_{dd}=1\\
\phi\_{ee}=1\\
$$

Yes, it is true that $\phi\_{ab}=\phi\_{ba}$.