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Jeremy Tan
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&[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/519d6d19-e1a8-4755-8a3a-cf99e380e7e3
&[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/503ba912-bf2f-4211-bb15-a94623fea736
Specifying the image of one vector only removes two degrees of freedom from the three present in the general 3D rotation, so your system of equations is underconstrained and does not have a unique solution.