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I would like to find the rotation transform between two 3D Cartesian coordinate frames knowing only the magnitude and direction of a single vector shown in both frames. The vector passes through the origin and therefore the origins are coincident.


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There is not enough information. Suppose one of the coordinate systems is the "standard" one, and our vector is $e_1=(1,0,0)$. Suppose the same vector is also represented as $(1,0,0)$ in the other system of coordinates. Then all we can conclude is that the coordinate systems are related by a rotation about the $x$-axis. We do not know the angle of rotation.

You need two non-collinear vectors instead of one.

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