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.
Help!
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.