Rotate a unit sphere such as to align it two orthogonal unit vectors I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors).
I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed to $d$, where $c$ and $d$ are two other orthogonal unit vectors. 
It feels a very similar problem to this question:, but I'm not quite seeing how to obtain a rotation matrix for the sphere from the answer given there (which I'm sure is due to my limited mathematical abilities!).

Of course, if you can do it with two or more rotations, you can do it with one: the composition of rotations is a rotation.
Conceptually the strategy should be clear: $a$ and $c$ lie in a plane, and you can find a rotation $R_1$ that turns that plane carrying $a$ onto $c$. After that, $R_1(b)$ and $d$ lie in the plane perpendicular to $R_1(a)=c$, and you just need to rotate around $c$ through some angle with $R_2$ to get $b$ to lie on $d$. Then the rotation $R_2R_1$ carries the first orthonormal pair onto the second pair.
It would be great exercise for you to work the details of this strategy out with matrices. Just start with a rotation around $a\times c$ and follow up with a rotation around $c$.
I'm not sure if you're familiar with doing such things in terms of quaternions, but that would also be a great exercise.
