Encoding quaternions as matrices?

I've a problem from my Group theory module that's been puzzling me, which is asking me to rotate a vector x (1,0,0) around a vector Ov, where v is (1/sqrt(3), -1/sqrt(3), 1/sqrt(3)), by 60 degrees by representing x as a matrix Z (encoding (0,1,0,0) as a matrix) and computing

UZU*,

where U is the unitary matrix encoding the quaternion

cos30 + vsin30.

Now, I know how to rotate it using quaternion conjugates, and how to do it using the cross product method, but this terminology is alien to me. What is a Unitary matrix and how is it calculated? Our lecture notes don't cover it, nor do they mention "encoding quaternions as matrices". I have no idea where to start. Any pointers?

Thanks for any help that is forthcoming.