My first question is how do you define the sense of rotation about an arbitrary axis?
Rotations are usually counterclockwise and when referring to rotation with respect to the $x$,$y$ or $z$ axis they happen as you look towards the origin when standing on the positive side of the axis. E.g. if you were to rotate the xy plane by $\theta$ then the rotation would be counterclockwise ($\theta>0$) if you were looking at the xy plane from the positive branch of the $z$ axis.
But if I were to ask you to rotate your coordinate system around some arbitrary axis $y'$ how would you know the sense of this rotation?
The next part of my question is related to a homework problem, and is quite short.
I'm asked to have a rotation $\alpha$ in the xy plane. I've done so using the mentality above. Then I'm asked to rotate (in another instance, that is, not doing the rotations consecutively) my coordinate system by $\beta$ in the $x'$-$z'$ plane (where $x',y'$ and $z'$ were my axis after the first rotation).
I only know how to do this if I imagine the rotations done consecutively. But I think my professor is asking me to consider the rotations separately.
How is this done separately? How is the sense of rotation determined here?
My intuition is that it's the same as doing the rotations consecutively.
Also: Is "rotating the xy plane" the same as "having a rotation in the xy plane"?