Coordinate Systems I have two axes: X,Y,Z and x',y',z'. I am given 4 relations: X is 60 degrees from x', Y is 90 degrees from x', Y is 120 degrees from z', Y is 30 degrees from y'. Knowing all of this, how do I find the rotation matrix relating the XYZ axis to the x'y'z' axis?

If you are asking for a rotation matrix, we are speaking of two sets of orthogonal systems.
First step, determine $Y$. Its position is clear, it is on the $y,z$ plane, and with the given angles it cannot be but$(0,\sqrt {3} /2, -1/2)$.
Second step, determine $X$, by imposing that its dot product with $x=(1,0,0)$ be $1/2$, and then that $X \cdot Y=0$, and finally that $|X|=1$.
Finally determine $Z=X \times Y$.
Now you have the matrix and can see if it is actually a rotation. Then I suppose you know how to proceed.
