# How do I get irreducible representations?

I'm currently trying to do a problem where it asks me to "show that the irreducible representations spanned by the atomic x-, y-, and z- coordinates is independent of the choice of local coordinate system. Use the diagrams below."

I know the general process is to determine the irreps for the separate coordinates. So, when I attempted to do the first one, by doing the symmetry operations to the x vectors from the D3h character table. However, it did not work as well. So now I don't know where to go with it. Please help and thank you.

If the angles are not 180 then it is necessary to use a rotation matrix and determine the trace (sum of diagonal elements). (This is explained in a good symmetry text book such as R. Carter, 'Molecular Symmetry and Group Theory'. Of course this has all been worked out and so it is necessary only to use a formula for $$C_n$$ and $$S_n$$ operations.
These are $$\displaystyle \chi(C_n)=1+2\cos\left(\frac{2\pi}{n}\right)$$ for $$n$$ fold rotations and $$\displaystyle \chi(S_n)=-1+2\cos\left(\frac{2\pi}{n}\right)$$ for rotation reflections, e.g. $$\chi(S_3)=-2\;; \chi(C_5)=(1+\sqrt{5})/2$$,
$$\begin{array}{ccc} \text{operation} & E & \sigma & i \\ \chi(n) & 3 & 1 & -3 \end{array}$$