Given a non normalized state, find the direction of the state $\chi$?: $$ \chi = (1+i)\chi_{+}^{z}-(1+i\sqrt{3})\chi_{-}^{z}$$ Where $\chi_{+}^{z}, \chi_{-}^{z}$ are eigenstates of $S_{z}$.
I know the eigenstates of $S_{z}$ are given as $$ |\chi_{+}^{z}\rangle = \cos\theta/2 |+\rangle + e^{i\phi}\sin\theta/2|- \rangle$$ $$ |\chi_{-}^{z}\rangle = e^{-i\phi}\sin\theta/2 |+\rangle - \cos\theta/2|- \rangle$$ After solving $S_{n}$ = $\vec{S} \cdot \hat{n}$ where $$ \hat{n} = \sin\theta\cos\phi \hat{i} + \sin\theta\sin\phi \hat{j} + \cos\theta \hat{k} $$ I was thinking of setting the state equal to up and down states and see if they equal each other but I'm not really able to solve the equation without knowing the correct $\phi$ and $\theta$