Let $Z=2e^{\frac{10\pi i}{21}}$ and $W=e^{\frac{\pi i}{7}}$ and if ${(Z-W)}^6=r.e^{i\theta },r\ge 0,$$-\pi <\theta \le \pi$  then the value of $\frac{\theta }{r}$ equals

$$\begin{gathered} {\text{z-w=e}}^{\frac{\text{iπ}}{\text{7}}}\left({\text{2e}}^{\frac{\text{iπ}}{3}}\text{-1}\right){\text{=e}}^{\frac{\text{iπ}}{\text{7}}}\text{.}\left(\sqrt{\text{3}}\text{i}\right) \\ =\sqrt{3}i(\cos \frac{\pi }{7}+i\sin \frac{\pi }{7}) \\ {(Z-W)}^6=27(-1)(\cos \frac{6\pi }{7}+i\sin \frac{6\pi }{7})=27(-1)(-\cos \frac{\pi }{7}+i\sin \frac{\pi }{7}) \\ =27.(\cos \frac{\pi }{7}-i\sin \frac{\pi }{7}) \Rightarrow r=27, \theta =\frac{-\pi }{7} \\ \frac{\text{θ}}{\text{r}}\text{=}\frac{\text{-π}}{\text{7×27}}\text{=-}\frac{\text{π}}{\text{189}} \end{gathered}$$