$\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$are two mutually perpendicular unit vectors and $\overrightarrow{\mathrm{c}}$ is a unit vector inclined at an angle $\mathrm{\theta }$ to both $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$. If $\overrightarrow{\mathrm{c}}=\mathrm{x}\overrightarrow{\mathrm{a}}+\mathrm{x}\overrightarrow{\mathrm{b}}+\mathrm{y}\left(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right),$ where $\mathrm{x},\mathrm{y}\in \mathrm{R}$, t h e n exhaustive range of $\text{'}\mathrm{\theta }\text{'}$ is

$$\begin{gathered} \left|\overline{\mathrm{a}}\right|=\left|\overline{\mathrm{b}}\right|=1 \overline{\mathrm{a}}·\overline{\mathrm{b}}=\cos \mathrm{\theta }, \left|\overline{\mathrm{c}}\right|=1 \\ \overline{\mathrm{c}}·\overline{\mathrm{a}}=\cos \mathrm{\theta } \overline{\mathrm{c}}·\overline{\mathrm{b}} =\cos \mathrm{\theta } \\ \overline{\mathrm{c}} =\mathrm{x} \overline{\mathrm{a}}+\mathrm{x} \overline{\mathrm{b}}+\mathrm{y}(\overline{\mathrm{a}}\times \overline{\mathrm{b}}) \\ \overline{\mathrm{c}}·\overline{\mathrm{a}} =\mathrm{x} \Rightarrow =\cos \mathrm{\theta } \end{gathered}$$

$$\begin{gathered} {\left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right]}^2=\left|\begin{array}{ccc}1& 0& \cos \mathrm{\theta }\\ 0& 1& \cos \mathrm{\theta }\\ \cos \mathrm{\theta }& \cos \mathrm{\theta }& 1\end{array}\right| \\ =1(1-{\cos }^2 \mathrm{\theta })+\cos \mathrm{\theta }(0-\cos \mathrm{\theta }) \\ =1-2{\cos }^2\mathrm{\theta }=-\cos 2\mathrm{\theta }>0 \\ \cos 2\mathrm{\theta }<0 \Rightarrow 2\mathrm{\theta }\in \left(\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right., 3\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \\ \mathrm{\theta }\in \left(\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right., 3\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right) \end{gathered}$$