If the vector  $\overline{b}=\left(\tan \alpha ,-1,2\sqrt{\sin \frac{\alpha }{2}}\right)$   and  $(\tan \alpha ,\tan \alpha ,\frac{-3}{\sqrt{\sin \frac{\alpha }{2}}})$ are orthogonal and vector  $\overline{a}=(1,3,\sin 2\alpha )$   makes an obtuse angle with the Z-axis. Then $\alpha$  =________

Since  $\overline{a}=(1,3,\sin 2\alpha )$  makes an obtuse angle with z-axis and its z-component is –ve. $\Rightarrow -1\le \sin 2\alpha \le 0$
but   $$\begin{gathered} \overline{b}.\overline{c}= \Rightarrow {\tan }^2\alpha -\tan \alpha -6=0 \Rightarrow (\tan \alpha -3)(\tan \alpha +2)=0 \\ \Rightarrow \tan \alpha =3,-2 \tan \alpha =-2 \end{gathered}$$     
           then   $\sin 2\alpha =\frac{-4}{5} \Rightarrow \tan 2\alpha >0$     .