Question

If the vectors $\vec{b}=\left(\tan \alpha ,-1,2\sqrt{\sin \frac{\alpha }{2}}\right)$ and $\vec{c}=\left(\tan \alpha ,\tan \alpha ,\frac{-3}{\sqrt{\sin \frac{\alpha }{2}}}\right)$ are orthogonal and a vector $\vec{a}=(1,3,\sin 2\alpha )$ makes an obtuse angle with Z-axis, then find the value of $5\sin 2\alpha$.

Answer 1

Answer: -4

Given: $\vec{a}=(1,3,\sin 2\alpha )$ makes an obtuse angle with Z-axis.
$\Rightarrow \sin 2\alpha <0...(i)$
$\vec{b}$ and $\vec{c}$ are orthogonal.
$\Rightarrow \vec{b}\cdot \vec{c}=0$
$\Rightarrow {\tan }^2\alpha -\tan \alpha -6=0$
$\Rightarrow \tan \alpha =3$ or $-2.$
If $\tan \alpha =3$
$\sin 2\alpha =\frac{2\tan \alpha }{1+{\tan }^2\alpha }=\frac{3}{5}>0$ (Not possible)
$\Rightarrow \tan \alpha =-2$
$\Rightarrow \sin 2\alpha =\frac{2(-2)}{1+(-2{)}^2}$
$\Rightarrow \sin 2\alpha =\frac{-4}{5}<0$
$\Rightarrow 5\sin 2\alpha =-4$