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

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)
$\Rightarrow \tan \alpha=-2$
$\Rightarrow \sin 2 \alpha=\frac{2(-2)}{1+(-2)^{2}}$
$\Rightarrow \sin 2 \alpha=\frac{-4}{5}<0$
$\Rightarrow 5 \sin 2 \alpha=-4$