Question

If $\bar{a}+\bar{b}+\bar{c}=\overline{0},|\bar{a}|=|\bar{b}|=|\bar{c}|=2, \theta$ is an angle between $\vec{b}$ and $\vec{c}$. Evaluate $\sec ^{2} \theta+\tan ^{2} \theta$

Answer 1

$\bar{a}+\bar{b}+\bar{c}=\overline{0} \text { and }|\bar{a}|=|\bar{b}|=|\bar{c}|=2$...[Given]
$\Rightarrow \bar{a}=-\bar{b}-\bar{c}$
$\Rightarrow|\bar{a}|^{2}=|\bar{b}|^{2}+|\bar{c}|^{2}+2|\bar{b}||\bar{c}| \cos \theta$
$\Leftrightarrow 4=4+4+2 \times 2 \times 2 \cos \theta$
$\Leftrightarrow 3 \cos \theta=-4 $
or $ \cos \theta=\frac{-1}{2}$
$\Rightarrow \theta=\frac{2 \pi}{3}$
$\Rightarrow \sec ^{2} \theta+\tan ^{2} \theta=\sec ^{2}\left(\frac{2 \pi}{3}\right)+\tan ^{2}\left(\frac{2 \pi}{3}\right)$
$=(-2)^{2}+(-\sqrt{3})^{2}$
$=4+3$
$=7$