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

Answer: 7

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