Question

If $P=(0,1,2),\,\,Q=(4,-2,1),O=(0,0,0),$ then $\angle POQ$ is equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

$\frac{\pi }{2}$

Answer 2

Direction ratios of $OP=(0-0,\,1-0,\,\,2-0)=(0,1,2)$ Direction ratios of
$OQ=(4-0,-2-0,1-0)=(4,-2,1)$
Let $\angle POQ=\theta ,$
then $\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$
$\Rightarrow$ $\cos \theta =\frac{0-2\times 1+2\times 1}{\sqrt{0+1+4}\sqrt{16+4+1}}$
$=\frac{0-2+2}{\sqrt{5}\sqrt{21}}=0$
$\Rightarrow$ $\theta =\frac{\pi }{2}$
$\therefore$ $\angle POQ=\frac{\pi }{2}$