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

• Correct answer
  A

  $\frac{\pi}{2}$
• B

  $\frac{\pi}{4}$
• C

  $\frac{\pi}{6}$
• D

  $\frac{\pi}{3}$

Solution

We have,
Direction cosines of $O Q$ are $4-0,-2-0,1-0$ i.e. $4,-2,1$ Direction cosines of $O P$ are $0-0,1-0,2-0$ i.e. $0,1,2$ Then, $\begin{aligned} \cos \theta & =l_1 l_2+m_1 m_2+n_1 n_2 \\ & =4 \cdot 0+(-2) \cdot 1+1 \cdot 2 \\ & =0-2+2=0 \\ \cos \theta & =0=\cos \frac{\pi}{2} \\ \theta & =\frac{\pi}{2}\end{aligned}$

Hence, the correct answer is option: 'A'.