JEE 2024 — Mathematics PYQ

P(3, 2, 3), Q(4, 6, 2) & R(7, 3, 2) be the vertices of PQR

$\begin{array}{rl}<br>\angle QPR& amp;=\theta \\ <br>\overrightarrow{OP}& amp;=-\hat{i}-4\hat{j}+\hat{k}\\ <br>\overrightarrow{RP}& amp;=-4\hat{i}-\hat{j}+\hat{k}\\ <br>\therefore & amp;\\ <br>\cos \theta & amp;=\frac{\left|\overrightarrow{QP}\cdot \overrightarrow{RP}\right|}{\left|\overrightarrow{QP}\right|\left|\overrightarrow{RP}\right|}\\ <br>& amp;=\frac{|4+4+1|}{\sqrt{18}\sqrt{18}}\\ <br>& amp;=\frac{9}{18}=\frac{1}{2}\\ <br>Q& amp;=\frac{\pi }{3}<br>\end{array}$