A tetrahedron has vertices at \(\mathrm{P}(2,1,3)\), \(\mathrm{Q}(-1,1,2), \mathrm{R}(1,2,1)\) and \(\mathrm{O}(0,0,0)\), then angle between the faces OPQ and PQR is

\(\begin{aligned} O P \times O Q & =\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & 1 & 3 \\ -1 & 1 & 2\end{array}\right| \\ & =-\hat{\mathrm{i}}-7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}\end{aligned}\)
\(\begin{aligned} P Q \times P R & =\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ -3 & 0 & -1 \\ -1 & 1 & -2\end{array}\right| \\ & =\hat{\mathrm{i}}-5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}\end{aligned}\)
Angle between the faces OPQ and PQR is
\(\cos \theta=\frac{(-\hat{\mathrm{i}}-7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \cdot(\hat{\mathrm{i}}-5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})}{\sqrt{(-1)^2+(-7)^2+3^2} \sqrt{1^2+(-5)^2+(-3)^2}}\)
\(=\frac{-1+35-9}{\sqrt{59} \sqrt{35}}\)
\(\begin{aligned} & \Rightarrow \cos \theta=\frac{25}{\sqrt{59} \sqrt{35}} \\ & \Rightarrow \theta=\cos ^{-1}\left(\frac{25}{\sqrt{59} \sqrt{35}}\right)\end{aligned}\)