Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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