question_answer

A tetrahedron has vertices \[P(1,2,1),Q(2,1,3),\]\[R(-1,1,2)\]and \[O(0,0,0).\]The angle between the faces OPQ and PQR is [JEE Main Online Paper Held On 12-Jan-2019 Morning]

A) \[{{\cos }^{-1}}\left( \frac{7}{31} \right)\]                                 

B) \[{{\cos }^{-1}}\left( \frac{17}{31} \right)\]

C) \[{{\cos }^{-1}}\left( \frac{19}{35} \right)\]                    

D)   \[{{\cos }^{-1}}\left( \frac{9}{35} \right)\]

Correct Answer: C

Solution :

Here, \[\overrightarrow{OP}\times \overrightarrow{OQ}=(\hat{i}+2\hat{j}+\hat{k})\times (2\hat{i}+\hat{j}+3\hat{k})\] \[=5\hat{i}-\hat{j}-3\hat{k}\] Again,\[\overrightarrow{PQ}\times \overrightarrow{PR}=(\hat{i}-\hat{j}+2\hat{k})\times (-2\hat{i}-\hat{j}+\hat{k})\] \[=\hat{i}-5\hat{j}-3\hat{k}\] Let angle between faces OPQ and PQR is \[\theta \] \[\therefore \] \[\cos \theta =\frac{5+5+9}{{{(\sqrt{25+9+1})}^{2}}}=\frac{19}{35}.\]