JEE Main 12 Jan 2019 Morning Question 30
Question: 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]
Options:
A) $ {{\cos }^{-1}}( \frac{7}{31} ) $
B) $ {{\cos }^{-1}}( \frac{17}{31} ) $
C) $ {{\cos }^{-1}}( \frac{19}{35} ) $
D) $ {{\cos }^{-1}}( \frac{9}{35} ) $
Show Answer
Answer:
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}. $
