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 :

• A. ${\cos }^{-1}\left(\frac{9}{35}\right)$
• B. ${\cos }^{-1}\left(\frac{19}{35}\right)$
• C. ${\cos }^{-1}\left(\frac{17}{31}\right)$
• D. ${\cos }^{-1}\left(\frac{7}{31}\right)$

Answer 1

Answer: (B)

$\overrightarrow{OP}\times \overrightarrow{OQ}=\left(\hat{i}+2\hat{j}+\hat{k}\right)\times \left(2\hat{i}+\hat{j}+3\hat{k}\right)$
$5\hat{i}-\hat{j}-3\hat{k}$
$\overrightarrow{PQ}\times \overrightarrow{PR}=\left(\hat{i}-\hat{j}+2\hat{k}\right)\times \left(-2\hat{i}-\hat{j}+\hat{k}\right)$
$\hat{i}-5\hat{j}-3\hat{k}$
$\cos \theta =\frac{5+5+9}{{\left(\sqrt{25+9+1}\right)}^2}=\frac{19}{35}$