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

The correct option is B ${\cos }^{-1}\left(\frac{19}{35}\right)$

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