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} $