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

Engineering

Mathematics

Plane and Its Different Forms

Question

$\cos^{- 1} (\frac{17}{31})$

$\cos^{- 1} (\frac{9}{35})$

$\cos^{- 1} (\frac{19}{35})$

$\cos^{- 1} (\frac{7}{31})$

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Normal to face OPQ is ${\vec{n}}_{1} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 2 & 1 & 3 \end{matrix} | \equiv$ < 5, – 1, – 3 >

Normal to face PQR is ${\vec{n}}_{2} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 1 & 3 \\ - 2 & - 1 & 1 \end{matrix} | \equiv$ < 1, – 5, – 3 >

$cosθ = | \frac{5 + 5 + 9}{35} | = \frac{19}{35}$