A tetrahedron has vertices P(1,2,1), Q(2,1,3), R(-1,1,2) and O(0,0,0).

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:

Accepted Answer

$$\overrightarrow {OP}  \times \overrightarrow {OQ}  = \left( {\widehat i + 2\widehat j + \widehat k} \right) \times \left( {2\widehat i + \widehat j + 3\widehat k} \right)$$

= 5$$\widehat i$$ - $$\widehat j$$ - 3$$\widehat k$$

$$\overrightarrow {PQ}  \times \overrightarrow {PR}  = \left( {\widehat i - \widehat j + 2\widehat k} \right) \times \left( { - 2\widehat i - \widehat j + \widehat k} \right)$$

= $$\widehat i$$ - 5$$\widehat j$$ - 3$$\widehat k$$

cos$$\theta $$ = $${{5 + 5 + 9} \over {{{\left( {\sqrt {25 + 9 + 1} } \right)}^2}}} = {{19} \over {35}}$$

Solutions · 1