If $p = \frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, q = \frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}$ and $\overline{r} = \frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}$ where $\overline{a}, \overline{b}$ and $\overline{c}$ are three non-coplanar vectors, then the value of the expression $(\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{p} + \overline{q} + \overline{r})$ is

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Detailed Solution

$(\overline{a} + \overline{b} + \overline{c}) \cdot (\overline{p} + \overline{q} + \overline{r}) \overline{a} \cdot \overline{p} + \overline{b} \cdot \overline{q} + \overline{c} \cdot \overline{r} = 3 (\sin c e a \cdot q = a \cdot r = 0, s i m i l a r l y r e m a i n i n g a n d \overline{a} \cdot \overline{p} = a . p = a \cdot \frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]} = \frac{[\overline{a} \overline{b} \overline{c}]}{[\overline{a} \overline{b} \overline{c}]} = 1, s i m i l a r l y r e m a i n i n g)$