If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and $\bar{p}=\frac{\bar{b} \times \bar{c}}{[\bar{a}…

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and $\bar{p}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{q}=\frac{\bar{c} \times \bar{a}}{[a} \bar{b} \overline{c]}, \bar{r}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}$, then $\bar{a} \cdot \bar{p}+\bar{b} \cdot \bar{q}+\bar{c} \cdot \bar{r}=$
  1. 2
  2. 1
  3. 0
  4. 3

Solution

Given that, $\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{\left[\begin{array}{ccc}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]}, \quad \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{\left[\begin{array}{ccc}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right], \quad \mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]}$ $\bar{a} \cdot \bar{p}=\frac{\bar{a} \cdot(\bar{b} \times \bar{c})}{\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]}=1$...(1) Similarly, $\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}=\frac{\mathrm{b} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}=1$...(2) and $\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\frac{(\overline{\mathrm{a}} \times \overline{\mathrm{b}})}{\left[\begin{array}{ccc}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]}=1$..(3) $\therefore \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=1+1+1=3 \quad \ldots[$ from $(1),(2) \&(3)]$

Asked in: MHT CET 2020 (19 Oct Shift 2)

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