If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are non-coplanar vectors and…

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

Solution

$\begin{aligned} & 2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}=\frac{2(\overline{\mathrm{~b}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}=\frac{2[\overline{\mathrm{~b}} \overline{\mathrm{c}} \mathrm{a}]}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}=2 . \\ & \overline{\mathrm{b}} \cdot \overline{\mathrm{q}}=\frac{(\overline{\mathrm{c}} \times \overline{\mathrm{a}}) \cdot \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}=\frac{[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{~b}}]}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}=1 \\ & \overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\frac{(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \cdot \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}=\frac{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}=1 \\ & \therefore \quad 2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=2+1+1=4 \end{aligned}$

Asked in: MHT CET 2024 (04 May Shift 2)

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