If $[\bar{a} \quad \bar{b} \quad \bar{c}]=4$, then volume of parallelopiped with coterminus edges $\bar{a}+2…

If $[\bar{a} \quad \bar{b} \quad \bar{c}]=4$, then volume of parallelopiped with coterminus edges $\bar{a}+2 \bar{b}$, $\bar{b}+2 \bar{c}, \quad \bar{c}+2 \bar{a}$ is
  1. 36 units $^{3}$
  2. 32 units $^{3}$
  3. 20 units $^{3}$
  4. 40 units $^{3}$

Solution

Given $\bar{a} \cdot(\bar{b} \times \bar{c})=4$ $\therefore$ Volume of parallelopiped $\begin{array}{l} =(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}}+2 \overline{\mathrm{c}}) \times(\overline{\mathrm{c}}+2 \overline{\mathrm{a}})] \\ =(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+2(\overline{\mathrm{c}} \times \overline{\mathrm{c}})+4(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \\ =[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]+[2 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})]+[4 \overline{\mathrm{a}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})]+ \\ =[2 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]+[4 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})]+[8 \overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \\ =[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]+0+[8 \overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \\ =9[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]=9(4)=36 \end{array}$

Asked in: MHT CET 2020 (12 Oct Shift 1)

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