If $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ are coterminous edges of a parallelepiped, then its…

If $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ are coterminous edges of a parallelepiped, then its volume is
  1. 0
  2. $4[\overline{\mathrm{b}} \overline{\mathrm{a}} \overline{\mathrm{c}}]$
  3. $3[\bar{a} \bar{c} \bar{b}]$
  4. $2[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]$

Solution

The volume of required parallelepiped $\begin{aligned} & =(\bar{a}+\bar{b}) \cdot[(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})] \\ & =(\bar{a}+\bar{b}) \cdot[(\bar{b} \times \bar{c})+(\bar{b} \times \bar{a})+(\bar{c} \times \bar{c})+(\bar{c} \times \bar{a})] \\ & =[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{b} \times \bar{a})]+0+[\bar{a} \times(\bar{c} \times \bar{a})] \\ & +[\bar{b} \cdot(\bar{b} \times \bar{a})]+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\ & =[\bar{a} \cdot(\bar{b} \times \bar{c})]+0+0+0+0+0+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\ & =2 \bar{a} \cdot(\bar{b} \times \bar{c}) \end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 2)

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