For any non-zero vectors $\bar{a}, \bar{b}, \bar{c}$, the value $\overline{\mathrm{a}}…

For any non-zero vectors $\bar{a}, \bar{b}, \bar{c}$, the value $\overline{\mathrm{a}} \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})]$ is
  1. $2[\bar{a} \bar{b} \bar{c}]$
  2. $[\bar{a} \bar{b} \bar{c}]$
  3. $[\overline{\mathrm{a}} \overline{\mathrm{c}} \overline{\mathrm{b}}]$
  4. 0

Solution

$\begin{aligned} & \bar{a} \cdot[(\bar{b} \times \bar{c}) \times(\bar{a}+\bar{b}+\bar{c})] \\ & =\bar{a} \cdot[(\bar{b} \times \bar{a})+(\bar{b} \times \bar{b})+(\bar{b} \times \bar{c})+(\bar{c} \times \bar{a})+(\bar{c} \times \bar{b})+(\bar{c} \times \bar{c})] \\ & =\bar{a} \cdot(\bar{b} \times \bar{a})+\bar{a} \cdot(0)+\bar{a} \cdot(\bar{b} \times \bar{c})+\bar{a} \cdot(\bar{c} \times \bar{a})+\bar{a} \cdot(\bar{c} \times \bar{b})+\bar{a} \cdot(0) \\ & =0+\bar{a} \cdot(\bar{b} \times \bar{c})+0-\bar{a} \cdot(\bar{b} \times \bar{c})=0\end{aligned}$

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

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