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

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are three non-coplanar vectors, then $(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}) \cdot[(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{c}})]$ equals
  1. 0
  2. $[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]$
  3. $2[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]$
  4. $-[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]$

Solution

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

Asked in: MHT CET 2024 (09 May Shift 1)

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