If \(\bar{a}, \bar{b}, \bar{c}\) are non \(-\) coplanar vectos and \((\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{b}}+\overline{\mathrm{b}} \times \overline{\mathrm{c}}+\overline{\mathrm{c}} \times \overline{\mathrm{a}})=\mathrm{k}[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]\),
then value of \(\mathrm{k}\) is

\((\bar{a}+\bar{b}+\bar{c}) \cdot[(\bar{a} \times \bar{b})+(\mathrm{b} \times \bar{c})+(\bar{c} \times \bar{a})] \)
\(=[\bar{a} \cdot(\bar{a} \times \bar{b})]+[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{c} \times \bar{a})]+[\bar{b} \cdot(\bar{a} \times \bar{b})]\) \(+~[\bar{b} \cdot(\bar{b} \times \bar{c})] \)
\(+[\bar{b} \cdot(\bar{c} \times \bar{a})]+[\bar{c} \cdot(\bar{a} \times \bar{b})]+[\bar{c} \cdot(\bar{b} \times \bar{c})]+[\bar{c} \cdot(\bar{c} \times \bar{a})] \)
\(= 0+[\bar{a} \cdot(\bar{b} \times \bar{c})]+0+0+0+[\bar{b} \cdot(\bar{c} \times \bar{a})]+[\bar{c} ~\cdot\) \((\bar{a} \times \bar{b})]+0+0 \)
\(= 3[\bar{a} \cdot(\bar{b} \times \bar{c})]=3\left[\begin{array}{lll}a \bar{b} \bar{c}\end{array}\right] \Rightarrow \mathrm{k}=3\)