• # question_answer If $\left[ \vec{a}\times \vec{b}\,\vec{b}\times \vec{c}\,\vec{c}\times \vec{a} \right]=\lambda {{\left[ \vec{a}\,\vec{b}\,\,\vec{c} \right]}^{2}}$then$\lambda$is equal to:   JEE Main  Solved  Paper-2014 A) 2                                              B) 3 C) 0                                              D) 1

$\left[ \overline{a}\times \overline{b}\,\,\overline{b}\times \overline{c}\,\,\overline{c}\times \overline{a} \right]$Let$\overline{u}=\overline{b}\times \overline{c}$ $=\left( \vec{a}\times \vec{b} \right).\left\{ \left( \vec{b}\times \overline{c} \right)\times \left( \overline{c}\times \vec{a} \right) \right\}$ $=\left( \vec{a}\times \vec{b} \right).\left\{ \vec{u}\times (\overline{c}\times \overline{a}) \right\}$ $=\left( \vec{a}\times b \right).\left\{ \left( \overline{u}\cdot \vec{a} \right)\vec{c}-(\overline{u}\cdot \overline{c})\overline{a} \right\}$ $=\left( \vec{a}\times \overline{b} \right).\left\{ \left[ \overline{b}\,\overline{c}\,\overline{a} \right]\vec{c}-\left[ \overline{b}\overline{c}\overline{c} \right]\vec{a} \right\}$ $=\left[ \overline{b}\,\overline{c}\,\overline{a} \right]\left[ \vec{a}\,\overline{b}\,\overline{c} \right]$                $={{\left[ \overline{b}\,\overline{c}\,\overline{a} \right]}^{2}}$$\Rightarrow$$\lambda =1$