Question

Prove that:-
$\overline{d}.\left[\overline{a}\times \{\overline{b}\times (\overline{c}\times \overline{d}\left)\right\}\right]=(\overline{b}.\overline{d})\left[\overline{a}\overline{c}\overline{d}\right]$

Answer 1

$\begin{array}{c}\overrightarrow{d}.\left[\overrightarrow{a}\times \left\{\overrightarrow{b}\times \left(\overrightarrow{c}\times \overrightarrow{d}\right)\right\}\right]\\ =\overrightarrow{d}.\left[\overrightarrow{a}\times \left(\left(\overrightarrow{b}\times \overrightarrow{d}\right)\overrightarrow{c}-\left(\overrightarrow{b}\times \overrightarrow{c}\right)\overrightarrow{d}\right)\right]\\ =\overrightarrow{d}.\left[\left(\overrightarrow{b}\times \overrightarrow{d}\right)\left(\overrightarrow{a}\times \overrightarrow{c}\right)-\left(\overrightarrow{b}\times \overrightarrow{c}\right)\left(\overrightarrow{a}\times \overrightarrow{d}\right)\right]\\ =\left(\overrightarrow{b}\times \overrightarrow{d}\right)\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]\\ Henceproved.\end{array}$