Question

If four points $A\left(\overline{a}\right),B\left(\overline{b}\right),C\left(\overline{c}\right)$ and $D\left(\overline{d}\right)$ are coplanar then show that
$\left[\overline{a}\overline{b}\overline{c}\right]+\left[\overline{b}\overline{c}\overline{d}\right]+\left[\overline{c}\overline{a}\overline{d}\right]=\left[\overline{a}\overline{b}\overline{c}\right]$.

Answer 1

$\begin{array}{c}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]+\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]+\left[\overrightarrow{c}\overrightarrow{a}\overrightarrow{d}\right]=\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\\ L.H.S\\ \Rightarrow \overrightarrow{a}\left(\overrightarrow{b}\times \overrightarrow{c}\right)+\overrightarrow{b}\left(\overrightarrow{c}\times \overrightarrow{d}\right)+\overrightarrow{c}\left(\overrightarrow{a}\times \overrightarrow{d}\right)\\ \Rightarrow 0+0+0=0\to \left(i\right)\\ R.H.S\\ \overrightarrow{a}\left(\overrightarrow{b}\times \overrightarrow{c}\right)=0\to \left(ii\right)\end{array}$

From (i) and (ii)

proved LHS = RHS

Reason {Vectors are co-planar & scalar triple product that's y 0}