Question

If $\overline{a},\overline{b},\overline{c}$ are coplanar, prove that
$\left(\overline{a}+\overline{b}\right).\left[\left(\left(\overline{b}+\overline{c}\right)\times \overline{a}\right)+\left(\left(\overline{b}+\overline{a}\right)\times \overline{b}\right)\right]=0$

Answer 1

$as\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}arecoplanar,hence\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=0$

$nowLHS=(\overrightarrow{a}+\overrightarrow{b})\cdot \left[\right(\overrightarrow{b}\times \overrightarrow{a})+(\overrightarrow{c}\times \overrightarrow{a})+(\overrightarrow{b}\times \overrightarrow{b})+(\overrightarrow{a}\times \overrightarrow{b}\left)\right]$

$=(\overrightarrow{a}+\overrightarrow{b})\cdot \left[\right(\overrightarrow{b}\times \overrightarrow{a})+(\overrightarrow{c}\times \overrightarrow{a})+(\overrightarrow{a}\times \overrightarrow{b}\left)\right]=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{a})+\overrightarrow{a}.(\overrightarrow{c}\times \overrightarrow{a})+\overrightarrow{a}.(\overrightarrow{a}\times \overrightarrow{b})+\overrightarrow{b}.(\overrightarrow{b}\times \overrightarrow{a})+\overrightarrow{b}.(\overrightarrow{c}\times \overrightarrow{a})+\overrightarrow{b}.(\overrightarrow{a}\times \overrightarrow{b})$

$=0+0+0+0+0+0=0$