Question

If vectors $\overline{b},\overline{c},\overline{d}$ are not coplanar then prove that
$(\overline{a}\times \overline{b})\times (\overline{c}\times \overline{d})+(\overline{a}\times \overline{c})\times (\overline{d}\times \overline{b})+(\overline{a}\times \overline{d})\times (\overline{b}\times \overline{c})$ is parallel to $\overline{a}$

Answer 1

We know

$(\overrightarrow{A}\times \overrightarrow{B})\times (\overrightarrow{C}\times \overrightarrow{D})=\left[\overrightarrow{A}\overrightarrow{C}\overrightarrow{D}\right]\overrightarrow{B}-\left[\overrightarrow{B}\overrightarrow{C}\overrightarrow{D}\right]\overrightarrow{A}$

$\therefore (\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{b}-\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}$ ___ (i)

$(\overrightarrow{a}\times \overrightarrow{c})\times (\overrightarrow{d}\times \overrightarrow{b})=\left[\overrightarrow{a}\overrightarrow{d}\overrightarrow{b}\right]\overrightarrow{c}-\left[\overrightarrow{c}\overrightarrow{d}\overrightarrow{b}\right]\overrightarrow{a}$ ___ (ii)

$(\overrightarrow{a}\times \overrightarrow{d})\times (\overrightarrow{b}\times \overrightarrow{c})=\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{d}-\left[\overrightarrow{d}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{a}$ ___ (iii)

$=-\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{b}-\left[\overrightarrow{a}\overrightarrow{d}\overrightarrow{b}\right]\overrightarrow{c}$ ___ (iv)

$\overrightarrow{a}=(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})+(\overrightarrow{a}\times \overrightarrow{c})\times (\overrightarrow{d}\times \overrightarrow{b})+(\overrightarrow{a}\times \overrightarrow{d})\times (\overrightarrow{b}\times \overrightarrow{c})$

$=\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{b}-\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}+\left[\overrightarrow{a}\overrightarrow{d}\overrightarrow{b}\right]\overrightarrow{c}-\left[\overrightarrow{c}\overrightarrow{d}\overrightarrow{b}\right]\overrightarrow{a}-\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{b}-\left[\overrightarrow{a}\overrightarrow{d}\overrightarrow{b}\right]\overrightarrow{c}$

$=-\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}-\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}$

$\overrightarrow{u}=-2\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}$

$\therefore \overrightarrow{u}=\lambda \overrightarrow{a}$

Hence, $\overrightarrow{u}$ & $\overrightarrow{a}$ are parallel vector.

$\therefore$ Given vector is paralle to $\overrightarrow{a}$ (proved)