Question

The pairs $\overline{a},\overline{b}$ and $\overline{c},\overline{d}$ each determine a plane. Then the planes are parallel if
$\left(\overline{a}\times \overline{c}\right)\times \left(\overline{b}\times \overline{d}\right)=\overline{0}2)\left(\overline{a}\times \overline{c}\right).\left(\overline{b}\times \overline{d}\right)=0$
$\left(\overline{a}\times \overline{b}\right)\times \left(\overline{c}\times \overline{d}\right)=\overline{0}4)\left(\overline{a}\times \overline{b}\right).\left(\overline{c}\times \overline{d}\right)=0$

Answer 1

Given,

$\overline{a},\overline{b}and\overline{c},\overline{d}$ are two planes, both are parallel to each other,

$(\overline{a}\times \overline{b})and(\overline{c}\times \overline{d})$ are two vector products.

$(\overline{a}\times \overline{b})\times (\overline{c}\times \overline{d})=0$ cross of two vector,

$\sin \theta =0$

so that, $\theta =0$ the planes are parallel.