Question

â cap + b cap + c cap = 0 . find â×b + b×c + c×â = ?

Answer 1

$\vec{a} + \vec{b} = - \vec{c}$

vector product is anti commutative:  $\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}$

vector product is distributive over vector addition.
$\vec{a} \times (\vec{b} +\vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

vector  multiplied by itself is null vector.

so now LHS =
$\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec {c} \times \vec{a} \\\\ \vec{a} \times \vec{b} + \vec{b} \times (-\vec{a} - \vec{b}) + (-\vec{a} - \vec{b}) \times \vec{a} \\\\ \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b} + -\vec{a} \times \vec{a} - \vec{b} \times \vec{a} \\\\ = 3 \vec{a} \times \vec{b} =3 \vec{b} \times \vec{c} =3\vec{c} \times \vec{a}$