• # question_answer If $2\text{ }\vec{a}+3\text{ }\vec{b}+\vec{c}=0,$ then $\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}$is equal to A)  $6\,(\vec{b}\times \vec{c})$        B)  $3\,(\vec{b}\times \vec{c})$ C)  $2\,(\vec{b}\times \vec{c})$         D)  $\vec{0}$

Given, $2\vec{a}+3\vec{b}+\vec{c}=\vec{0}$ $\Rightarrow$ $2\vec{a}+3\vec{b}=-\vec{c}$ Taking cross product with $\vec{a}$ and $\vec{b}$ respectively, we get $2(\vec{a}\times \vec{a})+3(\vec{a}\times \vec{b})=-\vec{a}\times \vec{c}$ $\Rightarrow$ $3(\vec{a}\times \vec{b})=\vec{c}\times \vec{a}$ ?..(i) and $2(\vec{b}\times \vec{a})+3(\vec{b}\times \vec{b})=-\vec{b}\times \vec{c}$ $\Rightarrow$ $2(\vec{a}\times \vec{b})=\vec{b}\times \vec{c}$ ?(ii) Now, $\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}$ $=\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+3(\vec{a}\times \vec{b})$   [using Eq. (i)] $=4(\vec{a}\times \vec{b})+\vec{b}\times \vec{c}$ $=2(\vec{b}\times \vec{c})+\vec{b}\times \vec{c}$ [using Eq. (ii)] $=3(\vec{b}\times \vec{c})$