Let $\vec{a}, \vec{b}, \vec{c}$ be three non zero vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ Then $\lambda \vec{b} \times \vec{a}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\vec{0}$, where $\lambda$ is equal to:

-2

Clearly $\vec{a}, \vec{b}$  and  $\vec{c}$ represents the sides of a triangle.

It’s area vector,

$=\frac{1}{2} \vec{a} \times \vec{b}$

$=\frac{1}{2} \vec{c} \times \vec{d}=\frac{1}{2} \vec{a} \times \vec{c}$

Thus, $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=-\vec{a} \times \vec{b}$

$\Rightarrow 2 \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=0$

Hence (4) is correct answer.