Let $\overline{b}=3\overline{i}-2\overline{j}+\overline{k}$ and $\overline{c}=\overline{i}-\overline{j}-\overline{k}$ be two vectors. If $\overline{a}$ is a vector such that $\overline{a}+\overline{b}+\overline{c}=0$ then $|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|=$

$\overline{b}=3\overline{i}-2\overline{j}+\overline{k},\overline{c}=\overline{i}-\overline{j}-\overline{k}$ and $\overline{a}+\overline{b}+\overline{c}=0$
$\overline{a}=-\overline{b}-\overline{c}=-3\overline{i}+2\overline{j}-\overline{k}-\overline{i}+\overline{j}+\overline{k}$
$=-4\overline{i}+3\overline{j}$
Now, $\overline{a}\times \overline{b}=\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ -4& 3& 0\\ 3& -2& 1\end{array}\right|=3\overline{i}+4\overline{j}-\overline{k}$
$\overline{b}\times \overline{c}=\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ 3& -2& 1\\ 1& -1& -1\end{array}\right|=3\overline{i}+4\overline{j}-\overline{k}$ and $\overline{c}\times \overline{a}=\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ 1& -1& -1\\ -4& 3& 0\end{array}\right|=3\overline{i}+4\overline{j}-\overline{k}$
So $\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}=9\overline{i}+12\overline{j}-3\overline{k}$ and $\left|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}\right|=\sqrt{81+144+9}=\sqrt{234}$