Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}.$   $\mathrm{If} \mathrm{\lambda }=\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}·\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}·\overrightarrow{\mathrm{a}}\text{ and}$$\overrightarrow{d}=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$  then the ordered pair, $\left(\lambda ,d\right)$ is equal to : 

$\overline{a}+\overline{b}+\overline{c}=0$

$\left(i\right)then\overline{a}\times \overline{b}=\overline{b}\times \overline{c}=\overline{c}\times \overline{a}$

$\overline{d}=\overline{a}\times \overline{b}+\overline{a}\times \overline{b}+\overline{a}\times \overline{b}=3\overline{a}\times \overline{b}$

$\left(ii\right)\overline{a}+\overline{b}+\overline{c}=0$

$S.O.B.S$

$|\overline{a}{|}^2+|\overline{b}{|}^2+|\overline{c}{|}^2+2\overline{a}.\overline{b}+2\overline{b}.\overline{c}+2\overline{c}.\overline{c}=0$

$\overline{a}.\overline{b}+\overline{b}.\overline{c}+\overline{c}.\overline{a}=\frac{-3}{2}$

$\lambda =-\frac{3}{2}$

$(\lambda ,d)=\left(\frac{-3}{2},3\overline{a}\times \overline{b}\right)$