$$\begin{gathered} \mathrm{Let} \stackrel{\rightharpoonup }{\mathrm{a}}, \stackrel{\rightharpoonup }{\mathrm{b}} \mathrm{and} \stackrel{\rightharpoonup }{\mathrm{c}} \mathrm{be} \mathrm{three} \mathrm{unit} \mathrm{vectors} \mathrm{such} \\ \mathrm{that} \stackrel{\rightharpoonup }{\mathrm{a}}+\stackrel{\rightharpoonup }{\mathrm{b}}+\stackrel{\rightharpoonup }{\mathrm{c}}=\stackrel{\rightharpoonup }{0}. \mathrm{If} \overline{\mathrm{a}}.\overline{\mathrm{b}}+\overline{\mathrm{b}}.\overline{\mathrm{c}}+\overline{\mathrm{c}}.\overline{\mathrm{a}} \mathrm{and} \\ \stackrel{\rightharpoonup }{\mathrm{d}}=\overline{\mathrm{a}}\times \overline{\mathrm{b}}+\overline{\mathrm{b}}\times \overline{\mathrm{c}}+\overline{\mathrm{c}}\pm \overline{\mathrm{a}} \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair}, \\ \left(\mathrm{\lambda }, \stackrel{\rightharpoonup }{\mathrm{d}}\right) \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

$\begin{array}{rcl}\left|\overline{\mathrm{a}}\right|& =& \left|\overline{\mathrm{b}}\right|=\left|\overline{\mathrm{c}}\right|=1\\ \overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}& =& \overline{\mathrm{O}} \Rightarrow {\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\right|}^2=0\\ & \Rightarrow & 1+1+1+2(\overline{\mathrm{a}}.\overline{\mathrm{b}}+\overline{\mathrm{b}}.\overline{\mathrm{c}}+\overline{\mathrm{c}}.\overline{\mathrm{a}})=0\\ & \Rightarrow & 3+2\mathrm{\lambda }=0 \Rightarrow \mathrm{\lambda }=\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ \overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}& =& \overline{\mathrm{o}} \Rightarrow \overline{\mathrm{a}}\times \overline{\mathrm{b}}=\overline{\mathrm{b}}\times \overline{\mathrm{c}}=\overline{\mathrm{c}}\times \overline{\mathrm{a}}\\ \mathrm{d}& =& \overline{\mathrm{a}}\times \overline{\mathrm{b}}+\overline{\mathrm{b}}\times \overline{\mathrm{c}}+\overline{\mathrm{c}}\times \overline{\mathrm{a}}=3(\overline{\mathrm{a}}\times \overline{\mathrm{b}})=3(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\\ & =& 3(\overline{\mathrm{c}}\times \overline{\mathrm{a}})\end{array}$