Question

If $\overline{a}+\overline{b}+\overline{c}=0$ prove that $\overline{a}\times \overline{b}=\overline{b}\times \overline{c}=\overline{c}\times \overline{a}$

Answer 1

$\begin{array}{c}given\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0\\ a\times \left(a+b+c\right)=0\\ a\times a+a\times b+a\times c=0\\ a\times b=-a\times c=c\times a.......\left(1\right)\\ similarly,b\times \left(a+b+c\right)=0\\ From\left(1\right)and\left(2\right)\\ a\times b=b\times c=c\times a\\ Hence,\\ proved.\end{array}$