Prove that $\left|\begin{array}{ccc}{{v}_{2}}^{2}& -{v}_{1}{v}_{2}& {{v}_{1}}^{2}\\ E& F& G\\ L& M& N\end{array}\right|=0$ then $\overset{\to }{v}$ is principal vector where nonzero tangent vector $\overset{\to }{v}={v}_{1}{x}_{u}+{v}_{2}{x}_{v}$. ( $E$,$F$,$G$= first fundamental form and $L$, $M$ , $N$ =second fundamental form)
I can't show that this vector is principal. I mean I took determinant but I can not see the relation between determinant and principal vector? Any help will be appreciated.