If $\overline{\mathrm{a}} \mathrm{and} \overline{\mathrm{b}}$ are two non-collinear vectors, then the points ${\mathrm{l}}_1\overline{\mathrm{a}}+{\mathrm{m}}_1\overline{\mathrm{b}},{\mathrm{l}}_2\overline{\mathrm{a}}+{\mathrm{m}}_2\overline{\mathrm{b}}\text{ and }{\mathrm{l}}_3\overline{\mathrm{a}}+{\mathrm{m}}_3\overline{\mathrm{b}}$ are collinear if

Given $\overline{\mathrm{a}} \mathrm{and} \overline{\mathrm{b}}$ are non-collinear. Then ${\mathrm{l}}_1\overline{\mathrm{a}}+{\mathrm{m}}_1\overline{\mathrm{b}},{\mathrm{l}}_2\overline{\mathrm{a}}+{\mathrm{m}}_2\overline{\mathrm{b}},{\mathrm{l}}_3\overline{\mathrm{a}}+{\mathrm{m}}_3\overline{\mathrm{b}}$ are collinear if
$\left|\begin{array}{ccc}{\mathrm{l}}_1& {\mathrm{l}}_2& {\mathrm{l}}_3\\ {\mathrm{m}}_1& {\mathrm{m}}_2& {\mathrm{m}}_3\\ 1& 1& 1\end{array}\right|=0\Rightarrow {\mathrm{l}}_1\left({\mathrm{m}}_2-{\mathrm{m}}_3\right)+{\mathrm{l}}_2\left({\mathrm{m}}_3-{\mathrm{m}}_1\right)+{\mathrm{l}}_3\left({\mathrm{m}}_1-{\mathrm{m}}_3\right)=0\Rightarrow \sum {\mathrm{l}}_1\left({\mathrm{m}}_2-{\mathrm{m}}_3\right)=0$