For what value of m, the points with position vectors  $10\stackrel{\wedge }{\mathrm{i}}+3\stackrel{\wedge }{\mathrm{j}},12\stackrel{\wedge }{\mathrm{i}}-5\stackrel{\wedge }{\mathrm{j}}$and $\mathrm{m}\stackrel{\wedge }{\mathrm{i}}+11\stackrel{\wedge }{\mathrm{j}}$ are collinear?

The vectors $10\stackrel{\wedge }{\mathrm{i}}+3\stackrel{\wedge }{\mathrm{j}},12\stackrel{\wedge }{\mathrm{i}}-5\stackrel{\wedge }{\mathrm{j}}$and $m\stackrel{\wedge }{\mathrm{i}}+11\stackrel{\wedge }{\mathrm{j}}$ are

collinear, if area of triangle formed by their position vectors is zero. 

$\Rightarrow \left|\begin{array}{ccc}10& 3& 1\\ 12& -5& 1\\ \mathrm{m}& 11& 1\end{array}\right|=0$

Applying ${\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_1$ and ${\mathrm{R}}_3\to {\mathrm{R}}_3-{\mathrm{R}}_1$

$\begin{array}{l}\Rightarrow \left|\begin{array}{cccc}10& 3& 1& \\ 2& -8& 0& \\ & \mathrm{m}-10& 8& 0\end{array}\right|=0\\ \Rightarrow 16+8\mathrm{m}-80=0\\ \Rightarrow 8\mathrm{m}-64=0\\ \Rightarrow \mathrm{m}=8\end{array}$