From Mechanics

\displaystyle \begin{align} & m(5\mathbf{i}+V\mathbf{j})+\lambda m(2\mathbf{i}-V\mathbf{j})=(m+\lambda m)(3\mathbf{i}-2\mathbf{j}) \\ & (5m+2\lambda m-3m-3\lambda m)\mathbf{i}+(mV-\lambda mV+2m+2\lambda m)\mathbf{j}=0\mathbf{i}+0\mathbf{j} \\ \end{align}

Consider the \displaystyle \mathbf{i} component. That is the \displaystyle \mathbf{i} component on the left hand side must be the same as the \displaystyle \mathbf{i} component on the right hand side, that is zero.

\displaystyle \begin{align} & 5m+2\lambda m-3m-3\lambda m=0 \\ & 2-\lambda =0 \\ & \lambda =2 \\ \end{align}