Mathematics - Vector Question with Solution | TestHub

Since $\vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=\lambda \hat{i}+\hat{j}+(2\lambda -1\hat{k})$ are coplanar therefore $[\vec{a}\vec{b}\vec{c}]=0$ $$\begin{array}{rl}& \text{ i.e., }\left|\begin{array}{ccc}1& 2& \lambda \\ -2& 3& 1\\ 3& -1& 2\lambda -1\end{array}\right|=0\\ & \Rightarrow 1(6\lambda -2)-2(-4\lambda -1)+\lambda (-7)=0\\ & \Rightarrow (6\lambda -2)+8\lambda +2+2+2\lambda -9\lambda =0\\ & \Rightarrow 7\lambda =0\Rightarrow \lambda =0\end{array}$$