question_answer

Find \[\lambda \] so that the vectors \[\vec{a}=2\hat{i}-\hat{j}+\hat{k},\] \[\vec{b}=\hat{i}+2\hat{j}-3\hat{k}\] and \[\vec{c}=3\hat{i}+\lambda \hat{j}+5\hat{k}\] are coplanar.\

Answer:

We know that vector \[\vec{a},\vec{b},\vec{c}\] are coplanar iff             \[[\begin{matrix}    {\vec{a}} & {\vec{b}} & {\vec{c}}  \\ \end{matrix}]=0.\] It is given that \[\vec{a},\vec{b},\vec{c}\] are coplanar. \[\therefore \] \[[\begin{matrix}    {\vec{a}} & {\vec{b}} & {\vec{c}}  \\ \end{matrix}]=0\] \[\Rightarrow \] \[\left| \begin{matrix}    2 & -\,1 & 1  \\    1 & 2 & -\,3  \\    3 & \lambda  & 5  \\ \end{matrix} \right|=0\] \[\Rightarrow \,\,2\,(10+3\lambda )+1\,(5+9)+(\lambda -6)=0\] \[\Rightarrow \]   \[7\lambda +28=0\] \[\Rightarrow \] \[\lambda =-\,4\]