Question

Find the value of λ so that the following vector is coplanar:

\[\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}\]

Answer 1

Given:

\[ \vec{a} =\hat { i} + 3 \hat{j} \]

\[ \vec{b} = 5 \hat {k} \]

\[ \vec{c} = \lambda \hat {i} - \hat {j} \]

\[\text { We know that vectors } \vec{a} , \vec{b} , \vec{c}\text {  are coplanar iff } \left[ \vec{a} \vec{b} \vec{c} \right] = 0 . \]

\[{\text { It is given that }} \vec{a} , \vec{b} , \vec{c} { \text { are coplanar} } . \]

\[ \therefore \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]

\[ \Rightarrow \begin{vmatrix}1 & 3 & 0 \\ 0 & 0 & 5 \\ \lambda & - 1 & 0\end{vmatrix} = 0 \]

\[ \Rightarrow 1\left( 0 + 5 \right) - 3\left( 0 - 5\lambda \right) + 0\left( 0 - 0 \right) = 0\]

\[ \Rightarrow 5 + 15\lambda = 0 \]

\[ \Rightarrow \lambda = - \frac{1}{3}\]