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Question
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = 2 \hat{i} - \hat {j} + \hat {k} , \vec{b} = \hat {i} + 2 \hat {j} - 3 \hat {k} , \vec{c} = \lambda \hat {i} + \lambda \hat {j} + 5 \hat {k}\]
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Solution
Given:
\[ \vec{a} = 2 \hat{i} -\hat{ j} + \hat{k} \]
\[ \vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k} \]
\[ \vec{c} = \lambda\hat{i} + \lambda j + 5 \hat { k}\]
\[\text { We know that vectors } \vec{a} , \vec{b} , \vec{c}\text { are coplanar iff } \left[ \vec{a} \vec{b} \vec{c} \right] = 0 . \]
\[\text { It is given that} \vec{a} , \vec{b} , \vec{c} \text { are coplanar } . \]
\[ \therefore \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]
\[ \Rightarrow \begin{vmatrix}2 & - 1 & 1 \\ 1 & 2 & - 3 \\ \lambda & \lambda & 5\end{vmatrix} = 0 \]
\[ \Rightarrow 2\left( 10 + 3\lambda \right) + 1\left( 5 + 3\lambda \right) + 1\left( \lambda - 2\lambda \right) = 0\]
\[ \Rightarrow 8\lambda + 25 = 0 \]
\[ \Rightarrow \lambda = - \frac{25}{8}\]
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