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Question
Show that the lines \[\vec{r} = \left( 2 \hat{j} - 3 \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)\] are coplanar. Also, find the equation of the plane containing them.
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Solution
\[\text{ We know that the lines } \vec{r} = \vec{a_1} + \lambda \vec{b_1} \text{ and } \vec{r} = \vec{a_2} + \mu \vec{b_2} \text{ are coplanar if} \]
\[ \vec{a_1} . \left( \vec{b_1} \times \vec{b_2} \right) = \vec{a_2} . \left( \vec{b_1} \times \vec{b_2} \right) \text{ and the equation of the plane containing them is } \]
\[ \vec{r} . \left( \vec{b_1} \times \vec{b_2} \right) = \vec{a_1} . \left( \vec{b_1} \times \vec{b_2} \right) . \]
\[\text{ Here } ,\]
\[ \vec{a_1} = 0 \hat{i} + 2 \hat{j} - 3 \hat{k} ; \vec{b_1} = \hat{i} + 2 \hat{j} + 3 \hat{k} ; \vec{a_2} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} ; \vec{b_2} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \]
\[ \vec{b_1} \times \vec{b_2} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 2 & 3 & 4\end{vmatrix} = - \hat{i} + 2 \hat{j} - \hat{k} \]
\[ \vec{a_1} . \left( \vec{b_1} \times \vec{b_2} \right) = \left( 0 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) . \left( - \hat{i} + 2 \hat{j} - \hat{k} \right) = 0 + 4 + 3 = 7\]
\[ \vec{a_2} . \left( \vec{b_1} \times \vec{b_2} \right) = \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right) . \left( - \hat{i} + 2 \hat{j} - \hat{k} \right) = - 2 + 12 - 3 = 7\]
\[\text{ Clearly } , \vec{a_1} . \left( \vec{b_1} \times \vec{b_2} \right) = \vec{a_2} . \left( \vec{b_1} \times \vec{b_2} \right)\]
\[\text{ Hence, the given lines are coplanar.} \]
\[\text{ The equation of the plane containing the given lines is } \]
\[ \vec{r} . \left( \vec{b_1} \times \vec{b_2} \right) = \vec{a_1} . \left( \vec{b_1} \times \vec{b_2} \right)\]
\[ \Rightarrow \vec{r} . \left( - \hat{i} + 2 \hat{j} - \hat{k} \right) = \left( 0 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) . \left( - \hat{i} + 2 \hat{j} - \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( - \hat{i} + 2 \hat{j} - \hat{k} \right) = 7\]
\[ \Rightarrow \vec{r} . \left( \hat{i} - 2 \hat{j} + \hat{k} \right) + 7 = 0\]
