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Question
Find the equation of the line passing through the point \[\hat{i} + \hat{j} - 3 \hat{k} \] and perpendicular to the lines \[\overrightarrow{r} = \hat{i} + \lambda\left( 2 \hat{i} + \hat{j} - 3 \hat{k} \right) \text { and } \overrightarrow{r} = \left( 2 \hat{i} + \hat{j} - \hat{ k} \right) + \mu\left( \hat{i} + \hat{j} + \hat{k} \right) .\]
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Solution
The required line is perpendicular to the lines parallel to the vectors \[\overrightarrow{b_1} = 2 \hat{i} + \hat{j} - 3 \hat{k} \text{ and } \overrightarrow{b_2} = \hat{ i} + \hat{j}+ \hat{k} \] So, the required line is parallel to the vector
\[\overrightarrow{b} = \overrightarrow{b_1} \times \overrightarrow{b_2}\]
Now,
\[\overrightarrow{b} = \overrightarrow{b_1} \times \overrightarrow{b_2} \]
\[ = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & - 3 \\ 1 & 1 & 1\end{vmatrix}\]
\[ = 4 \hat{i} - 5 \hat{j} + \hat{k}\]
Equation of the required line passing through the point
\[\left( \hat{i} + \hat{j} - 3 \hat{k} \right)\] and parallel to
\[\left( 4 \hat{i} - 5 \hat{j} + \hat{k} \right)\] is
\[\overrightarrow{r} = \left( \hat{i} + \hat{j} - 3 \hat{k} \right) + \lambda\left( 4 \hat{i} - 5 \hat{j} + \hat{k} \right)\]
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