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Question
By computing the shortest distance determine whether the following pairs of lines intersect or not : \[\overrightarrow{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \overrightarrow{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\]
Solution 1
\[\overrightarrow{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \overrightarrow{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\]
Comparing the given equations with the equations
\[\overrightarrow{r} = \overrightarrow{a_1} + \lambda \overrightarrow{b_1} \text{ and } \overrightarrow{r} = \overrightarrow{a_2} + \mu \overrightarrow{b_2}\]
We get ,
\[\overrightarrow{a_1} = \hat{i} - \hat{j} \]
\[ \overrightarrow{a_2} = 2 \hat{i} - \hat{j} \]
\[ \overrightarrow{b_1} = 2 \hat{i} + \hat{k} \]
\[ \overrightarrow{b_2} = \hat{i} + \hat{j} - \hat{k} \]
\[ \therefore \overrightarrow{a_2} - \overrightarrow{a_1} = \hat{i} \]
\[\text{ and } \overrightarrow{b_1} \times \overrightarrow{b_2} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 2 & 0 & 1 \\ 1 & 1 & - 1\end{vmatrix}\]
\[ = - \hat{i}+ 3 \hat{j} + 2 \hat{k} \]
\[\left( \overrightarrow{a_2} - \overrightarrow{a_1} \right) . \left( \vec{b_1} \times \overrightarrow{b_2} \right) = \left( \hat{i} \right) . \left( - \hat{i} + 3 \hat{j} + 2 \hat{k} \right)\]
\[ = - 1\]
\[\text{ We observe } \]
\[\left( \overrightarrow{a_2} - \overrightarrow{a_1} \right) . \left( \overrightarrow{b_1} \times \overrightarrow{b_2} \right) \neq 0\]
\[\text{ Thus, the given lines do not intersect } .\]
Solution 2
\[\vec{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\]
Comparing the given equations with the equations
\[\vec{r} = \vec{a_1} + \lambda \vec{b_1} \text{ and } \vec{r} = \vec{a_2} + \mu \vec{b_2}\]
\[\vec{a_1} = \hat{i} - \hat{j} \]
\[ \vec{a_2} = 2 \hat{i} - \hat{j} \]
\[ \vec{b_1} = 2 \hat{i} + \hat{k} \]
\[ \vec{b_2} = \hat{i} + \hat{j} - \hat{k} \]
\[ \therefore \vec{a_2} - \vec{a_1} = \hat{i} \]
\[\text{ and } \vec{b_1} \times \vec{b_2} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 2 & 0 & 1 \\ 1 & 1 & - 1\end{vmatrix}\]
\[ = - \hat{i}+ 3 \hat{j} + 2 \hat{k} \]
\[\left( \vec{a_2} - \vec{a_1} \right) . \left( \vec{b_1} \times \vec{b_2} \right) = \left( \hat{i} \right) . \left( - \hat{i} + 3 \hat{j} + 2 \hat{k} \right)\]
\[ = - 1\]
\[\text{ We observe } \]
\[\left( \vec{a_2} - \vec{a_1} \right) . \left( \vec{b_1} \times \vec{b_2} \right) \neq 0\]
\[\text{ Thus, the given lines do not intersect } .\]
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