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Question
Determine whether the following pair 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
\[\overrightarrow{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \overrightarrow{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\]
The position vectors of two arbitrary points on the given lines are
\[\left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) = \left( 1 + 2\lambda \right) \hat{i} - \hat{j}+ \lambda \hat{k} \]
\[\left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right) = \left( 2 + \mu \right) \hat{i} + \left( - 1 + \mu \right) \hat{j} - \mu \hat{k} \]
If the lines intersect, then they have a common point. So, for some values of \[\lambda \text{ and } \mu\]
we must have ,
\[\left( 1 + 2\lambda \right) \hat{i} + - \hat{j} + \lambda \hat{k} = \left( 2 + \mu \right) \hat{i} + \left( - 1 + \mu \right) \hat{j} - \mu \hat{k}\]
Equating the coefficients of \[\hat{i} , \hat{j} \text{ and } \hat{k} \],
we get ,
\[1 + 2\lambda = 2 + \mu . . . (1)\]
\[ - 1 = - 1 + \mu . . . (2) \]
\[\lambda = - \mu . . . (3)\]
Solving (2) and (3), we get
\[\lambda = 0 \]
\[\mu = 0\]
Substituting the values
\[\lambda = 0 \text{ and } \mu = 0\] in (1), we get
\[LHS = 1 + 2\lambda\]
\[ = 1 + 2\left( 0 \right)\]
\[ = 1\]
\[RHS = 2 + \mu\]
\[ = 2 + 0\]
\[ = 2\]
\[ \Rightarrow LHS \neq RHS\]
\[\text{ Since } \lambda = 0 \text{ and } \mu = 0 \text{ do not satisfy (1), the given lines do not intersect } . \]
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