Question

Determine whether the following pair of lines intersect or not: 

\[\overrightarrow{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \overrightarrow{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\]

Answer 1

\[\overrightarrow{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \overrightarrow{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\] 

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

\[LHS = 1 + 2\lambda\]

\[ = 1 + 2\left( 0 \right)\]

\[ = 1\]

\[RHS = 2 + \mu\]

\[ = 2 + 0\]

\[ = 2\]

\[ \Rightarrow LHS \neq RHS\]

\[\text{ Since } \lambda = 0 \text{ and  } \mu = 0 \text{ do not satisfy (1), the given lines do not intersect } . \]