Question

Determine if the lines $\mathbf{r}_1 = ( \hat{i} + \hat{j} - \hat{k} ) + \lambda ( 3\hat{i} - \hat{j} )$ and $\mathbf{r}_2 = ( 4\hat{i} - \hat{k} ) + \mu ( 2\hat{i} + 3\hat{k} )$ intersect with each other.

Hint:

To check if two lines intersect, equate their components and solve the resulting system of equations for the parameters.

Answer 1

The given lines are: \[ \mathbf{r}_1 = ( \hat{i} + \hat{j} - \hat{k} ) + \lambda ( 3\hat{i} - \hat{j} ), \] \[ \mathbf{r}_2 = ( 4\hat{i} - \hat{k} ) + \mu ( 2\hat{i} + 3\hat{k} ). \] For intersection, $\lambda$ and $\mu$ must satisfy:
- $x$-component: $1 + 3\lambda = 4 + 2\mu$,
- $y$-component: $1 - \lambda = 0$,
- $z$-component: $-1 - \lambda = -1 + 3\mu$.
From the $y$-component, $\lambda = 1$. Substituting into the $x$-component: $1 + 3(1) = 4 + 2\mu \Rightarrow 4 = 4 + 2\mu \Rightarrow \mu = 0$. Substituting $\lambda = 1$ and $\mu = 0$ into the $z$-component: $-1 - 1 = -1 + 3(0) \Rightarrow -2 = -1$. This contradiction indicates the lines do not intersect.