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

We are given two lines: \[ \mathbf{r}_1 = ( \hat{i} + \hat{j} - \hat{k} ) + \lambda ( 3\hat{i} - \hat{j} ), \] \[ \mathbf{r}_2 = ( 4\hat{i} - \hat{k} ) + \mu ( 2\hat{i} + 3\hat{k} ). \] For the lines to intersect, there must exist values of $\lambda$ and $\mu$ such that the coordinates of points on both lines are equal. Equating the $i$, $j$, and $k$ components of the two lines:
- $x$-component: $1 + 3\lambda = 4 + 2\mu$,
- $y$-component: $1 - \lambda = 0$,
- $z$-component: $-1 - \lambda = -1 + 3\mu$.
From the $y$-component equation, we have $\lambda = 1$. Substituting $\lambda = 1$ into the $x$-component equation: \[ 1 + 3(1) = 4 + 2\mu \quad \Rightarrow \quad 4 = 4 + 2\mu \quad \Rightarrow \quad \mu = 0. \] Substituting $\lambda = 1$ and $\mu = 0$ into the $z$-component equation: \[ -1 - 1 = -1 + 3(0) \quad \Rightarrow \quad -2 = -1. \] This results in a contradiction, so the lines do not intersect.