Find the points of intersection of the following pairs of lines

Question

$\begin{bmatrix}x \\y \\z\end{bmatrix} = \begin{bmatrix}3 \\-1 \\2\end{bmatrix} + t\begin{bmatrix}1 \\1 \\-1\end{bmatrix}$

$\begin{bmatrix}x \\y \\z\end{bmatrix} = \begin{bmatrix}1 \\1 \\-2\end{bmatrix} + s\begin{bmatrix}2 \\0 \\3\end{bmatrix}$

My solution:

Eliminating $\begin{bmatrix}x \\y \\z\end{bmatrix}$ gives $\begin{bmatrix}3 \\-1 \\2\end{bmatrix} + t\begin{bmatrix}1 \\1 \\-1\end{bmatrix} = \begin{bmatrix}1 \\1 \\-2\end{bmatrix} + s\begin{bmatrix}2 \\0 \\3\end{bmatrix}$. Equating coefficients gives three equations for s and t:

$3 + t = 1 + 2s$

$-1 + t = 1$

$2 - t = -2 + 3s$

This is where I get stuck. I was expecting $s$ and $t$ to be in separate equations so I can simply solve it and then plug it in the original equation but I'm confused on how to go about this point.

Answer 1

First, let us take "known terms" to one side and $t$ and $s$ terms to the other, and rewrite your equation as:

$$\begin{bmatrix} 2 \\ -2 \\ 4 \end{bmatrix} = \begin{bmatrix}3 \\-1 \\2\end{bmatrix} - \begin{bmatrix}1 \\1 \\-2\end{bmatrix} = - t\begin{bmatrix}1 \\1 \\-1\end{bmatrix} + s\begin{bmatrix}2 \\0 \\3\end{bmatrix}$$

Translate that to a system:

$$\left\{\begin{array}{c} -t+2s=2 \\ -t=-2 \\ t+3s=4 \end{array}\right.$$

So $t=2$, and plugging in we get $-2+2s=2$ and $2+3s=4$, so $s=0$, but that does not satisfy the second equation, so there is no solution. It seems those two lines do not intersect. Being not parallel, they must belong to distinct planes in space. Or I have made a calculation mistake. If so, please point it out :).

Answer 2

Your problem seems to be that you did not expect to get more equations than variables.

Just pick to easy ones to determine $s$ and $t$ and then check if the third equation holds as well for these values. Otherwise there is no solution.

The easiest one is the middle one, it gives $t = 2$. Then the first gives $s = 2$ and the third gives $s = 2/3$. Or: the third equation would turn into $0 = 4$, so it is not fulfilled.

This means there is no solution to the problem. There is no intersection.

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