Line relationships — parallel, intersecting, or skew

By the end of this lesson

By the end you can decide whether two lines in 3D are parallel, intersecting or skew, and find the point where they meet when they do intersect.

Three possibilities, not two

On a flat page, two straight lines either run side by side for ever (parallel) or cross once. Space has a third option.

Picture a road and a bridge that carries another road over it. The two roads point in different directions, so they are not parallel. But one is higher than the other, so they never touch. Lines like that are called skew.

Relationship	Same direction?	Common point?
Parallel	yes	no (unless they are the same line)
Intersecting	no	exactly one
Skew	no	none

In a flat drawing, skew lines can look as if they cross. The dashed gap is the giveaway: one line is on the floor and the other is on the roof.

What the symbols mean

Each line is written in the form you met in the last lesson:

$L_1:\ \mathbf{r} = \mathbf{a}_1 + t\,\mathbf{b}_1 \qquad L_2:\ \mathbf{r} = \mathbf{a}_2 + s\,\mathbf{b}_2$

$\mathbf{a}_1$ and $\mathbf{a}_2$ are the position vectors of one known point on each line.
$\mathbf{b}_1$ and $\mathbf{b}_2$ are the direction vectors: which way each line points.
$t$ and $s$ are the parameters: numbers you can change to move along each line, with each value giving one point. Use different letters for the two lines, because the shared point (if there is one) is usually reached at different parameter values.

Test 1: are they parallel?

Two lines are parallel when their direction vectors point the same way, which means one direction vector is a multiple of the other: $\mathbf{b}_2 = k\,\mathbf{b}_1$ for some number $k$ .

To test it, divide matching components. If every ratio is the same number $k$ , the lines are parallel. If a component is $0$ in one direction vector, it must also be $0$ in the other, because $k \times 0 = 0$ . For example, $(1, 1, 0)$ and $(0, 1, 1)$ cannot be parallel: the $z$ -components would need $k \times 0 = 1$ .

Parallel lines can still be the same line written twice. To tell, take the known point of one line and test whether it lies on the other. If it does, the lines coincide; if not, they are distinct parallel lines.

Worked example: Example 1: parallel, but not the same line

$L_1:\ \mathbf{r} = \begin{pmatrix}1\\2\\-1\end{pmatrix} + t\begin{pmatrix}2\\-1\\3\end{pmatrix}$ and $L_2:\ \mathbf{r} = \begin{pmatrix}3\\-1\\2\end{pmatrix} + s\begin{pmatrix}4\\-2\\6\end{pmatrix}$ .

Are $L_1$ and $L_2$ parallel? Are they the same line?

Have a go first?

Check 1 — parallel?

Multiple choiceQuestion 1 · Apply

Answer the question above to continue.