Coplanar lines and skew lines

What you will be able to do

• sort two lines in space into parallel, intersecting or skew;
• test whether two lines are coplanar with a scalar triple product or a determinant;
• find the point where two intersecting lines meet.

Cables over a street

Two cables run across a street. If they are the same height and not parallel, they meet. But one cable is usually higher than the other. Then they are not parallel, and they still never touch. We call such lines skew.

In a plane two lines are either parallel or they meet. So skew lines can only exist in three dimensions. A flat picture cannot show the difference, because both lines seem to cross in the picture.

For two lines in space there are three cases.

• Parallel: the directions are proportional. The lines never meet unless they are the same line.
• Intersecting: the lines are not parallel and have exactly one common point.
• Skew: the lines are not parallel and have no common point.

Two lines are coplanar if one plane holds both. Parallel lines and intersecting lines are coplanar. Skew lines are not.

The test for coplanar lines

Let the lines be $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$. The vector $\vec{a}_2 - \vec{a}_1$ joins a point of the first line to a point of the second. If the lines are coplanar, then $\vec{a}_2 - \vec{a}_1$, $\vec{b}_1$ and $\vec{b}_2$ all lie in that one plane. So their scalar triple product is zero:

In Cartesian form, with the lines $\dfrac{x - x_1}{a_1} = \dfrac{y - y_1}{b_1} = \dfrac{z - z_1}{c_1}$ and $\dfrac{x - x_2}{a_2} = \dfrac{y - y_2}{b_2} = \dfrac{z - z_2}{c_2}$, the same test is the determinant

The triple product is the Vectors chapter's $\vec{u} \cdot (\vec{v} \times \vec{w})$. Some books write it as $[\vec{u}\ \vec{v}\ \vec{w}]$. The cross product $\vec{b}_1 \times \vec{b}_2$ is the zero vector exactly when the lines are parallel.

Sorting two lines

1. Check whether $\vec{b}_1 \times \vec{b}_2 = \vec{0}$. If so, the lines are parallel.
2. If not, work out the triple product. If it is $0$, the lines intersect.
3. If it is not $0$, the lines are skew.

To find the common point of intersecting lines, write both lines with their parameters and set the three coordinates equal. Use two equations to find $\lambda$ and $\mu$, and then check the third.

A worked example

Line 1 passes through $(1, 2, 3)$ with direction $(2, 3, 4)$. Line 2 passes through $(2, 6, 5)$ with direction $(1, -1, 2)$.

1. Not parallel. The directions are not proportional.
2. The cross product. $\vec{b}_1 \times \vec{b}_2 = (3 \times 2 - 4 \times (-1),\ 4 \times 1 - 2 \times 2,\ 2 \times (-1) - 3 \times 1) = (10, 0, -5)$.
3. The triple product. $\vec{a}_2 - \vec{a}_1 = (1, 4, 2)$, and $(1, 4, 2) \cdot (10, 0, -5) = 10 + 0 - 10 = 0$. The lines are coplanar, so they intersect.
4. The point. The $x$ and $y$ equations are $1 + 2\lambda = 2 + \mu$ and $2 + 3\lambda = 6 - \mu$. Adding them gives $3 + 5\lambda = 8$, so $\lambda = 1$ and $\mu = 1$. The $z$ equation gives $3 + 4 = 7$ and $5 + 2 = 7$. They agree. The lines meet at $(3, 5, 7)$.

Now lift line 2 by one unit, so that it passes through $(2, 6, 6)$. The triple product becomes $(1, 4, 3) \cdot (10, 0, -5) = -5$. It is not zero, so the lines are skew. A second example. The directions $(2, 3, 4)$ and $(-4, -6, -8)$ have the cross product $\vec{0}$, so those lines are parallel.

Practise with new numbers

Try it in Python

The program sorts two lines into parallel, intersecting or skew. It uses the cross product and the triple product.

Common mistakes

• Testing only the triple product. For parallel lines the cross product is $\vec{0}$, so the triple product is $0$ too. Check for parallel lines first.
• Using $\vec{a}_1 - \vec{a}_2$ and then mixing the order. The sign of the triple product changes, but zero stays zero. Keep one order inside one calculation.
• Trusting the picture. Skew lines look as if they cross in a flat drawing.
• Skipping the third equation. Two equations always give $\lambda$ and $\mu$. The third equation decides whether the lines really meet.

Remember these

• Skew lines are not parallel and do not meet. They need three dimensions.
• Coplanar lines: $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$.
• Parallel lines: $\vec{b}_1 \times \vec{b}_2 = \vec{0}$.
• To find the meeting point, solve for $\lambda$ and $\mu$ from two coordinates, and check the third.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two lines have direction ratios $(2, -4, 6)$ and $(-1, 2, -3)$. How are they related?

2. Can two lines drawn on one sheet of paper be skew?

3. The $x$-axis passes through the origin with direction $(1, 0, 0)$. Another line passes through $(0, 0, 3)$ with direction $(0, 1, 0)$. Are the lines coplanar?

4. Show that the lines $\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} + 3\hat{j} - \hat{k})$ and $\vec{r} = (4\hat{i} + 7\hat{j} - 4\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k})$ meet, and find the point.

5. Why can parallel lines never be skew?

References

• OpenStax, Calculus Volume 3, Chapter 2, section on equations of lines and planes in space (parallel, intersecting and skew lines).
• NCERT, Mathematics Part II for Class XII, Chapter 11, section on the shortest distance between two lines (coplanar and skew lines).