C. Geometry and Algebra of Vectors · Lines and Planes

Intersections of Lines in Three-Space: Intersecting, Parallel and Skew

C3.3 C4.7About 70 minutes

Learning goals

I can describe the four configurations of two lines in three-space: coincident, parallel and distinct, intersecting, and skew.
I can decide which configuration two given lines have.
I can find the point of intersection of two lines when it exists.
I can use the triple scalar product to test whether two lines are coplanar.

Words to know

skew lines
Lines in three-space that are not parallel and do not intersect.

coplanar lines
Lines that lie in a common plane. Coplanar lines are parallel, coincident or intersecting.

consistent system
A system of equations that has at least one solution.

inconsistent system
A system of equations with no solution.

triple scalar product
a · (b × c). It is 0 exactly when the three vectors are coplanar.

Roads that never meet

On a highway overpass, one road passes over another. The roads point in different directions, so they are not parallel, but they never meet either: one is always above the other. In two-space that is impossible: two non-parallel lines in a plane must cross. In three-space it is common, and such lines are called skew lines.

In this lesson you will sort any two lines in three-space into one of four cases, and find the point of intersection when there is one.

Four possible configurations

For two lines L₁: r = r₁ + tm₁ and L₂: r = r₂ + sm₂ in three-space:

Parallel directions (m₂ = km₁):

Coincident: a point of one line is on the other. Every point is shared.
Parallel and distinct: no point is shared.

Non-parallel directions:

Intersecting: they meet in exactly one point. The two lines lie in one plane.
Skew: they never meet. No plane contains both lines.

How to test. Use a different parameter for each line (t and s), and set the coordinates equal:

r₁ + tm₁ = r₂ + sm₂ gives three equations in only two unknowns.

Solve two of them for t and s, then substitute into the third. If it checks, the lines intersect; substitute t into L₁ to find the point. If it fails, the lines are skew.

Edges of a box show the possibilities. In each box, the top front edge is drawn thick together with one other edge: one parallel to it, one that intersects it at a corner, and one that is skew (it never meets the top front edge and points in a different direction). Dashed edges are hidden at the back.

A two-question test sorts any pair of lines in three-space into one of four cases.

A one-line test for skew lines

Two non-parallel lines intersect exactly when they lie in one plane. The vector P₁P₂ joining a point of each line must then lie in the plane spanned by m₁ and m₂, so it is perpendicular to m₁ × m₂. So the lines intersect if (r₂ − r₁) · (m₁ × m₂) = 0, and are skew if it is not 0. (This is the triple scalar product from Dot and Cross Products: the volume of the box made by the three vectors is zero when they are coplanar.)

Worked example

Example 1: Two lines that intersect

Find the intersection of L₁: r = [1, 2, 3] + t[1, −1, 2] and L₂: r = [4, 0, 6] + s[2, −1, 1].

The directions [1, −1, 2] and [2, −1, 1] are not multiples, so the lines intersect or are skew.
Set coordinates equal: 1 + t = 4 + 2s, 2 − t = −s, 3 + 2t = 6 + s.
From the second equation, s = t − 2. Substitute into the first: 1 + t = 4 + 2t − 4, so t = 1 and s = −1.
Check the third: 3 + 2(1) = 5 and 6 + (−1) = 5 ✓.
Point: t = 1 in L₁ gives (2, 1, 5). (Check in L₂ with s = −1: (4 − 2, 0 + 1, 6 − 1) = (2, 1, 5) ✓.)

The lines intersect at (2, 1, 5).

Worked example

Example 2: Skew lines

Classify L₁: r = [1, −1, 2] + t[2, 1, 0] and L₂: r = [0, 3, 1] + s[1, −1, 1].

Directions [2, 1, 0] and [1, −1, 1] are not parallel.
Equations: 1 + 2t = s, −1 + t = 3 − s, 2 = 1 + s.
The third gives s = 1. Then the first gives t = 0.
Check the second: −1 + 0 = −1, but 3 − 1 = 2. −1 ≠ 2.

No values of t and s work: the lines are skew.

Worked example

Example 3: Parallel lines

Classify L₁: r = [2, 0, 1] + t[1, −2, 3] and L₂: r = [0, 1, 1] + s[−2, 4, −6].

[−2, 4, −6] = −2[1, −2, 3], so the directions are parallel.
Is (0, 1, 1) on L₁? From x: 2 + t = 0, so t = −2. Then y = 0 − 2(−2) = 4, but the point has y = 1.

Parallel and distinct.

Worked example

Example 4: The triple product test

Use (r₂ − r₁) · (m₁ × m₂) to check Examples 1 and 2.

Example 1: r₂ − r₁ = [3, −2, 3]. m₁ × m₂ = [1, −1, 2] × [2, −1, 1] = [(−1)(1) − (2)(−1), (2)(2) − (1)(1), (1)(−1) − (−1)(2)] = [1, 3, 1]. Dot product: 3 − 6 + 3 = 0, so they intersect ✓.
Example 2: r₂ − r₁ = [−1, 4, −1]. m₁ × m₂ = [2, 1, 0] × [1, −1, 1] = [(1)(1) − (0)(−1), (0)(1) − (2)(1), (2)(−1) − (1)(1)] = [1, −2, −3]. Dot product: −1 − 8 + 3 = −6 ≠ 0, so they are skew ✓.

The test agrees: 0 for the intersecting lines, −6 (not 0) for the skew lines.

Just for fun

Skew lines are the ultimate introverts: they never meet, and they refuse to even head in the same direction.

Hands-on activity

Shoebox edges

You need: a shoebox (or any rectangular box) and three different coloured sticky notes or pieces of tape.

Choose one edge of the box and mark it with tape. Call it edge E.
Find every other edge parallel to E and mark each with a second colour.
Find every edge that meets E at a corner and mark each with a third colour.
The edges left over are skew to E. Count them.
Pick a skew pair and hold a ruler along each edge. Convince yourself that no flat sheet could contain both edges.

What to look for: a box has 12 edges. For any edge E, the other 11 split into 3 parallel, 4 intersecting (2 at each end) and 4 skew.

Your turn

Classify r = [1, 1, 1] + t[1, 2, 3] and r = [2, 3, 4] + s[2, 4, 6].

Classify r = t[1, 1, 1] and r = [1, 0, 0] + s[1, 1, 1].

Find the intersection of r = [1, 0, 0] + t[1, 1, 0] and r = [0, 1, 0] + s[1, −1, 0].

Show that the x-axis and the line r = [0, 1, 1] + s[0, 1, 0] are skew.

Find the intersection of L₁: x = 1 + t, y = 2t, z = 3 − t and L₂: x = 2s, y = 1 + s, z = 1 + s.

Find the acute angle between lines with directions [1, 2, 2] and [2, −1, 2].

Use the triple product test to decide whether r = [1, 0, 2] + t[1, 1, 1] and r = [0, 1, 0] + s[1, 0, 2] intersect.

More practice: warm-up

What are skew lines?

Can two lines in two-space be skew?

Are lines with directions [1, 2, 3] and [−2, −4, −6] parallel?

Two lines have non-parallel directions and the t, s system is consistent. What is their configuration?

On a shoebox, name a pair of skew edges.

More practice: on your own

Find the intersection of r = [3, −1, 2] + t[1, 1, −1] and r = [0, 2, −1] + s[2, −1, 1].

Show that r = [1, 2, 3] + t[2, 0, 1] and r = s[1, 1, 1] are skew.

Find k so that r = [1, 0, k] + t[1, 1, 1] and r = [0, 1, 0] + s[1, 0, 1] intersect.

Drone A follows r = [0, 0, 10] + t[2, 1, 0] and drone B follows r = [10, −5, 0] + t[0, 1, 1] (metres, t in seconds, same clock). Do their paths cross? Do they collide?

Find the acute angle between lines with directions [1, 0, 1] and [0, 1, 1].

Challenge

Explain why two non-parallel lines intersect exactly when (r₂ − r₁) · (m₁ × m₂) = 0.

Find a scalar equation of the plane containing the two intersecting lines of Example 1.

For which values of a do the lines r = t[1, a, 1] and r = [1, 0, 0] + s[0, 1, 1] intersect? What are they otherwise?

Check your understanding

Quiz

Skew lines are lines in three-space that:

Two lines have direction vectors [2, −1, 4] and [−4, 2, −8]. The lines must be:

The x-axis and the line x = 0, z = 1 (parallel to the y-axis through (0, 0, 1)) are:

For two non-parallel lines, you solve two of the equations and get t = 1 and s = 1, but the third equation fails. The lines are:

Where do r = [1, 2, 3] + t[1, −1, 2] and r = [4, 0, 6] + s[2, −1, 1] intersect?

Which configuration of two lines can happen in three-space but never in two-space?

For parents and teachers

Big idea (C3.3, C4.7): two lines in three-space are coincident, parallel and distinct, intersecting, or skew. Skew lines are new in three-space: they point in different directions and never meet. To decide, compare directions, then solve the three coordinate equations with two different parameters.

How to help: The shoebox activity is quick and memorable. When your teen solves for an intersection, ask: "Did you check the third equation?" Skipping that check is the single most common mistake.

Common difficulties:

Using the same parameter t for both lines. That only finds a common point reached at the same parameter value (a collision), not a crossing of paths.
Not checking the third equation, and reporting a "point" for skew lines.
Calling lines parallel because they do not meet. In three-space, they may be skew.
Arithmetic slips in the triple product. Check by also solving the system.