C. Geometry and Algebra of Vectors · Lines and Planes

Two Planes and Their Line of Intersection

C3.2 C4.2About 70 minutes

Learning goals

I can explain why two non-parallel planes meet in a line.
I can find the line of intersection of two planes by elimination or with a cross product.
I can recognize parallel and coincident planes from their equations.
I can represent a line in three-space as the intersection of two planes.

Words to know

line of intersection
The line of points that lie on both of two non-parallel planes.

coincident planes
Two equations that describe the same plane; one is a non-zero multiple of the other.

system of linear equations
Two or more linear equations that must be satisfied at the same time.

family of planes
All the planes through a given line, written (equation 1) + k(equation 2) = 0, plus the second plane itself.

Where two walls meet

Two walls of a room meet along a vertical edge. A wall meets the floor along a horizontal edge. The two halves of an open book meet at the spine. In each case, two planes that are not parallel meet in a line.

This gives the second way to describe a line in three-space (C4.2): as the set of points that satisfy two scalar equations at once. In this lesson you will find that line from the two plane equations, and go the other way, from a line to two planes that contain it.

Two planes that are not parallel meet in a line, like the two halves of an open book meet at the spine. The line lies in both planes, so its direction is perpendicular to both normals: m = n₁ × n₂.

Two equations in three unknowns

A system of two linear equations in x, y and z, such as

x + y + z = 1 and x − y + z = 0,

asks for the points on both planes. There are three possibilities:

Normals not parallel: the planes meet in a line. There are infinitely many solutions, described with one parameter.
Normals parallel, equations not multiples: the planes are parallel and distinct. Elimination gives a false statement like 0 = 5: no solution.
One equation is a multiple of the other: the planes coincide. Every point of the plane is a solution (two parameters).

The three possibilities for two planes. Non-parallel normals give a line. Parallel normals give either no common points (different planes) or a whole plane (the same plane written twice).

Finding the line of intersection

Method 1: elimination.

Eliminate one variable to get an equation in the other two.
Let one of the remaining variables be the parameter t.
Express the other two variables in terms of t.

Method 2: cross product.

The direction is m = n₁ × n₂ (perpendicular to both normals).
Find any one point on both planes: set one variable to 0 (often z = 0) and solve the 2-by-2 system. If that system has no solution, set a different variable to 0.
Write r = r₀ + tm.

Both methods give the same line, though the equations may look different.

From a line to two planes

Given a line in symmetric form (x − x₀)/a = (y − y₀)/b = (z − z₀)/c, each "equals sign" pair is a plane:

(x − x₀)/a = (y − y₀)/b gives b(x − x₀) − a(y − y₀) = 0,
(y − y₀)/b = (z − z₀)/c gives c(y − y₀) − b(z − z₀) = 0.

The line lies in both, so the two equations together describe it. If a direction component is 0, the matching coordinate is constant, and that constant equation is one of your planes.

Worked example

Example 1: Elimination

Find the line of intersection of x + y + z = 1 and x − y + z = 0.

Subtract the second equation from the first: 2y = 1, so y = 1/2.
Substitute into the first: x + z = 1/2.
Let z = t. Then x = 1/2 − t.
Line: r = [1/2, 1/2, 0] + t[−1, 0, 1].
Check with the cross product: [1, 1, 1] × [1, −1, 1] = [(1)(1) − (1)(−1), (1)(1) − (1)(1), (1)(−1) − (1)(1)] = [2, 0, −2], which is parallel to [−1, 0, 1] ✓.

r = [1/2, 1/2, 0] + t[−1, 0, 1], t ∈ ℝ. (Every point on this line has y = 1/2.)

Worked example

Example 2: Cross product method

Find the line of intersection of 2x − y + z = 3 and x + y − 2z = 0.

Direction: [2, −1, 1] × [1, 1, −2] = [(−1)(−2) − (1)(1), (1)(1) − (2)(−2), (2)(1) − (−1)(1)] = [1, 5, 3].
A point: let z = 0. Then 2x − y = 3 and x + y = 0. Add: 3x = 3, so x = 1 and y = −1. Point (1, −1, 0).
Line: r = [1, −1, 0] + t[1, 5, 3].
Check t = 1, the point (2, 4, 3): 4 − 4 + 3 = 3 ✓ and 2 + 4 − 6 = 0 ✓.

r = [1, −1, 0] + t[1, 5, 3]

Worked example

Example 3: Parallel planes

Solve the system x − 2y + 3z = 4 and −2x + 4y − 6z = 1.

Normals [1, −2, 3] and [−2, 4, −6] = −2[1, −2, 3] are parallel.
Multiply the first equation by 2 and add: 0 = 9. This is false.
So the planes are parallel and distinct.

No solution: the planes never meet.

Worked example

Example 4: A line as two planes

Represent the line through (3, 2, −1) and (0, 2, 1) by vector and parametric equations, and by the scalar equations of two intersecting planes.

Direction: [0 − 3, 2 − 2, 1 − (−1)] = [−3, 0, 2]. Vector: r = [3, 2, −1] + t[−3, 0, 2]. Parametric: x = 3 − 3t, y = 2, z = −1 + 2t.
The y-component is 0, so every point has y = 2. That is one plane.
From (x − 3)/(−3) = (z + 1)/2: 2(x − 3) = −3(z + 1), so 2x − 6 = −3z − 3, giving 2x + 3z − 3 = 0.
Check both points in 2x + 3z − 3 = 0: (3, 2, −1) gives 6 − 3 − 3 = 0 ✓; (0, 2, 1) gives 0 + 3 − 3 = 0 ✓.

The line is the intersection of the planes y = 2 and 2x + 3z − 3 = 0. (Many other pairs of planes also contain this line.)

Just for fun

Two non-parallel planes always find common ground. It is only a line, but it goes on forever, which is more than most group projects manage.

Hands-on activity

Interlocking cards

You need: two index cards (or two rectangles cut from a cereal box), scissors (use them carefully), and a ruler.

On each card, cut a straight slit from the middle of one long edge to the centre of the card.
Slide the cards together along the slits so they stand up in an X shape. The two cards are two planes. Where do they meet? (Along a line: the slits.)
Lay one card flat on the table and hold the other parallel to it, 2 cm above. Do they share any points? (No: parallel planes.)
Stack the two cards exactly on top of each other. How many points do they share? (All of them: coincident.)
Back in the X shape, hold a pencil along the line where the cards meet. The pencil is perpendicular to a pencil standing straight out of each card. That is why m = n₁ × n₂.

What to look for: two planes are never "just touching at one point"; they share a line, nothing, or everything.

Your turn

Describe the intersection of x + y − z = 2 and 2x + 2y − 2z = 4.

Describe the intersection of x + 2y − z = 1 and 2x + 4y − 2z = 5.

Find the line of intersection of x − y = 0 and y − z = 0.

Find the line of intersection of x + y + z = 6 and x − y + 2z = 1 by elimination.

Write the line r = [1, 0, 2] + t[2, 1, −1] as the intersection of two planes.

Describe the intersection of the planes z = 0 and x = 3.

Find k so that the planes 2x + ky + 4z = 1 and x − 3y + 2z = 5 are parallel.

More practice: warm-up

How many points do two parallel, distinct planes share?

What is the intersection of the xy-plane and the xz-plane?

Two planes have normals [1, 2, 3] and [2, 4, 6]. What can you say?

True or false: the line where two planes meet is perpendicular to both normals.

Do the planes z = 2 and z = 5 intersect?

More practice: on your own

Find the line of intersection of x − y + z = 2 and 2x + y − z = 1.

Represent the line through A(1, 2, 3) and B(4, 2, 5) as the intersection of two planes.

Does the line of intersection of x + y = 4 and y + z = 6 pass through (1, 3, 3)?

Find a third plane that contains the whole line of intersection of x + y + z = 1 and x − y + z = 0.

Find the acute angle between the planes x + y + z = 1 and x − y + z = 0.

Challenge

Show that the line r = [1, 2, 0] + t[1, −1, 2] is the line of intersection of x + y = 3 and 2x − z = 2.

Find the plane that contains the line of intersection of x + y + z = 1 and x − y + z = 0 and also passes through (1, 1, 1).

For which values of k do the planes x + 2y − z = 3 and 3x + 6y + kz = 9 (a) coincide, (b) meet in a line? Can they be parallel and distinct?

Check your understanding

Quiz

Two planes whose normals are not parallel intersect in:

Two planes have normals n₁ and n₂ that are not parallel. Which vector gives the direction of their line of intersection?

The planes 2x − y + 3z = 1 and 4x − 2y + 6z = 2 are:

Which pair of planes has the line x = 2, y = 3 + t, z = −1 as its intersection?

What is the intersection of the planes x + y + z = 6 and z = 0?

Which point lies on the line of intersection of x + y + z = 1 and x − y + z = 0?

For parents and teachers

Big idea (C3.2, C4.2): two linear equations in three variables are two planes. If their normals are not parallel, their common solutions form a line. That line can be found by elimination (letting one variable be a parameter) or with m = n₁ × n₂ and one common point. Going backwards, any line in three-space can be written as two plane equations.

How to help: The interlocking-card model makes the three cases obvious. When your teen finds a line of intersection, ask them to check a point from it (try t = 1) in both plane equations.

Common difficulties:

Expecting two planes to meet in a point. With three unknowns and only two equations, there is always at least one free parameter (or no solution).
When finding a point with z = 0, getting no solution and giving up. That just means the line is parallel to the xy-plane; set x = 0 or y = 0 instead.
Not recognizing that different-looking answers describe the same line. Compare directions (parallel?) and test one point.
Forgetting that a zero direction component gives a constant coordinate, which is itself one of the two planes.