C. Geometry and Algebra of Vectors · Lines and Planes

Linear Equations in Three Variables: Planes

C3.2About 70 minutes

Learning goals

I can explain why the solutions of Ax + By + Cz + D = 0 form a plane.
I can compare the shapes of the solutions of equations such as x = 0 and y = x in two-space and in three-space.
I can describe planes such as z = 5, y − z = 3 and x + z = 1 by reasoning about missing variables.
I can sketch a plane using intercepts and traces.

Words to know

plane
A flat surface that extends forever in every direction.

scalar equation of a plane
An equation Ax + By + Cz + D = 0 (A, B, C not all 0) whose solution points form a plane.

intercept (of a plane)
A point where the plane crosses a coordinate axis.

trace
The line where a plane meets a coordinate plane.

octant
One of the eight regions into which the three coordinate planes divide space. In the first octant x, y and z are all positive.

Flat surfaces everywhere

Look around the room. The floor, each wall, the ceiling and the top of a table are all pieces of planes: flat surfaces that, in mathematics, go on forever in every direction. In two-space, one linear equation gave a line. In three-space, one linear equation gives a plane. This lesson is about seeing why, and getting good at picturing planes from their equations.

One equation in three variables is a plane

An equation Ax + By + Cz + D = 0 (A, B, C not all zero) is a scalar equation of a plane. Its solution points (x, y, z) form a plane.

Why a plane and not a line? Suppose C ≠ 0. You may choose x and y freely, and then the equation decides z. Two free choices means two "directions of freedom", and a flat surface is exactly that. (A line has only one free parameter t.)

To find points on a plane, pick two coordinates and solve for the third. For 4x − y + 2z = 8:

x = 0, y = 0 gives z = 4: (0, 0, 4)
x = 1, y = 2 gives 4 − 2 + 2z = 8, so z = 3: (1, 2, 3)

Same equation, different space

An equation can mean different things in two-space and three-space:

Equation	In two-space (x, y)	In three-space (x, y, z)
x = 0	the y-axis (a line)	the yz-plane
y = 0	the x-axis (a line)	the xz-plane
y = x	a line through the origin	a vertical plane containing the z-axis
z = 5	not meaningful (there is no z)	a horizontal plane 5 units above the xy-plane

The same equation in two-space and in three-space.

Missing variable rule: if a variable is missing from the equation, it can take any value, so the plane is parallel to that variable's axis. For example, y − z = 3 has no x, so it is parallel to the x-axis; x + z = 1 has no y, so it is parallel to the y-axis.

The same equation y = x. In two-space its solutions form a line. In three-space z can be anything, so the line is swept straight up and down into a vertical plane that contains the z-axis.

Left: z = 5 is a horizontal plane 5 units above the xy-plane. Right: x + z = 1 has no y, so it runs parallel to the y-axis, like a ramp through (1, 0, 0) and (0, 0, 1). Only a piece of each plane is drawn.

Sketching a plane with intercepts and traces

Intercepts: set two variables to 0 and solve for the third. For 2x + 3y + 6z = 12: x-intercept 6, y-intercept 4, z-intercept 2.
Plot the three intercepts on the axes and join them. The triangle is the part of the plane in the first octant (where x, y, z ≥ 0).
Traces are the lines where the plane meets the coordinate planes. Set one variable to 0: the xy-trace of 2x + 3y + 6z = 12 is 2x + 3y = 12 (with z = 0).

If the plane passes through the origin, all the intercepts are 0. Then draw the traces instead.

Technology check: free 3-D graphing tools (for example, GeoGebra's 3D calculator) let you type an equation like y − z = 3 and turn the view with your finger or mouse. Use one to confirm your sketches, but make sure you can reason out each shape by hand.

The part of the plane 2x + 3y + 6z = 12 in the first octant is a triangle whose corners are the three intercepts. Its edges are the traces: the lines where the plane cuts the coordinate planes. The plane itself carries on forever beyond the triangle.

Worked example

Example 1: x = 0, y = 0 and y = x in two-space and three-space

Compare the solution sets of x = 0, y = 0 and y = x in two-space and in three-space.

x = 0: in two-space, the points (0, y) form the y-axis. In three-space, the points (0, y, z) form the yz-plane.
y = 0: in two-space, the x-axis. In three-space, the xz-plane.
y = x: in two-space, the line through the origin with slope 1. In three-space, z is free, so every point (a, a, z) is a solution: the line is swept up and down into a vertical plane containing the z-axis.
Pattern: each line in two-space becomes a plane in three-space, because the extra variable z is unrestricted.

In two-space each is a line; in three-space each is a plane.

Worked example

Example 2: z = 5, y − z = 3 and x + z = 1

Describe the shape of the solution points of z = 5, y − z = 3 and x + z = 1 in three-space.

z = 5: x and y are free, z is fixed. A horizontal plane 5 units above the xy-plane.
y − z = 3: x is missing, so the plane is parallel to the x-axis. In the yz-plane, its trace is the line y − z = 3, through (0, 3, 0) and (0, 0, −3). Slide that line along the x-direction.
x + z = 1: y is missing, so the plane is parallel to the y-axis. Its trace in the xz-plane is the line through (1, 0, 0) and (0, 0, 1). Slide it along the y-direction: a tilted ramp.

All three are planes: one horizontal, one parallel to the x-axis, one parallel to the y-axis.

Worked example

Example 3: Intercepts and traces

Find the intercepts and the traces of 2x + 3y + 6z = 12.

x-intercept: y = z = 0 gives 2x = 12, x = 6.
y-intercept: 3y = 12, y = 4. z-intercept: 6z = 12, z = 2.
xy-trace (z = 0): 2x + 3y = 12.
xz-trace (y = 0): 2x + 6z = 12, or x + 3z = 6.
yz-trace (x = 0): 3y + 6z = 12, or y + 2z = 4.

Intercepts (6, 0, 0), (0, 4, 0), (0, 0, 2); traces 2x + 3y = 12, x + 3z = 6 and y + 2z = 4.

Just for fun

Your bedroom floor is a piece of the plane z = 0, and mathematically that plane goes on forever in every direction. That would explain where all the missing socks go.

Hands-on activity

Cardboard plane on straw axes

You need: three drinking straws (or pencils), a lump of modelling clay or play dough, a ruler, a marker, a piece of thin cardboard (a cereal box works), and scissors (use them carefully).

We will build the first-octant piece of 2x + 3y + 6z = 12 at a scale of 2 cm per unit.

Push the three straws into the clay so they meet at one corner at right angles: two lying flat (x and y) and one standing up (z). This corner is the origin.
Mark the intercepts: 12 cm along the x-straw (x = 6), 8 cm along the y-straw (y = 4), 4 cm up the z-straw (z = 2).
Work out the triangle's side lengths with the distance formula, in centimetres: between (12, 0, 0) and (0, 8, 0) it is √(144 + 64) ≈ 14.4 cm; between (0, 8, 0) and (0, 0, 4) it is √(64 + 16) ≈ 8.9 cm; between (12, 0, 0) and (0, 0, 4) it is √(144 + 16) ≈ 12.6 cm.
Draw a triangle with these sides on the cardboard (draw the 14.4 cm side, then use the ruler to find where the other two sides meet) and cut it out.
Rest its corners on the three marks. It should fit snugly.
Now imagine the cardboard extended forever in every direction: that is the whole plane.

What to look for: the triangle tilts back toward the z-axis, and its edges lie along the three traces.

Your turn

Is (1, −2, 3) on the plane 2x + y + z = 3?

Find the intercepts of 3x + 2y − 6z = 12.

Describe x = −2 in two-space and in three-space.

Describe y = 4 in three-space.

Which axis is the plane 2x − 3z = 6 parallel to?

Find z so that (2, 1, z) lies on x + 3y − 2z = 7.

Find the xy-trace of x + 2y + 4z = 8.

More practice: warm-up

What shape is the solution set of 5x − y + 2z = 1 in three-space?

Write an equation of the xy-plane.

Is the origin on x + y + z = 0?

Find the z-intercept of x + y + 2z = 6.

In two-space, what is the graph of y = 2?

More practice: on your own

Find the intercepts of x + 2y + 3z = 6 and describe the first-octant piece of the plane.

Describe x + y = 4 in two-space and in three-space.

Write an equation of the plane parallel to the xy-plane through (3, −1, 7).

Write an equation of the plane that contains the z-axis and the point (2, 3, 0).

Show that 2x − y + 4z = 0 passes through the origin, and find its three traces.

Challenge

A plane has non-zero intercepts a, b and c. Show that its equation can be written x/a + y/b + z/c = 1, and use this to find the plane with intercepts 2, −3 and 6.

The plane x + y + z = 1 contains (1, 0, 0), (0, 1, 0) and (0, 0, 1). Does it contain any point whose three coordinates are all negative?

For which k does the plane kx + 2y − z = 4 contain the entire line x = t, y = 2 + t, z = 3t?

Check your understanding

Quiz

In three-space, the equation x = 3 describes:

What are the intercepts of 2x + 4y − z = 8?

The plane y − z = 3 is parallel to which axis?

Which point lies on the plane 3x − y + 2z = 5?

In two-space, y = x is a line. In three-space, y = x is:

What is the trace of 2x + 3y + 6z = 12 in the xz-plane?

For parents and teachers

Big idea (C3.2): in three-space one linear equation is a plane. Equations that were lines in two-space (x = 0, y = x) become planes, because the extra variable is free.

How to help: Point at real surfaces: "The floor is z = 0. What equation is the wall?" Ask your teen to explain what a missing variable in an equation means for the plane ("y is missing, so it is parallel to the y-axis").

Common difficulties:

Drawing y = x as a line in three-space. Ask: "Can z be anything? Then what shape do you get?"
Mixing up "parallel to the xz-plane" (y = constant) and "parallel to the y-axis" (y missing).
Sign errors in intercepts (for −z = 8, the z-intercept is −8).
Thinking the drawn triangle is the whole plane. It is only the first-octant piece.