Plane Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Plane.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A perfectly flat surface extending infinitely in all directions with zero thickness; defined by three non-collinear points.

An infinite sheet of paper with absolutely no thickness, extending forever in every direction.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A plane is a perfectly flat surface that extends forever in all directions and is fixed by three non-collinear points.

Common stuck point: The procedure for plane is the easy part; the trap is thinking two points fix a plane. Asking "Is it a flat surface extending infinitely in two dimensions, with no thickness?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is it a flat surface extending infinitely in two dimensions, with no thickness?

Worked Examples

Example 1

easy

What is the equation of the xy-plane in 3D space?

Answer

z = 0

First step

Step 1: The xy-plane contains all points where the z-coordinate is zero.

Full solution

2
Step 2: In the general plane equation $ax + by + cz = d$ , we want z = 0 for all points on this plane.
3
Step 3: Set $a=0, b=0, c=1, d=0$ : the equation is $z = 0$ .

The xy-plane is defined by all points

(x, y, 0)

— it is the familiar 2D coordinate plane embedded in 3D space. Its normal vector is

(0,0,1)

, pointing straight up.

Example 2

medium

Three points A

(0,0,0)

, B

(1,0,0)

, C

(0,1,0)

lie on a plane. Write the equation of that plane.

Example 3

easy

A plane passes through

(0, 0, 5)

and is parallel to the

xy

-plane. Write its equation.

Example 4

medium

Find the equation of the plane through

(1, 0, 0)

(0, 2, 0)

, and

(0, 0, 3)

Example 5

medium

Find the distance from the point

(1, 2, 3)

to the plane

x + 2y + 2z = 3

Example 6

medium

Does the line

\{(t, 2t, 3t) : t \in \mathbb{R}\}

lie in the plane

x + y - z = 0

Example 7

medium

Two planes have normals

\langle 1, 2, 2 \rangle

and

\langle 2, 1, -2 \rangle

. Find the angle between the planes.

Example 8

hard

Find the equation of the plane through the three points

A(1, 0, 0)

B(0, 1, 0)

, and

C(1, 1, 1)

Example 9

hard

Find the distance from the origin to the plane

3x - 4y + 12z = 26

Example 10

hard

Find the equation of the plane that contains the point

(1, 1, 1)

and the line

\{(t, 2t, 3t)\}

Example 11

challenge

Find the equation of the plane that is equidistant from the parallel planes

2x - y + 2z = 3

and

2x - y + 2z = 9

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

How many dimensions does a plane have, and how many coordinates are needed to specify a point on a plane?

Example 2

hard

Does the point

(2, 3, -1)

lie on the plane

2x - y + 3z = 4

Example 3

easy

How many dimensions does a plane have?

Example 4

easy

How many non-collinear points are needed to determine a unique plane?

Example 5

easy

Does a plane have any edges or boundaries?

Example 6

easy

A flat floor is a model of which geometric object?

Example 7

easy

A line and a plane meet at exactly one point. Is the line in the plane or crossing through it?

Example 8

easy

Two distinct planes intersect. What shape is their intersection?

Example 9

easy

Is the coordinate grid (the

xy

-plane) an example of a plane?

Example 10

easy

Three points are all on the same line. Do they determine a unique plane?

Example 11

medium

Why does a four-legged chair sometimes wobble but a three-legged stool never does?

Example 12

medium

Can two distinct planes intersect in exactly one point?

Example 13

medium

A line is parallel to a plane. How many points do they share?

Example 14

medium

How many planes can contain a single given line?

Example 15

medium

In the plane equation

2x + 3y + z = 6

, find the point where the plane crosses the

z

-axis.

Example 16

medium

A plane and a sphere intersect. What shape is the intersection (when they meet in more than one point)?

Example 17

medium

Two planes are both perpendicular to the same line. How are the two planes related?

Example 18

medium

Why can you say 'a plane has no thickness' even though any real surface does?

Example 19

challenge

Three planes pairwise intersect in three different lines, and no two planes are parallel. Describe the possible ways the three planes can meet.

Example 20

challenge

How many distinct planes are determined by 4 points in space, no three of which are collinear and not all four coplanar?

Example 21

challenge

A plane cuts all six edges meeting around one vertex of a cube is impossible — but a single plane can cut through a cube to expose a regular hexagon cross-section. Roughly how is the plane positioned to do this?

Example 22

challenge

Explain why a system of two linear equations in

x

y

z

\{x + y + z = 1,\ x + y + z = 2\}

has no solution, using planes.

Example 23

easy

What is the equation of the

xz

-plane in 3D space?

Example 24

easy

Does the point

(1, 2, 3)

lie on the plane

x + y + z = 6

Example 25

easy

A normal vector to the plane

2x - 3y + 4z = 5

is what?

Example 26

easy

True or false: the plane

z = 7

is parallel to the

xy

-plane.

Example 27

medium

Are the planes

2x + 3y - z = 4

and

4x + 6y - 2z = 9

parallel, identical, or intersecting?

Example 28

medium

What are the

x

y

-, and

z

-intercepts of the plane

3x + 4y + 6z = 12

Example 29

medium

Find the equation of the plane through

(2, 0, 0)

with normal vector

\langle 1, 1, 1 \rangle

Example 30

medium

What is the equation of the plane parallel to

2x - y + 3z = 7

passing through the origin?

Example 31

hard

Find a parametric form of the line of intersection of the planes

x + y + z = 6

and

x - y + 2z = 5

Example 32

hard

Find the angle (in degrees) between the planes

x + y = 2

and

y + z = 3

← Back to Plane Formula Explained Practice Mode

Related Concepts

Line Dimension

Background Knowledge

These ideas may be useful before you work through the harder examples.

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