Planes in 3D Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Planes in 3D.

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 flat, infinite surface in 3D space described by $ax + by + cz = d$ , where $\langle a, b, c \rangle$ is the normal vector.

Think of a plane as a perfectly flat, infinite floor that can be tilted at any angle in space. A horizontal floor is one plane; tilt it and you get another. To describe which tilt you have, imagine sticking a pole straight up out of the floor—that pole is the normal vector, and it captures the exact orientation of the surface. Any flat sheet in 3D, no matter how it's angled, is completely determined by where it sits and which direction its pole points.

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 flat infinite surface whose tilt is captured entirely by its normal vector.

Common stuck point: The procedure for planes in 3d is the easy part; the trap is using a direction vector instead of the normal vector. Asking "Is the object a flat surface fixed by a perpendicular (normal) vector and one equation $ax+by+cz=d$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the object a flat surface fixed by a perpendicular (normal) vector and one equation $ax+by+cz=d$ ?

Worked Examples

Example 1

easy

Write the equation of the plane with normal vector

\langle 2, -3, 1 \rangle

passing through the point

(1, 4, -2)

Answer

2 x - 3 y + z + 12 = 0

First step

The equation of a plane with normal

\langle A, B, C \rangle

through

(x_0, y_0, z_0)

A(x-x_0) + B(y-y_0) + C(z-z_0) = 0

Full solution

2
Substitute: $2(x-1) - 3(y-4) + 1(z+2) = 0$ .
3
Expand: $2x - 2 - 3y + 12 + z + 2 = 0$ , so $2x - 3y + z + 12 = 0$ .

A plane in 3D is determined by a point on the plane and a normal vector perpendicular to it. The normal vector's components become the coefficients of

x

y

, and

z

in the plane's equation. This generalizes the concept of a line (in 2D) having a perpendicular direction.

Example 2

medium

Find the equation of the plane through the points

A(1, 0, 0)

B(0, 2, 0)

, and

C(0, 0, 3)

Example 3

medium

Find the equation of the plane through

(1, -1, 2)

parallel to the plane

3x + y - 2z = 9

Example 4

medium

Find a plane through

A(1,0,2)

B(2,1,0)

C(0,-1,1)

Example 5

medium

Find the foot of the perpendicular from

(2, 3, 4)

to the plane

x + y + z = 0

Example 6

hard

Find the intersection point of the line

\vec{r}(t)=(1,0,2)+t\langle 2,1,-1\rangle

with the plane

x + 2y - z = 1

Example 7

hard

Find the equation of the plane containing the lines

\vec{r_1}(t)=(0,0,0)+t\langle 1,1,0\rangle

and

\vec{r_2}(s)=(0,0,0)+s\langle 0,1,1\rangle

Practice Problems

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

Example 1

medium

Find the distance from the point

(3, -1, 2)

to the plane

2x + y - 2z = 4

Example 2

hard

Find the angle between the planes

x + 2y - z = 5

and

2x - y + 3z = 1

Example 3

easy

Write the equation of the plane through

(0,0,0)

with normal

\langle 2,3,4 \rangle

Example 4

easy

What is the normal vector of the plane

3x - y + 2z = 7

Example 5

easy

Find

d

for the plane through

(1,1,1)

with normal

\langle1,2,3\rangle

Example 6

easy

\langle 1,1,0 \rangle

parallel to the plane

x + y + z = 5

(i.e., perpendicular to its normal)?

Example 7

easy

Are the planes

2x+y-z=3

and

4x+2y-2z=9

parallel?

Example 8

easy

Does the point

(2,0,1)

lie on the plane

x + 2y - z = 1

Example 9

easy

Find the

z

-intercept of the plane

2x + 3y + 4z = 12

Example 10

easy

The plane

z = 4

has what normal vector?

Example 11

medium

Find the equation of the plane through

(1,0,0)

(0,1,0)

(0,0,1)

Example 12

medium

Find the distance from the origin to the plane

2x + y + 2z = 9

Example 13

medium

Find the line of intersection direction of planes

x+y+z=1

and

x-y+z=3

Example 14

medium

Find the angle between the planes

x=0

and

x+y=0

via their normals.

Example 15

medium

A plane has normal

\langle1,2,2\rangle

and passes through

(3,0,0)

. Find its distance to the origin.

Example 16

medium

Where does the line

x=t,\ y=t,\ z=t

meet the plane

x+y+z=6

Example 17

medium

Is the line

x=1+t,\ y=2,\ z=3-t

parallel to the plane

x+z=5

Example 18

medium

Find the equation of the plane parallel to

2x - y + 3z = 5

passing through

(1,1,1)

Example 19

medium

Find the

x

-intercept of the plane

3x + 6y + 2z = 12

Example 20

challenge

Find the distance between the parallel planes

x+2y+2z=6

and

x+2y+2z=12

Example 21

challenge

Find the equation of the plane containing the point

(1,2,3)

and the line

x=t,\ y=t,\ z=0

Example 22

challenge

Three planes

x=0

y=0

z=0

— describe their common intersection.

Example 23

easy

Write the equation of the plane with normal

\langle 1, -2, 4 \rangle

passing through

(0, 0, 0)

Example 24

easy

Does the point

(1, 2, 3)

lie on the plane

2x - y + z = 3

Example 25

easy

Are the planes

x - 2y + z = 4

and

-2x + 4y - 2z = 1

parallel?

Example 26

easy

Find the equation of the plane through

(2, 1, 5)

with normal

\langle 1, 1, 1 \rangle

Example 27

medium

Find the distance from the point

(1, 1, 1)

to the plane

2x - 2y + z = 5

Example 28

medium

Find the angle between the planes

x + y + z = 1

and

x - y = 0

Example 29

medium

Find the direction vector of the line where

x + y + z = 6

and

2x - y + z = 3

intersect.

Example 30

medium

Find the equation of the plane perpendicular to the line

\vec{r}(t)=(1,2,3)+t\langle 4,-1,2\rangle

passing through

(0,1,0)

Example 31

medium

Find

d

so that the plane

2x - y + 2z = d

is tangent to the sphere

x^2 + y^2 + z^2 = 9

(i.e., distance 3 from origin).

Example 32

medium

Does the line

\vec{r}(t)=(0,0,0)+t\langle 1, 1, -2 \rangle

lie in the plane

x + y + z = 0

Example 33

hard

Find the distance from

(1, 2, 0)

to the plane through

(0,0,0)

(1,1,0)

(0,1,1)

Example 34

hard

Find the equation of the plane that contains the line

\vec{r}(t)=(1,2,3)+t\langle 1,0,1\rangle

and the point

(0, 0, 0)

Example 35

hard

Two parallel planes have equations

2x + 3y + 6z = 14

and

2x + 3y + 6z = -7

. Find the distance between them.

Example 36

hard

Find all values of

k

so that the planes

x + ky + z = 1

and

2x - y + 3z = 4

are perpendicular.

Example 37

challenge

Find the equation of the plane equidistant from the parallel planes

2x - y + 2z = 3

and

2x - y + 2z = 9

Example 38

challenge

Find the reflection of the point

(1, 2, 3)

across the plane

x + y + z = 0

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Related Concepts

Lines in 3D

Background Knowledge

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

lines in 3d