Planes in 3D Examples in Math

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=dax + by + cz = d, where ⟨a,b,c⟩\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=dax+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=dax+by+cz=d?

Worked Examples

Example 1

easy
Write the equation of the plane with normal vector ⟨2,−3,1⟩\langle 2, -3, 1 \rangle passing through the point (1,4,−2)(1, 4, -2).

Answer

2x−3y+z+12=02x - 3y + z + 12 = 0

First step

1
The equation of a plane with normal ⟨A,B,C⟩\langle A, B, C \rangle through (x0,y0,z0)(x_0, y_0, z_0) is A(x−x0)+B(y−y0)+C(z−z0)=0A(x-x_0) + B(y-y_0) + C(z-z_0) = 0.

Full solution

  1. 2
    Substitute: 2(x−1)−3(y−4)+1(z+2)=02(x-1) - 3(y-4) + 1(z+2) = 0.
  2. 3
    Expand: 2x−2−3y+12+z+2=02x - 2 - 3y + 12 + z + 2 = 0, so 2x−3y+z+12=02x - 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 xx, yy, and zz 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)A(1, 0, 0), B(0,2,0)B(0, 2, 0), and C(0,0,3)C(0, 0, 3).

Example 3

medium
Find the equation of the plane through (1,−1,2)(1, -1, 2) parallel to the plane 3x+y−2z=93x + y - 2z = 9.

Example 4

medium
Find a plane through A(1,0,2)A(1,0,2), B(2,1,0)B(2,1,0), C(0,−1,1)C(0,-1,1).

Example 5

medium
Find the foot of the perpendicular from (2,3,4)(2, 3, 4) to the plane x+y+z=0x + y + z = 0.

Example 6

hard
Find the intersection point of the line r⃗(t)=(1,0,2)+t⟨2,1,−1⟩\vec{r}(t)=(1,0,2)+t\langle 2,1,-1\rangle with the plane x+2y−z=1x + 2y - z = 1.

Example 7

hard
Find the equation of the plane containing the lines r1⃗(t)=(0,0,0)+t⟨1,1,0⟩\vec{r_1}(t)=(0,0,0)+t\langle 1,1,0\rangle and r2⃗(s)=(0,0,0)+s⟨0,1,1⟩\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)(3, -1, 2) to the plane 2x+y−2z=42x + y - 2z = 4.

Example 2

hard
Find the angle between the planes x+2y−z=5x + 2y - z = 5 and 2x−y+3z=12x - y + 3z = 1.

Example 3

easy
Write the equation of the plane through (0,0,0)(0,0,0) with normal ⟨2,3,4⟩\langle 2,3,4 \rangle.

Example 4

easy
What is the normal vector of the plane 3x−y+2z=73x - y + 2z = 7?

Example 5

easy
Find dd for the plane through (1,1,1)(1,1,1) with normal ⟨1,2,3⟩\langle1,2,3\rangle.

Example 6

easy
Is ⟨1,1,0⟩\langle 1,1,0 \rangle parallel to the plane x+y+z=5x + y + z = 5 (i.e., perpendicular to its normal)?

Example 7

easy
Are the planes 2x+y−z=32x+y-z=3 and 4x+2y−2z=94x+2y-2z=9 parallel?

Example 8

easy
Does the point (2,0,1)(2,0,1) lie on the plane x+2y−z=1x + 2y - z = 1?

Example 9

easy
Find the zz-intercept of the plane 2x+3y+4z=122x + 3y + 4z = 12.

Example 10

easy
The plane z=4z = 4 has what normal vector?

Example 11

medium
Find the equation of the plane through (1,0,0)(1,0,0), (0,1,0)(0,1,0), (0,0,1)(0,0,1).

Example 12

medium
Find the distance from the origin to the plane 2x+y+2z=92x + y + 2z = 9.

Example 13

medium
Find the line of intersection direction of planes x+y+z=1x+y+z=1 and x−y+z=3x-y+z=3.

Example 14

medium
Find the angle between the planes x=0x=0 and x+y=0x+y=0 via their normals.

Example 15

medium
A plane has normal ⟨1,2,2⟩\langle1,2,2\rangle and passes through (3,0,0)(3,0,0). Find its distance to the origin.

Example 16

medium
Where does the line x=t, y=t, z=tx=t,\ y=t,\ z=t meet the plane x+y+z=6x+y+z=6?

Example 17

medium
Is the line x=1+t, y=2, z=3−tx=1+t,\ y=2,\ z=3-t parallel to the plane x+z=5x+z=5?

Example 18

medium
Find the equation of the plane parallel to 2x−y+3z=52x - y + 3z = 5 passing through (1,1,1)(1,1,1).

Example 19

medium
Find the xx-intercept of the plane 3x+6y+2z=123x + 6y + 2z = 12.

Example 20

challenge
Find the distance between the parallel planes x+2y+2z=6x+2y+2z=6 and x+2y+2z=12x+2y+2z=12.

Example 21

challenge
Find the equation of the plane containing the point (1,2,3)(1,2,3) and the line x=t, y=t, z=0x=t,\ y=t,\ z=0.

Example 22

challenge
Three planes x=0x=0, y=0y=0, z=0z=0 — describe their common intersection.

Example 23

easy
Write the equation of the plane with normal ⟨1,−2,4⟩\langle 1, -2, 4 \rangle passing through (0,0,0)(0, 0, 0).

Example 24

easy
Does the point (1,2,3)(1, 2, 3) lie on the plane 2x−y+z=32x - y + z = 3?

Example 25

easy
Are the planes x−2y+z=4x - 2y + z = 4 and −2x+4y−2z=1-2x + 4y - 2z = 1 parallel?

Example 26

easy
Find the equation of the plane through (2,1,5)(2, 1, 5) with normal ⟨1,1,1⟩\langle 1, 1, 1 \rangle.

Example 27

medium
Find the distance from the point (1,1,1)(1, 1, 1) to the plane 2x−2y+z=52x - 2y + z = 5.

Example 28

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

Example 29

medium
Find the direction vector of the line where x+y+z=6x + y + z = 6 and 2x−y+z=32x - y + z = 3 intersect.

Example 30

medium
Find the equation of the plane perpendicular to the line r⃗(t)=(1,2,3)+t⟨4,−1,2⟩\vec{r}(t)=(1,2,3)+t\langle 4,-1,2\rangle passing through (0,1,0)(0,1,0).

Example 31

medium
Find dd so that the plane 2x−y+2z=d2x - y + 2z = d is tangent to the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9 (i.e., distance 3 from origin).

Example 32

medium
Does the line r⃗(t)=(0,0,0)+t⟨1,1,−2⟩\vec{r}(t)=(0,0,0)+t\langle 1, 1, -2 \rangle lie in the plane x+y+z=0x + y + z = 0?

Example 33

hard
Find the distance from (1,2,0)(1, 2, 0) to the plane through (0,0,0)(0,0,0), (1,1,0)(1,1,0), (0,1,1)(0,1,1).

Example 34

hard
Find the equation of the plane that contains the line r⃗(t)=(1,2,3)+t⟨1,0,1⟩\vec{r}(t)=(1,2,3)+t\langle 1,0,1\rangle and the point (0,0,0)(0, 0, 0).

Example 35

hard
Two parallel planes have equations 2x+3y+6z=142x + 3y + 6z = 14 and 2x+3y+6z=−72x + 3y + 6z = -7. Find the distance between them.

Example 36

hard
Find all values of kk so that the planes x+ky+z=1x + ky + z = 1 and 2x−y+3z=42x - y + 3z = 4 are perpendicular.

Example 37

challenge
Find the equation of the plane equidistant from the parallel planes 2x−y+2z=32x - y + 2z = 3 and 2x−y+2z=92x - y + 2z = 9.

Example 38

challenge
Find the reflection of the point (1,2,3)(1, 2, 3) across the plane x+y+z=0x + 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