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: Planes are two-dimensional—infinite extent in two directions, zero thickness in the third.

Common stuck point: Three non-collinear points determine exactly one unique plane—two points alone cannot define a plane.

Sense of Study hint: Try placing three pencil tips on a table (not in a line). Notice only one flat surface passes through all three.

Worked Examples

Example 1

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

Solution

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

Answer

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.

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?
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Related Concepts

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Background Knowledge

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

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