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 three-dimensional space described by the equation ax + by + cz = d, where the vector \langle a, b, c \rangle is normal (perpendicular) to the plane.

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.

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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 in 3D is determined by a point and a normal vector. The normal vector \langle a, b, c \rangle appears directly as the coefficients in the equation ax + by + cz = d. Two planes are parallel if and only if their normal vectors are parallel.

Common stuck point: Two planes intersect in a LINE (not a point), unless they are parallel. To find the intersection line, solve the two plane equations simultaneously—you'll get parametric equations for a line.

Sense of Study hint: Read the normal vector directly from the coefficients: ax + by + cz = d has normal (a, b, c). Use a known point to find 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).

Solution

  1. 1
    The equation of a plane with normal \langle A, B, C \rangle through (x_0, y_0, z_0) is A(x-x_0) + B(y-y_0) + C(z-z_0) = 0.
  2. 2
    Substitute: 2(x-1) - 3(y-4) + 1(z+2) = 0.
  3. 3
    Expand: 2x - 2 - 3y + 12 + z + 2 = 0, so 2x - 3y + z + 12 = 0.

Answer

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).

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.
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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