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Writing the equation of a plane in 3D

2026-10-02Spatial GeometryMr Wong

Build Ax + By + Cz + D = 0 from a normal vector and one known point.

Key formulas

Ax + By + Cz + D = 0
A(x − x₀) + B(y − y₀) + C(z − z₀) = 0

Read the normal

In Ax + By + Cz + D = 0, the vector (A, B, C) is normal to the plane. Any point on the plane determines D. Keep ordered triples neat and expand only after the signs look right.

Worked example

The plane passes through M(1, 2, 0) with normal n = (3, −1, 4). Start with 3(x − 1) − 1(y − 2) + 4(z − 0) = 0. Expand: 3x − 3 − y + 2 + 4z = 0, so 3x − y + 4z − 1 = 0. Check by substituting M: 3 − 2 + 0 − 1 = 0.

Second example

Through (2, 0, 1) with normal (1, 2, −1): 1(x − 2) + 2(y − 0) − 1(z − 1) = 0 → x − 2 + 2y − z + 1 = 0 → x + 2y − z − 1 = 0.

Worked example

Worked example

Problem. A plane passes through M(1, 2, 0) and has normal vector n = (3, −1, 4). Write its equation.

Solution. Use n · (r − M) = 0: 3(x − 1) − 1(y − 2) + 4(z − 0) = 0 → 3x − y + 4z − 1 = 0. Check: substituting M gives 3 − 2 + 0 − 1 = 0.

Verification habit

Always substitute the given point back into the finished equation. A non-zero result means a sign error.

Review quiz

A plane through (5,6,−7) with normal (8,−5,9) has equation:

Practice questions

  1. Write the plane through (0, 3, −1) with normal (2, −2, 1).
  2. Show that (1, 1, 2) lies on x + 2y − z − 1 = 0.

Study notes

Can I multiply the equation by a constant?

Yes. 3x − y + 4z − 1 = 0 and 6x − 2y + 8z − 2 = 0 describe the same plane.

What if I only have three points?

Form two direction vectors, take their cross product as the normal, then use one point.

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