Plane Through a Point with a Normal Vector (3D) Calculator

Published on: May 10, 2026

This Plane Through a Point with a Normal Vector (3D) Calculator helps you find the equation of a plane in three-dimensional space from a given point and normal vector. Use the point as a known position on the plane and the normal vector to determine the plane’s orientation. Then substitute those values into the point-normal form and simplify to get the equation. It is a simple way to check answers, understand the method clearly, and practise 3D coordinate geometry step by step.

Step-by-step method

  1. Identify the point P₀ = (x₀,y₀,z₀) and normal vector n = ⟨a,b,c⟩.
  2. Use the point-normal plane equation.
  3. Substitute the point and normal vector, then simplify to standard form.

Formula:

a(x − x₀) + b(y − y₀) + c(z − z₀) = 0
ax + by + cz = d, where d = ax₀ + by₀ + cz₀

Example 1: (1,2,3),(2,-1,4)

Step 1 - Identify the point and normal vector.

In this problem: Read the point P₀ and the normal vector n.

P₀ = ( 1, 2, 3 )
n = ⟨2, -1, 4⟩

Step 2 - Write the point-normal form.

In this problem: Use a(x − x₀) + b(y − y₀) + c(z − z₀) = 0.

a(x − x₀) + b(y − y₀) + c(z − z₀) = 0

Step 3a - Substitute values.

In this problem: Substitute the point and normal vector.

2(x − 1) + -1(y − 2) + 4(z − 3) = 0

Step 3b - Write standard form.

In this problem: Rearrange into ax + by + cz = d.

−y + 2x + 4z = 12

Final answer: n=<2,-1,4>, plane: 2x+-1y+4z=12

Example 2: (1/2,0,-3),(6,-3,9)

Step 1 - Identify the point and normal vector.

In this problem: Read the point P₀ and the normal vector n.

P₀ = (
1
2
, 0, -3 )
n = ⟨6, -3, 9⟩

Step 1b - Simplify the normal vector.

In this problem: Use a simpler equivalent normal vector.

n = ⟨2, -1, 3⟩

Step 2 - Write the point-normal form.

In this problem: Use a(x − x₀) + b(y − y₀) + c(z − z₀) = 0.

a(x − x₀) + b(y − y₀) + c(z − z₀) = 0

Step 3a - Substitute values.

In this problem: Substitute the point and normal vector.

2(x −
1
2
) + -1y + 3(z + 3) = 0

Step 3b - Write standard form.

In this problem: Rearrange into ax + by + cz = d.

−y + 2x + 3z = -8

Final answer: n=<2,-1,3>, plane: 2x+-1y+3z=-8