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

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 P(x₀, y₀, z₀) and the normal vector n = ⟨a, b, c⟩.
2. Write the point-normal plane equation.
3. Substitute values and simplify.

Formula:

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

Step 1 - Identify the point and normal components.

In this problem: Extract x₀, y₀, z₀ and a, b, c from the input.

Step 2 - Write the point-normal plane equation.

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

Step 3a - Substitute values.

In this problem: Substitute your point and normal values into the formula.

Step 3b - Expand.

In this problem: Expand and combine like terms.

Step 3c - Write standard form.

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

Final answer: a(x-x0)+b(y-y0)+c(z-z0)=0 with a,b,c=(2,-1,4) and (x0,y0,z0)=(1,2,3)

Example 2: (1/2,0,-3),(3,2,-4)

Step 1 - Identify the point and normal components.

In this problem: Extract x₀, y₀, z₀ and a, b, c from the input.

Step 2 - Write the point-normal plane equation.

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

Step 3a - Substitute values.

In this problem: Substitute your point and normal values into the formula.

Step 3b - Expand.

In this problem: Expand and combine like terms.

Step 3c - Write standard form.

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

Final answer: a(x-x0)+b(y-y0)+c(z-z0)=0 with a,b,c=(3,2,-4) and (x0,y0,z0)=(1/2,0,-3)