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

Published on: September 29, 2026

This Plane Through a Point with a Normal Vector (3D) Calculator writes the equation of a plane from one point on it and a vector perpendicular to it, and shows the working line by line. Every vector lying in the plane is at right angles to the normal, so their dot product is zero, and that condition is the equation. The normal supplies the coefficients directly. Following the same pattern every time keeps the point and the normal in their separate roles.

Step-by-step method

  1. Set up the problem.
  2. Use the point-normal plane equation.
  3. Substitute the point and normal vector, then simplify to standard form.
See Example 1 Hide Example 1

Problem

\(P_{0} = \left(1,\; 2,\; 3\right),\; \vec{n} = \left\langle 2,\; -3,\; 4 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use the point-normal plane equation.
  3. Substitute the point and normal vector, then simplify to standard form.

Step 1

Step 1 - Set up the problem.

In this problem: The point is \(P_{0} = \left(1,\; 2,\; 3\right)\) and the normal is \(\vec{n} = \left\langle 2,\; -3,\; 4 \right\rangle\).

\(P_{0} = \left(1,\; 2,\; 3\right),\; \vec{n} = \left\langle 2,\; -3,\; 4 \right\rangle\)

Step 2

Step 2 - Use the point-normal plane equation.

In this problem: The plane contains every point whose displacement from P₀ is perpendicular to n.

\(a\left(x - x_{0}\right) + b\left(y - y_{0}\right) + c\left(z - z_{0}\right) = 0\)

Step 3

Step 3 - Substitute the point and normal vector, then simplify to standard form.

In this problem: Expanding gives the standard form \(2x - 3y + 4z = 8\).

\(2\left(x - 1\right) + \left(-3\right)\left(y - 2\right) + 4\left(z - 3\right) = 0 \;\Rightarrow\; 2x - 3y + 4z = 8\)

Final Answer

\(2x - 3y + 4z = 8\)
See Example 2 Hide Example 2

Problem

\(P_{0} = \left(0,\; 0,\; 0\right),\; \vec{n} = \left\langle 1,\; 1,\; 1 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use the point-normal plane equation.
  3. Substitute the point and normal vector, then simplify to standard form.

Step 1

Step 1 - Set up the problem.

In this problem: The point is \(P_{0} = \left(0,\; 0,\; 0\right)\) and the normal is \(\vec{n} = \left\langle 1,\; 1,\; 1 \right\rangle\).

\(P_{0} = \left(0,\; 0,\; 0\right),\; \vec{n} = \left\langle 1,\; 1,\; 1 \right\rangle\)

Step 2

Step 2 - Use the point-normal plane equation.

In this problem: The plane contains every point whose displacement from P₀ is perpendicular to n.

\(a\left(x - x_{0}\right) + b\left(y - y_{0}\right) + c\left(z - z_{0}\right) = 0\)

Step 3

Step 3 - Substitute the point and normal vector, then simplify to standard form.

In this problem: Expanding gives the standard form \(x + y + z = 0\).

\(1\left(x\right) + 1\left(y\right) + 1\left(z\right) = 0 \;\Rightarrow\; x + y + z = 0\)

Final Answer

\(x + y + z = 0\)
See Example 3 Hide Example 3

Problem

\(P_{0} = \left(2,\; -1,\; 4\right),\; \vec{n} = \left\langle 1,\; 3,\; -2 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use the point-normal plane equation.
  3. Substitute the point and normal vector, then simplify to standard form.

Step 1

Step 1 - Set up the problem.

In this problem: The point is \(P_{0} = \left(2,\; -1,\; 4\right)\) and the normal is \(\vec{n} = \left\langle 1,\; 3,\; -2 \right\rangle\).

\(P_{0} = \left(2,\; -1,\; 4\right),\; \vec{n} = \left\langle 1,\; 3,\; -2 \right\rangle\)

Step 2

Step 2 - Use the point-normal plane equation.

In this problem: The plane contains every point whose displacement from P₀ is perpendicular to n.

\(a\left(x - x_{0}\right) + b\left(y - y_{0}\right) + c\left(z - z_{0}\right) = 0\)

Step 3

Step 3 - Substitute the point and normal vector, then simplify to standard form.

In this problem: Expanding gives the standard form \(x + 3y - 2z = -9\).

\(1\left(x - 2\right) + 3\left(y + 1\right) + \left(-2\right)\left(z - 4\right) = 0 \;\Rightarrow\; x + 3y - 2z = -9\)

Final Answer

\(x + 3y - 2z = -9\)
See Example 4 Hide Example 4

Problem

\(P_{0} = \left(-1,\; 0,\; 2\right),\; \vec{n} = \left\langle 4,\; -1,\; 1 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use the point-normal plane equation.
  3. Substitute the point and normal vector, then simplify to standard form.

Step 1

Step 1 - Set up the problem.

In this problem: The point is \(P_{0} = \left(-1,\; 0,\; 2\right)\) and the normal is \(\vec{n} = \left\langle 4,\; -1,\; 1 \right\rangle\).

\(P_{0} = \left(-1,\; 0,\; 2\right),\; \vec{n} = \left\langle 4,\; -1,\; 1 \right\rangle\)

Step 2

Step 2 - Use the point-normal plane equation.

In this problem: The plane contains every point whose displacement from P₀ is perpendicular to n.

\(a\left(x - x_{0}\right) + b\left(y - y_{0}\right) + c\left(z - z_{0}\right) = 0\)

Step 3

Step 3 - Substitute the point and normal vector, then simplify to standard form.

In this problem: Expanding gives the standard form \(4x - y + z = -2\).

\(4\left(x + 1\right) + \left(-1\right)\left(y\right) + 1\left(z - 2\right) = 0 \;\Rightarrow\; 4x - y + z = -2\)

Final Answer

\(4x - y + z = -2\)