Plane Through Three Points (3D) Calculator

Published on: May 17, 2026
Final Answer: Free • Full Steps: Pro

This Plane Through Three Points (3D) Calculator writes the equation of the plane containing three given points, and shows the working line by line. Two vectors are built inside the plane first, and their cross product is perpendicular to both, which makes it the normal. One of the original points then fixes where the plane sits. Following the same pattern every time keeps the normal and the point as separate findings.

Step-by-step method

  1. Set up the problem.
  2. Find two direction vectors in the plane and compute the normal vector n = v × w.
  3. Use the point-normal plane equation and simplify.
See Example 1 Hide Example 1

Problem

\(P_{1} = \left(1,\; 0,\; 0\right),\; P_{2} = \left(0,\; 1,\; 0\right),\; P_{3} = \left(0,\; 0,\; 1\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Find two direction vectors in the plane and compute the normal vector n = v × w.
  3. Use the point-normal plane equation and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: The points are \(P_{1} = \left(1,\; 0,\; 0\right)\), \(P_{2} = \left(0,\; 1,\; 0\right)\), and \(P_{3} = \left(0,\; 0,\; 1\right)\).

\(P_{1} = \left(1,\; 0,\; 0\right),\; P_{2} = \left(0,\; 1,\; 0\right),\; P_{3} = \left(0,\; 0,\; 1\right)\)

Step 2

Step 2 - Write the normal vector formula.

In this problem: Two in-plane vectors give the normal through their cross product.

\(\vec{n} = \left(P_{2} - P_{1}\right) \times \left(P_{3} - P_{1}\right)\)

Step 3

Step 3 - Find two direction vectors in the plane and compute the normal vector n = v × w.

In this problem: The direction vectors are \(\left\langle -1,\; 1,\; 0 \right\rangle\) and \(\left\langle -1,\; 0,\; 1 \right\rangle\), and their cross product is \(\left\langle 1,\; 1,\; 1 \right\rangle\).

\(\vec{v} = \left\langle -1,\; 1,\; 0 \right\rangle,\; \vec{w} = \left\langle -1,\; 0,\; 1 \right\rangle,\; \vec{n} = \vec{v} \times \vec{w} = \left\langle 1,\; 1,\; 1 \right\rangle\)

Step 4

Step 4 - Use the point-normal plane equation and simplify.

In this problem: Using \(P_{1}\) with the normal gives \(x + y + z = 1\).

\(x + y + z = 1\)

Final Answer

\(x + y + z = 1\)
See Example 2 Hide Example 2

Problem

\(P_{1} = \left(1,\; 2,\; 3\right),\; P_{2} = \left(2,\; 3,\; 1\right),\; P_{3} = \left(3,\; 1,\; 2\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Find two direction vectors in the plane and compute the normal vector n = v × w.
  3. Use the point-normal plane equation and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: The points are \(P_{1} = \left(1,\; 2,\; 3\right)\), \(P_{2} = \left(2,\; 3,\; 1\right)\), and \(P_{3} = \left(3,\; 1,\; 2\right)\).

\(P_{1} = \left(1,\; 2,\; 3\right),\; P_{2} = \left(2,\; 3,\; 1\right),\; P_{3} = \left(3,\; 1,\; 2\right)\)

Step 2

Step 2 - Write the normal vector formula.

In this problem: Two in-plane vectors give the normal through their cross product.

\(\vec{n} = \left(P_{2} - P_{1}\right) \times \left(P_{3} - P_{1}\right)\)

Step 3

Step 3 - Find two direction vectors in the plane and compute the normal vector n = v × w.

In this problem: The direction vectors are \(\left\langle 1,\; 1,\; -2 \right\rangle\) and \(\left\langle 2,\; -1,\; -1 \right\rangle\), and their cross product is \(\left\langle -3,\; -3,\; -3 \right\rangle\).

\(\vec{v} = \left\langle 1,\; 1,\; -2 \right\rangle,\; \vec{w} = \left\langle 2,\; -1,\; -1 \right\rangle,\; \vec{n} = \vec{v} \times \vec{w} = \left\langle -3,\; -3,\; -3 \right\rangle\)

Step 4

Step 4 - Use the point-normal plane equation and simplify.

In this problem: Using \(P_{1}\) with the normal gives \(-3x - 3y - 3z = -18\).

\(-3x - 3y - 3z = -18\)

Final Answer

\(-3x - 3y - 3z = -18\)

Example Problems

Click one to load it into the box and solve it.