Line Through Two Points (3D) Calculator

Published on: April 19, 2026
Final Answer: Free • Full Steps: Pro

This Line Through Two Points (3D) Calculator writes the equation of the line joining two points in space, and shows the working line by line. Subtracting one point from the other gives a direction vector, and either point then serves as a place to start from. A parameter slides along that direction to reach every point on the line. Following the same pattern every time keeps the starting point and the direction distinct.

Step-by-step method

  1. Set up the problem.
  2. Compute the direction vector P₂ − P₁.
  3. Write the vector form r = r₁ + t(P₂ − P₁) and substitute values.
  4. Write the parametric form (and symmetric form if applicable).
See Example 1 Hide Example 1

Problem

\(P_{1} = \left(1,\; 2,\; 3\right),\; P_{2} = \left(4,\; 5,\; 6\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Compute the direction vector P₂ − P₁.
  3. Write the vector form r = r₁ + t(P₂ − P₁) and substitute values.
  4. Write the parametric form (and symmetric form if applicable).

Step 1

Step 1 - Set up the problem.

In this problem: The points are \(P_{1} = \left(1,\; 2,\; 3\right)\) and \(P_{2} = \left(4,\; 5,\; 6\right)\).

\(P_{1} = \left(1,\; 2,\; 3\right),\; P_{2} = \left(4,\; 5,\; 6\right)\)

Step 2

Step 2 - Write the line formula.

In this problem: The direction comes from subtracting the two points.

\(\vec{r} = \vec{r}_{1} + t\left(P_{2} - P_{1}\right)\)

Step 3

Step 3 - Compute the direction vector P₂ − P₁.

In this problem: The direction vector is \(\left\langle 3,\; 3,\; 3 \right\rangle\).

\(\vec{d} = \left\langle 4 - 1,\; 5 - 2,\; 6 - 3 \right\rangle = \left\langle 3,\; 3,\; 3 \right\rangle\)

Step 4

Step 4 - Write the vector form r = r₁ + t(P₂ − P₁) and substitute values.

In this problem: Substitute the base point and direction.

\(\vec{r} = \left\langle 1,\; 2,\; 3 \right\rangle + t\left\langle 3,\; 3,\; 3 \right\rangle\)

Step 5

Step 5 - Write the parametric form (and symmetric form if applicable).

In this problem: Read off the parametric equations and solve for t to get the symmetric form.

\(\begin{gathered} x = 1 + 3t \\ y = 2 + 3t \\ z = 3 + 3t \\ \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \end{gathered}\)

Final Answer

\(\frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3}\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Compute the direction vector P₂ − P₁.
  3. Write the vector form r = r₁ + t(P₂ − P₁) and substitute values.
  4. Write the parametric form (and symmetric form if applicable).

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the line formula.

In this problem: The direction comes from subtracting the two points.

\(\vec{r} = \vec{r}_{1} + t\left(P_{2} - P_{1}\right)\)

Step 3

Step 3 - Compute the direction vector P₂ − P₁.

In this problem: The direction vector is \(\left\langle 1,\; 0,\; 0 \right\rangle\).

\(\vec{d} = \left\langle 1 - 0,\; 0 - 0,\; 0 - 0 \right\rangle = \left\langle 1,\; 0,\; 0 \right\rangle\)

Step 4

Step 4 - Write the vector form r = r₁ + t(P₂ − P₁) and substitute values.

In this problem: Substitute the base point and direction.

\(\vec{r} = \left\langle 0,\; 0,\; 0 \right\rangle + t\left\langle 1,\; 0,\; 0 \right\rangle\)

Step 5

Step 5 - Write the parametric form (and symmetric form if applicable).

In this problem: Read off the parametric equations.

\(\begin{gathered} x = t \\ y = 0 \\ z = 0 \end{gathered}\)

Final Answer

\(\vec{r} = \left\langle 0,\; 0,\; 0 \right\rangle + t\left\langle 1,\; 0,\; 0 \right\rangle\)

Example Problems

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