Line Through a Point (3D) Notes

Published on: September 29, 2026

This Line Through a Point (3D) Calculator writes the equation of the line through a given point in a given direction, and shows the working line by line. The point fixes where the line sits and the direction vector fixes which way it runs, so together they pin it down completely. A parameter slides along the direction to reach every other point. Following the same pattern every time keeps position and direction from being confused.

Step-by-step method

  1. Set up the problem.
  2. Write the vector form of the line and substitute values.
  3. Write the parametric form and substitute values.
  4. Write the symmetric form when every direction component is nonzero.
See Example 1 Hide Example 1

Problem

\(P_{0} = \left(1,\; 2,\; 3\right),\; \vec{d} = \left\langle 4,\; 5,\; 6 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the vector form of the line and substitute values.
  3. Write the parametric form and substitute values.
  4. Write the symmetric form when every direction component is nonzero.

Step 1

Step 1 - Set up the problem.

In this problem: The point is \(P_{0} = \left(1,\; 2,\; 3\right)\) and the direction is \(\vec{d} = \left\langle 4,\; 5,\; 6 \right\rangle\).

\(P_{0} = \left(1,\; 2,\; 3\right),\; \vec{d} = \left\langle 4,\; 5,\; 6 \right\rangle\)

Step 2

Step 2 - Write the line formula.

In this problem: Every point on the line is the base point plus a multiple of the direction vector.

\(\vec{r} = \vec{r}_{0} + t\vec{d}\)

Step 3

Step 3 - Write the vector form of the line and substitute values.

In this problem: Substitute the point and direction into the vector form.

\(\vec{r} = \left\langle 1,\; 2,\; 3 \right\rangle + t\left\langle 4,\; 5,\; 6 \right\rangle\)

Step 4

Step 4 - Write the parametric form and substitute values.

In this problem: Read off one parametric equation per coordinate.

\(\begin{gathered} x = 1 + 4t \\ y = 2 + 5t \\ z = 3 + 6t \end{gathered}\)

Step 5

Step 5 - Write the symmetric form when every direction component is nonzero.

In this problem: All direction components are nonzero, so solve each parametric equation for t.

\(\frac{x - 1}{4} = \frac{y - 2}{5} = \frac{z - 3}{6}\)

Final Answer

\(\frac{x - 1}{4} = \frac{y - 2}{5} = \frac{z - 3}{6}\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the vector form of the line and substitute values.
  3. Write the parametric form and substitute values.
  4. Write the symmetric form when every direction component is nonzero.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the line formula.

In this problem: Every point on the line is the base point plus a multiple of the direction vector.

\(\vec{r} = \vec{r}_{0} + t\vec{d}\)

Step 3

Step 3 - Write the vector form of the line and substitute values.

In this problem: Substitute the point and direction into the vector form.

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

Step 4

Step 4 - Write the parametric form and substitute values.

In this problem: Read off one parametric equation per coordinate.

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

Step 5

Step 5 - Write the symmetric form when every direction component is nonzero.

In this problem: A direction component is zero, so the symmetric form is not available; the parametric form is the final answer.

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

Final Answer

\(\vec{r} = \left\langle 0,\; 0,\; 0 \right\rangle + t\left\langle 1,\; 0,\; 0 \right\rangle\)
See Example 3 Hide Example 3

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the vector form of the line and substitute values.
  3. Write the parametric form and substitute values.
  4. Write the symmetric form when every direction component is nonzero.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the line formula.

In this problem: Every point on the line is the base point plus a multiple of the direction vector.

\(\vec{r} = \vec{r}_{0} + t\vec{d}\)

Step 3

Step 3 - Write the vector form of the line and substitute values.

In this problem: Substitute the point and direction into the vector form.

\(\vec{r} = \left\langle 2,\; -1,\; 4 \right\rangle + t\left\langle 3,\; 1,\; -2 \right\rangle\)

Step 4

Step 4 - Write the parametric form and substitute values.

In this problem: Read off one parametric equation per coordinate.

\(\begin{gathered} x = 2 + 3t \\ y = -1 + t \\ z = 4 - 2t \end{gathered}\)

Step 5

Step 5 - Write the symmetric form when every direction component is nonzero.

In this problem: All direction components are nonzero, so solve each parametric equation for t.

\(\frac{x - 2}{3} = \frac{y + 1}{1} = \frac{z - 4}{-2}\)

Final Answer

\(\frac{x - 2}{3} = \frac{y + 1}{1} = \frac{z - 4}{-2}\)
See Example 4 Hide Example 4

Problem

\(P_{0} = \left(-1,\; 0,\; 5\right),\; \vec{d} = \left\langle 2,\; 4,\; 1 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the vector form of the line and substitute values.
  3. Write the parametric form and substitute values.
  4. Write the symmetric form when every direction component is nonzero.

Step 1

Step 1 - Set up the problem.

In this problem: The point is \(P_{0} = \left(-1,\; 0,\; 5\right)\) and the direction is \(\vec{d} = \left\langle 2,\; 4,\; 1 \right\rangle\).

\(P_{0} = \left(-1,\; 0,\; 5\right),\; \vec{d} = \left\langle 2,\; 4,\; 1 \right\rangle\)

Step 2

Step 2 - Write the line formula.

In this problem: Every point on the line is the base point plus a multiple of the direction vector.

\(\vec{r} = \vec{r}_{0} + t\vec{d}\)

Step 3

Step 3 - Write the vector form of the line and substitute values.

In this problem: Substitute the point and direction into the vector form.

\(\vec{r} = \left\langle -1,\; 0,\; 5 \right\rangle + t\left\langle 2,\; 4,\; 1 \right\rangle\)

Step 4

Step 4 - Write the parametric form and substitute values.

In this problem: Read off one parametric equation per coordinate.

\(\begin{gathered} x = -1 + 2t \\ y = 4t \\ z = 5 + t \end{gathered}\)

Step 5

Step 5 - Write the symmetric form when every direction component is nonzero.

In this problem: All direction components are nonzero, so solve each parametric equation for t.

\(\frac{x + 1}{2} = \frac{y}{4} = \frac{z - 5}{1}\)

Final Answer

\(\frac{x + 1}{2} = \frac{y}{4} = \frac{z - 5}{1}\)