Parametric Equations from Symmetric Form (3D) Notes

Published on: September 29, 2026

This Parametric Equations from Symmetric Form (3D) Calculator rewrites a line given in symmetric form as three parametric equations, and shows the working line by line. Each fraction in the symmetric form is set equal to the same parameter, and rearranging each one gives a coordinate in terms of it. The point and the direction fall straight out of the form. Following the same pattern every time keeps the numerators and denominators in their roles.

Step-by-step method

  1. Set up the problem.
  2. Write the parametric equations.
  3. Substitute values.
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the parametric equations.
  3. Substitute values.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the parametric equations.

In this problem: Each fraction equals the parameter t, so each coordinate can be solved for.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; x = x_{0} + at,\; y = y_{0} + bt,\; z = z_{0} + ct\)

Step 3

Step 3 - Substitute values.

In this problem: Substitute the point and direction components.

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the parametric equations.
  3. Substitute values.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the parametric equations.

In this problem: Each fraction equals the parameter t, so each coordinate can be solved for.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; x = x_{0} + at,\; y = y_{0} + bt,\; z = z_{0} + ct\)

Step 3

Step 3 - Substitute values.

In this problem: Substitute the point and direction components.

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

Final Answer

\(\begin{gathered} x = \frac{1}{2} + 3t \\ y = -1 + 2t \\ z = 5 - 4t \end{gathered}\)
See Example 3 Hide Example 3

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the parametric equations.
  3. Substitute values.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the parametric equations.

In this problem: Each fraction equals the parameter t, so each coordinate can be solved for.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; x = x_{0} + at,\; y = y_{0} + bt,\; z = z_{0} + ct\)

Step 3

Step 3 - Substitute values.

In this problem: Substitute the point and direction components.

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the parametric equations.
  3. Substitute values.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the parametric equations.

In this problem: Each fraction equals the parameter t, so each coordinate can be solved for.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; x = x_{0} + at,\; y = y_{0} + bt,\; z = z_{0} + ct\)

Step 3

Step 3 - Substitute values.

In this problem: Substitute the point and direction components.

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

Final Answer

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