Direction Vector from Symmetric Equation (3D) Notes

Published on: September 29, 2026

This Direction Vector from Symmetric Equation (3D) Calculator reads the direction a line runs in straight off its symmetric form, and shows the working line by line. The three denominators are the components of the direction vector, because each one says how far that coordinate moves for one step of the parameter. The numerators fix a point instead. Following the same pattern every time keeps the denominators from being mistaken for coordinates.

Step-by-step method

  1. Set up the problem.
  2. Use d = ⟨a, b, c⟩.
  3. Substitute values to write the direction vector.
See Example 1 Hide Example 1

Problem

\(a = 2,\; b = -1,\; c = 4\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use d = ⟨a, b, c⟩.
  3. Substitute values to write the direction vector.

Step 1

Step 1 - Set up the problem.

In this problem: The denominators of the symmetric form are the direction components.

\(a = 2,\; b = -1,\; c = 4\)

Step 2

Step 2 - Use d = ⟨a, b, c⟩.

In this problem: The direction vector collects the three denominators.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; \vec{d} = \left\langle a, b, c \right\rangle\)

Step 3

Step 3 - Substitute values to write the direction vector.

In this problem: The direction vector is \(\left\langle 2,\; -1,\; 4 \right\rangle\).

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

Final Answer

\(\vec{d} = \left\langle 2,\; -1,\; 4 \right\rangle\)
See Example 2 Hide Example 2

Problem

\(a = 3,\; b = 2,\; c = -4\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use d = ⟨a, b, c⟩.
  3. Substitute values to write the direction vector.

Step 1

Step 1 - Set up the problem.

In this problem: The denominators of the symmetric form are the direction components.

\(a = 3,\; b = 2,\; c = -4\)

Step 2

Step 2 - Use d = ⟨a, b, c⟩.

In this problem: The direction vector collects the three denominators.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; \vec{d} = \left\langle a, b, c \right\rangle\)

Step 3

Step 3 - Substitute values to write the direction vector.

In this problem: The direction vector is \(\left\langle 3,\; 2,\; -4 \right\rangle\).

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

Final Answer

\(\vec{d} = \left\langle 3,\; 2,\; -4 \right\rangle\)
See Example 3 Hide Example 3

Problem

\(a = 5,\; b = -2,\; c = 1\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use d = ⟨a, b, c⟩.
  3. Substitute values to write the direction vector.

Step 1

Step 1 - Set up the problem.

In this problem: The denominators of the symmetric form are the direction components.

\(a = 5,\; b = -2,\; c = 1\)

Step 2

Step 2 - Use d = ⟨a, b, c⟩.

In this problem: The direction vector collects the three denominators.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; \vec{d} = \left\langle a, b, c \right\rangle\)

Step 3

Step 3 - Substitute values to write the direction vector.

In this problem: The direction vector is \(\left\langle 5,\; -2,\; 1 \right\rangle\).

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

Final Answer

\(\vec{d} = \left\langle 5,\; -2,\; 1 \right\rangle\)
See Example 4 Hide Example 4

Problem

\(a = 1,\; b = 3,\; c = -6\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use d = ⟨a, b, c⟩.
  3. Substitute values to write the direction vector.

Step 1

Step 1 - Set up the problem.

In this problem: The denominators of the symmetric form are the direction components.

\(a = 1,\; b = 3,\; c = -6\)

Step 2

Step 2 - Use d = ⟨a, b, c⟩.

In this problem: The direction vector collects the three denominators.

\(\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}\;\Rightarrow\; \vec{d} = \left\langle a, b, c \right\rangle\)

Step 3

Step 3 - Substitute values to write the direction vector.

In this problem: The direction vector is \(\left\langle 1,\; 3,\; -6 \right\rangle\).

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

Final Answer

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