Direction Vector from Symmetric Equation (3D) Calculator

This Direction Vector from Symmetric Equation (3D) Calculator helps you find the direction vector of a line in three-dimensional space from its symmetric equation. Read the denominators of the symmetric form to identify the direction values, and treat any constant coordinate as having a zero component in the direction vector. This gives the vector that shows the line’s direction in x, y, and z. It is a simple way to check answers, understand the method clearly, and practise 3D coordinate geometry step by step.

Step-by-step method

1. Identify the direction components a, b, c from the symmetric form.
2. Use d = ⟨a, b, c⟩.
3. Substitute values to write the direction vector.

Formula:

Example 1:

Step 1 - Identify the direction components a, b, c from the symmetric form.

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

Step 2 - Use d = ⟨a, b, c⟩.

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

Step 3 - Substitute values to write the direction vector.

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

Final answer:

Example 2:

Step 1 - Identify the direction components a, b, c from the symmetric form.

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

Step 2 - Use d = ⟨a, b, c⟩.

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

Step 3 - Substitute values to write the direction vector.

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

Final answer:

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

1. Identify the direction components a, b, c from the symmetric form.
2. Use d = ⟨a, b, c⟩.
3. Substitute values to write the direction vector.

Step 1

Step 1 - Identify the direction components a, b, c from the symmetric form.

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. Identify the direction components a, b, c from the symmetric form.
2. Use d = ⟨a, b, c⟩.
3. Substitute values to write the direction vector.

Step 1

Step 1 - Identify the direction components a, b, c from the symmetric form.

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

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