Vector Normalization Calculator
Transform vectors into unit vectors and visualize them instantly. Understand vector normalization with our interactive tool.
Enter Your Vector
Comma-separated values
Enter the vector components separated by commas.
Normalized Vector
Vector Visualization
Normalization Formula
The normalized vector \( \mathbf{\hat{v}} \) is obtained by dividing the original vector \( \mathbf{v} \) by its magnitude \( ||\mathbf{v}|| \).
$$ \mathbf{\hat{v}} = \frac{\mathbf{v}}{||\mathbf{v}||} $$
- \( \mathbf{\hat{v}} \): Normalized vector (unit vector).
- \( \mathbf{v} \): Original vector.
- \( ||\mathbf{v}|| \): Magnitude of vector \( \mathbf{v} \).
Understanding Vector Normalization
Vector normalization scales a vector to have a length of 1, creating a unit vector. This process preserves the vector's direction but removes its magnitude, which is crucial in many applications.
Why Normalize Vectors?
- Focus on Direction: Unit vectors represent direction without magnitude influence, simplifying directional comparisons.
- Cosine Similarity: Used in machine learning to measure the similarity between vectors based on their direction.
- Graphics and Physics: Essential for lighting calculations, surface normals in 3D graphics, and direction vectors in physics simulations.
Example
For vector \( \mathbf{v} = [3, 4] \), the magnitude is \( ||\mathbf{v}|| = \sqrt{3^2 + 4^2} = 5 \). The normalized vector is \( \mathbf{\hat{v}} = [\frac{3}{5}, \frac{4}{5}] = [0.6, 0.8] \).