Unit Vector Calculator
Calculate the unit vector (normalized vector) in the direction of a given 2D, 3D, or n-dimensional vector. Get the magnitude, each normalized component, direction angles, step-by-step normalization process, and visual verification that the result has length 1.
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About Unit Vector Calculator
The Unit Vector Calculator computes the normalized vector (unit vector) in the direction of any given 2D, 3D, or n-dimensional vector using the formula \(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\). Enter your vector components to instantly get the unit vector, magnitude, direction angles, scale factor, and a step-by-step normalization process with visual verification that the resulting vector has length 1.
What Is a Unit Vector?
A unit vector is a vector whose magnitude (length) is exactly 1. It preserves only the direction of the original vector, stripping away the magnitude. Unit vectors are denoted with a "hat" symbol: \(\hat{v}\) (read as "v-hat"). Every non-zero vector has a unique unit vector pointing in the same direction.
Standard Basis Unit Vectors
Any vector can be expressed as a linear combination of these basis unit vectors: \(\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}\).
Unit Vector Formula
| Property | Formula | Description |
|---|---|---|
| Unit Vector | \(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\) | Divide each component by the magnitude |
| Magnitude | \(|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\) | Euclidean norm (length) of the vector |
| Verification | \(|\hat{v}| = 1\) | The unit vector always has length 1 |
| Direction Cosines | \(\cos\alpha = \hat{v}_x, \; \cos\beta = \hat{v}_y, \; \cos\gamma = \hat{v}_z\) | Components of the unit vector are the direction cosines |
| Identity | \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\) | Sum of squared direction cosines always equals 1 |
Real-World Applications
How to Use the Unit Vector Calculator
- Select the dimension: Choose 2D, 3D, or Custom for higher dimensions. Or click a quick example to auto-fill a sample vector.
- Enter the vector: Type the components separated by commas (e.g., 3, 4 for 2D or 1, 2, 3 for 3D).
- Watch the live preview: The diagram updates in real-time, showing both the original vector and unit vector on a unit circle.
- Click Normalize Vector: Press the button to get the full results including the unit vector, direction angles, component breakdown, and step-by-step verification.
- Explore the animation: Click the Animate button to watch the normalization process — the original vector smoothly shrinks to the unit circle.
Properties of Unit Vectors
- Magnitude is always 1: \(|\hat{v}| = 1\) by definition — this is the key verification for any normalization.
- Same direction as original: \(\hat{v}\) points in the exact same direction as \(\vec{v}\).
- Scalar relationship: \(\vec{v} = |\vec{v}| \cdot \hat{v}\), so any vector equals its magnitude times its unit vector.
- Direction cosines: The components of a unit vector are exactly the cosines of the angles with each coordinate axis.
- Dot product relation: \(\hat{a} \cdot \hat{b} = \cos\theta\), where θ is the angle between the unit vectors.
FAQ
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"Unit Vector Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-10
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