Unit Vector Calculator
MiniWebtool’s Unit Vector Calculator normalizes a nonzero vector by its magnitude. Enter two, three, or up to fifty components to see the unit vector and applicable direction-angle information.
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Enter a 2D, 3D or longer vector below to get its magnitude, its unit vector and its direction angles.
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.
At a Glance
Unit Vector Calculator normalizes a nonzero vector by dividing each component by its magnitude. It accepts 2D and 3D vectors, or up to 50 components, and shows the unit vector with direction angles where they apply.
| Feature | What you get | Good for |
|---|---|---|
| Dimensions | Choose 2D, 3D or Custom and type the components separated by commas, such as 3, 4 or 1, 2, 3. Custom takes up to 50 components. | Vectors from a school problem up to higher-dimensional data |
| Formula | Computes v̂ = v / |v|, where |v| is the Euclidean magnitude, and checks that the result has length 1. | Verifying a normalization by hand |
| Direction angles | Shows the angles α, β and γ with the x, y and z axes, found with arccosine of each unit-vector component. | Physics and engineering problems that ask for direction cosines |
| Step-by-step working | Lists the magnitude, the scale factor and each component division. | Learning how normalization is done |
| Live diagram | A unit-circle diagram shows the original and unit vectors as you type, with a Grid view and an Animate button that shrinks the vector onto the circle. | Seeing that direction stays the same while length becomes 1 |
| Zero vector check | Detects the zero vector, which has no unit vector, and displays a message instead of a result. | Catching an invalid input |
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
Reference this content, page, or tool as:
"Unit Vector Calculator" at https://MiniWebtool.com/unit-vector-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-10
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