# Unit Vector Calculator

> Normalise a vector by dividing each component by its magnitude to get the unit vector in the same direction, with the 2D direction angle.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/unit-vector-calculator

A unit vector keeps a vector’s direction but has length 1. Divide every component by the vector’s magnitude.

Also called: Vector normalisation calculator, Normalize vector calculator.

## Inputs
- Vector (default 3, 4)

## Results
- Unit vector û
- Original magnitude |v|
- Direction angle (2D only)
- Number of components

## Formula

```
û = v ÷ |v| = (v₁/|v|, v₂/|v|, …, vₙ/|v|);  |v| = √(Σ vᵢ²)
```

Where:
- v: the vector
- |v|: its magnitude
- û: the unit vector

## Worked example
1. |v| = √(3² + 4²) = √25 = 5.
2. Divide each component by 5: 3 ÷ 5 = 0.6 and 4 ÷ 5 = 0.8.
3. Check: 0.6² + 0.8² = 0.36 + 0.64 = 1.

**Result:** The unit vector in the direction of (3, 4) is (0.6, 0.8).

## What the result means

The components of a unit vector are the direction cosines: the cosines of the angles it makes with each axis. Multiply the unit vector by any length to get a vector of that length in the same direction.

## Assumptions and limits
- Components are in an ordinary Cartesian basis.
- The zero vector has no direction, so it cannot be normalised.
- Components are shown to 6 decimal places, so their squares may sum to 1 only after rounding.

## Questions

### How do I get a vector of length 10 in the same direction?

Multiply each component of the unit vector by 10.

### Is the unit vector unique?

For a given direction, yes. The opposite direction gives −û.

## Related calculators
- [Vector Magnitude Calculator](https://calculators.geekmidst.com/calculators/vector-magnitude-calculator)
- [Vector Angle Calculator](https://calculators.geekmidst.com/calculators/vector-angle-calculator)
- [Slope Calculator](https://calculators.geekmidst.com/calculators/slope-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/unit-vector-calculator