How to use this calculator
- Enter or choose vector.
- The unit vector û updates as you type. Use Copy result or Share to keep it.
Method
û = v ÷ |v| = (v₁/|v|, v₂/|v|, …, vₙ/|v|); |v| = √(Σ vᵢ²)
- v
- the vector
- |v|
- its magnitude
- û
- the unit vector
Worked example
- |v| = √(3² + 4²) = √25 = 5.
- Divide each component by 5: 3 ÷ 5 = 0.6 and 4 ÷ 5 = 0.8.
- Check: 0.6² + 0.8² = 0.36 + 0.64 = 1.
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.