Unit Vector Calculator

This unit vector calculator divides a vector by its magnitude to give the unit vector in the same direction — for example ⟨3, 4⟩ becomes ⟨3/5, 4/5⟩. It works for 2D, 3D and 4D vectors, keeps answers exact (with simplified square roots), and checks that the result has length 1.

Formula

\hat{\mathbf{v}} = \frac{\mathbf{v}}{\lVert \mathbf{v} \rVert}

Divide each component by the magnitude of the vector.

\lVert \langle a, b, c \rangle \rVert = \sqrt{a^2 + b^2 + c^2}

The magnitude (length) of a vector, by Pythagoras.

How to use this calculator

Type the components of the vector separated by commas, such as 3, 4 or 1, 2, 2. Fractions and decimals work too.
Press Calculate to get the unit vector, its decimals, and the magnitude of the original vector.
The steps show the magnitude found by Pythagoras, then each component divided by it.

Worked examples

Easy: a 2D vector

|v| = √(3² + 4²) = √25 = 5.

Divide each component by 5: ⟨3/5, 4/5⟩.

Medium: a 3D vector

|v| = √(1 + 4 + 4) = 3.

The unit vector is ⟨1/3, 2/3, 2/3⟩.

Hard: a square root in the magnitude

|v| = √(4 + 1) = √5.

Divide and rationalise: 2/√5 = 2√5/5 and −1/√5 = −√5/5.

Unit vector ≈ ⟨0.894427, −0.447214⟩.

Common mistakes

Forgetting to square the negative components: (−1)² is 1, not −1.
Dividing by the sum of the components instead of the magnitude.
Trying to normalise the zero vector — it has no direction, so it has no unit vector.
Rounding the magnitude before dividing; keep √5 exact until the end.

Frequently asked questions

What is a unit vector?

A vector of length 1. It keeps only the direction of the original vector, which is why it is used to describe directions.

How do I find the unit vector of a vector?

Find its magnitude with Pythagoras, √(a² + b² + c²), then divide every component by that magnitude.

How do I check a unit vector?

Square its components and add them: the total must be exactly 1. The calculator does this check automatically.

Result