Vector Magnitude Calculator

How it works

1. 1Enter the x and y components (and z for 3D) of your vector.
2. 2The magnitude |v| is computed as the Euclidean norm: √(x² + y² + z²).
3. 3The unit vector v̂ = v / |v| is shown alongside — it points in the same direction with magnitude 1.

Use cases

• Physics: find the speed from a velocity vector or the force magnitude from component forces.
• Computer graphics: normalize direction vectors before lighting calculations or camera transforms.
• Engineering: determine the resultant displacement or force from its x, y, z components.

Frequently asked questions

What is the magnitude of a vector?

The magnitude (or length) |v| of a vector v = (x, y, z) is the straight-line distance from the origin to the tip of the vector: |v| = √(x² + y² + z²). For 2D vectors it simplifies to √(x² + y²).

What is a unit vector?

A unit vector v̂ is a vector that points in the same direction as v but has a magnitude of exactly 1. It is computed by dividing each component by |v|: v̂ = v / |v|. The result is undefined for the zero vector since it has no direction.

Why is the unit vector useful?

Unit vectors let you separate a vector’s direction from its magnitude. In graphics, physics simulations, and navigation, you often need to know which way something points without caring how far — that’s the job of the unit vector.