Vector Magnitude Calculator
How Long Is a Vector?
A vector's magnitude is simply its length — how far its arrowhead sits from the origin, regardless of which direction it points. It's the same Pythagorean idea used for distance between two points, applied to a vector's own components instead.
The Formula
2D: |V| = √(x² + y²)
3D: |V| = √(x² + y² + z²)
3D: |V| = √(x² + y² + z²)
Worked Examples
| Vector | Calculation | Magnitude |
|---|---|---|
| (3, 4) | √(3²+4²) = √25 | 5 |
| (2, 3, 6) | √(2²+3²+6²) = √49 | 7 |
Where This Matters
- Speed from velocity — a velocity vector's magnitude is the scalar speed, stripped of direction information.
- Force strength — the magnitude of a force vector gives its strength independent of which way it points.
- Normalizing directions — dividing a vector by its own magnitude produces a unit vector, a standard step before combining directions in graphics or physics.
How to Use This Calculator
- Choose the dimension: 2D Vector or 3D Vector.
- Enter the Vector x, y (and z for 3D) components.
- Select Calculate to get the vector's magnitude.
Related Calculations
Need the direction too, not just the length? Use the Angle Between Vectors Calculator, or combine vectors first with the Vector Addition Calculator.