Free Vector Magnitude Calculator

Last updated October 10, 2026 · Report an error

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A vector's magnitude is its length — how "big" it is, independent of the direction it points — found via the Pythagorean theorem extended to however many components the vector has. This single number is essential in physics (the magnitude of a velocity vector is speed; the magnitude of a force vector is how strong that force is), engineering (structural load calculations), and computer graphics (distance and speed calculations for objects and cameras).

How it works

Enter the vector's components (2D or 3D). The calculator squares each component, adds the squares together, and takes the square root of the sum: |v|=√(x²+y²(+z²)). This is exactly the same operation as finding the straight-line distance from the origin to the point (x, y, z), since a vector's magnitude is really just its distance from the tail (usually placed at the origin) to its tip.

Enter dimension.
Enter vector — x.
Enter vector — y.
Enter vector — z (3D only).
Click Calculate to see your results.

Examples

(3, 4, 0)

|v|=√(9+16+0)=√25=5 — the classic 3-4-5 triangle.

(2, -2, 1)

|v|=√(4+4+1)=√9=3.

Common mistakes to avoid

Forgetting to square each component before summing (adding raw values instead).
Forgetting a component entirely in 3D (treating a 3D vector as if it were 2D).
Expecting a negative result when one or more components are negative — squaring removes the sign before the square root is taken.
Confusing a vector's magnitude with one of its individual components — magnitude combines all of them together.

Frequently asked questions

No — magnitude is always a non-negative square root by definition, since it represents a physical length and squares eliminate any sign from the individual components.

The vector itself carries both direction and size (e.g. "5 units northeast"); the magnitude is just the size part on its own, stripped of any directional information (just "5 units").

Only when every single component is zero — that is, only the zero vector itself has zero magnitude. Any vector with at least one nonzero component has a positive magnitude.

Dividing a vector by its own magnitude produces a unit vector — a vector pointing the same direction but with magnitude exactly 1. This is the core operation used by the Unit Vector Calculator.

No — since every component is squared before summing, swapping the x, y, and z values around doesn't change the total magnitude, though it would change the direction the vector actually points.

A velocity vector's magnitude gives its speed, a force vector's magnitude gives its strength, and a displacement vector's magnitude gives the straight-line distance traveled — magnitude strips away direction to leave just the "how much" part of any vector quantity.

Not really a shortcut, just one fewer term — 2D magnitude uses √(x²+y²) while 3D magnitude adds a z² term inside the same square root, so the 2D case is simply the 3D formula with z set to 0.