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Vector Magnitude Calculator

Find the length of a vector from its components.

How it is calculated

The magnitude is the Euclidean length of the vector: |v| = √(x² + y² + z²), where x, y and z are the components along the three axes, all in the same unit. The result is expressed in that unit too, so a velocity in m/s gives a speed in m/s. Each component is squared, which removes the sign, then the squares are summed and the square root is taken. The result is shown with two decimals. All three fields are required; to get a 2D length enter 0 for z. Negative components are fine, because squaring makes them positive, and the magnitude is never negative.

Worked example

Take the vector (3, 4, 12). Squares: 3² = 9, 4² = 16, 12² = 144. Their sum is 9 + 16 + 144 = 169. The square root of 169 is 13, so the magnitude is 13.00, displayed to two decimals. This is a neat case because 3, 4, 12 and 13 form a whole-number quadruple, which makes the answer easy to check by hand.

Try this example →

Common mistakes

  • Mixing units between components, for example x in metres and y in centimetres; convert them first.
  • Forgetting the z component when the vector is 3D, which gives only the length of its projection onto the xy plane.
  • Dropping a minus sign and thinking it changes the answer: the signs vanish when squared, so (-3, 4, 12) has the same length as (3, 4, 12).
  • Confusing magnitude with the sum of components; 3 + 4 + 12 = 19 is not the length.

About this calculator

This calculator finds the magnitude (length) of a 3D vector from its x, y, and z components using the Euclidean norm, √(x² + y² + z²) — the direct generalization of the Pythagorean theorem into three dimensions. Leaving z at zero reduces it to the familiar 2D case.

Physics and engineering students use it to find the resultant size of a force, velocity, or displacement vector, while game developers and 3D graphics programmers use the same formula constantly to normalize direction vectors and measure distances between points in a scene.

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