Free · Coordinate geometry calculators 📈

Euclidean Vector Magnitude: 2D to 5D

Find the Euclidean length of a vector with two, three, four or five real components.

  • Formula & worked example
  • Private in your browser
  • No signup
Go to tool

Calculator inputs

Loading calculator…

How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

The vector (2, −3, 6) has magnitude 7 because 2² + (−3)² + 6² = 49 and √49 = 7.

Magnitude = √(x₁² + x₂² + … + xₙ²), for n = 2, 3, 4 or 5

Worked example

Enter these known values and leave the other values blank.

First component (x)
2
Second component (y)
-3
Third component (z)
6
3D magnitude
7

Assumptions and limitations

  • Select the number of components first. Each component can be positive, negative or zero.
  • This is an ordinary Euclidean norm using perpendicular axes and equally scaled components. It is not a relativistic spacetime interval or a weighted norm.
  • Components must be expressed in compatible units. No unit conversion is performed; the result has the same unit as each component.
  • Reverse-solving a missing component returns its nonnegative square-root branch. A nonzero negative component can have the same magnitude, so the norm alone does not determine its sign.
  • If the supplied magnitude is smaller than the combined contribution of the known components, no real missing component exists.

Common questions

Can negative components produce a positive magnitude?

Yes. Each component is squared before addition. For example, (−3, 4) and (3, 4) both have magnitude 5.

Why does solving backwards return a positive component?

The inverse uses the principal square root. If the magnitude only determines a component’s square, both signs may be possible; the calculator displays the nonnegative one.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

Explore coordinate geometry calculators 📈 or browse all math calculators.