Vector Magnitude Calculator
Compute the Euclidean magnitude (length) of a vector with any number of components.
Euclidean norm (length) of the vector
- 1
Sum of squared components
3² + 4² + 12² = 169 - 2
Magnitude |v|
√(169) = 13Square root of the sum of squares — the Euclidean length of the vector.
How does this calculator work?
A vector magnitude is the length of the vector, computed as the square root of the sum of the squares of its components: |v| = √(v₁² + … + vₙ²). Enter components separated by spaces or commas and the calculator returns |v| plus the squared magnitude for vectors of any dimension.
Formula
How this is calculated
Enter the components of your vector v = (v₁, v₂, …, vₙ) in the text box, separated by spaces or commas. The calculator accepts a vector of any dimension — 2D, 3D, or higher — and ignores empty entries between separators.
The magnitude (also called the length, norm, or Euclidean norm) is found by squaring every component, summing those squares, and taking the square root of the total: |v| = √(Σ vᵢ²). This is a direct generalization of the Pythagorean theorem to n dimensions. The squared magnitude Σ vᵢ² is reported separately because it appears often in physics and statistics (for example in kinetic energy or variance) and avoids the square root.
The result is always non-negative and is unitless unless your components share a physical unit, in which case |v| carries that same unit. The zero vector has magnitude 0. Non-numeric entries make the input invalid; sign of components does not matter because each is squared.
Frequently asked questions
It is the length of the vector — the straight-line distance from the origin to the point defined by its components. Geometrically it is the hypotenuse generalized to any number of dimensions.
Yes. Enter as many components as you like separated by spaces or commas. The Euclidean norm formula |v| = √(Σ vᵢ²) works in 2D, 3D, and higher-dimensional spaces.
No. Each component is squared before summing, so negative and positive components of the same size contribute equally. The magnitude is always zero or positive.
Also known as
TG we-Calculate Editorial Team. (2026). Vector Magnitude Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/vector-magnitude-calculator
TG we-Calculate Editorial Team. "Vector Magnitude Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/vector-magnitude-calculator.
TG we-Calculate Editorial Team, "Vector Magnitude Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/vector-magnitude-calculator
@misc{wecalculate_vector_magnitude_calculator, title = {Vector Magnitude Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/vector-magnitude-calculator}}, year = {2026}, note = {TG we-Calculate} }
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