Vector Magnitude Calculator

Last updated: 2026-05-19

What is vector magnitude?

The magnitude of a vector is its length — the straight-line distance from the origin to the tip of the arrow. For a vector v = (v₁, v₂, …, vₙ) in n-dimensional space, the Euclidean magnitude is:

$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} = \sqrt{\sum_{i=1}^{n} v_i^2}$

This calculator accepts any number of comma-separated components (e.g. 3, 4 for 2D or 1, 2, 3 for 3D) and returns the magnitude, the number of dimensions, and the unit vector pointing in the same direction.

Where the formula comes from

The formula is the Pythagorean theorem generalised to n dimensions. In 2D, a vector (a, b) forms the hypotenuse of a right triangle with legs a and b, so $|\mathbf{v}| = \sqrt{a^2 + b^2}$. In 3D, a vector (a, b, c) is the space diagonal of a rectangular box, giving $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$. The same logic extends to any number of dimensions: square each component, sum them, take the square root.

Worked example

Problem: Find the magnitude and unit vector of v = (3, 4, 12).

Step 1 — square and sum the components: $3^2 + 4^2 + 12^2 = 9 + 16 + 144 = 169$

Step 2 — take the square root: $|\mathbf{v}| = \sqrt{169} = 13$

Step 3 — divide each component by the magnitude to get the unit vector: $\hat{\mathbf{v}} = \left(\frac{3}{13},\ \frac{4}{13},\ \frac{12}{13}\right) \approx (0.230769,\ 0.307692,\ 0.923077)$

Interpretation: The vector (3, 4, 12) is 13 units long. The unit vector describes its direction — it points "3 parts across, 4 parts up, and 12 parts forward" per unit of distance travelled.

The unit vector

A unit vector has magnitude exactly 1. It preserves the direction of the original vector while stripping away its scale. It is computed by dividing every component by |v|:

$\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} = \left(\frac{v_1}{|\mathbf{v}|},\ \frac{v_2}{|\mathbf{v}|},\ \ldots,\ \frac{v_n}{|\mathbf{v}|}\right)$

Unit vectors are ubiquitous:

• Physics: force and velocity directions, surface normals.
• Computer graphics: lighting calculations, camera orientation.
• Machine learning: cosine similarity, normalised feature vectors.
• Navigation: bearing from a displacement vector.

If the input is a zero vector (all components zero), its magnitude is 0 and no unit vector exists — the direction is undefined.

Other norms

The Euclidean norm (L²) is the most common, but two others frequently appear:

This calculator computes the L² (Euclidean) norm, which is the standard interpretation of "length" in most scientific and engineering contexts.

Input format and edge cases

Components are entered as a comma-separated list, such as 3, 4, 1, 2, 3, or 0.5, -1.2, 0.8, 2.0. Negative components are valid, since squaring removes the sign. Fractional components are also valid: 1.5, 2.5 gives |v| ≈ 2.915476. A single component (e.g. 5) yields magnitude 5 and unit vector (1), consistent with the formula.

Applications

• Physics: the speed (magnitude of velocity) or force magnitude from component form.
• Computer graphics: normalising a surface normal before lighting calculations.
• Data science: L2-normalising feature vectors before cosine-similarity search.
• Robotics: end-effector reach from joint displacement vectors.
• GPS / mapping: converting (Δeast, Δnorth, Δup) displacements into a total distance.