Vector Magnitude Calculator

Calculate the magnitude (length or Euclidean norm) of a 2D, 3D, or n-dimensional vector. Get step-by-step computation showing each squared component, the sum under the radical, and the final square root result, with an interactive vector diagram.

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About Vector Magnitude Calculator

The Vector Magnitude Calculator computes the length (Euclidean norm) of a vector in any number of dimensions. Enter your vector components and instantly get the magnitude, unit vector, direction angles, a detailed step-by-step square root computation, component contribution analysis, and an interactive diagram showing the vector and its projections.

The Magnitude Formula

For a vector $\vec{v} = \langle v_1, v_2, \ldots, v_n \rangle$, the magnitude (or Euclidean norm) is:

$$|\vec{v}| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$

This is a direct generalization of the Pythagorean theorem. In 2D, $|\vec{v}| = \sqrt{x^2 + y^2}$ gives the hypotenuse of the right triangle formed by the components. In 3D, it extends to $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$.

Magnitude by Dimension

2D Vector

$|\vec{v}| = \sqrt{x^2 + y^2}$
The length of a vector in the xy-plane. Equivalent to the distance from the origin to point (x, y).

3D Vector

$|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$
Used in physics, engineering, and 3D graphics for spatial calculations.

Unit Vector

$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$
A vector with magnitude 1 pointing in the same direction. Used to represent pure direction.

n-Dimensional

$|\vec{v}| = \sqrt{\sum_{i=1}^{n} v_i^2}$
The same formula works in any dimension, used in data science and machine learning.

Key Formulas Reference

Formula	Expression	Description
Magnitude	$\|\vec{v}\| = \sqrt{\sum v_i^2}$	Length of the vector (L2 norm)
Unit Vector	$\hat{v} = \vec{v} / \|\vec{v}\|$	Normalized direction vector
Direction Cosine	$\cos \alpha_i = v_i / \|\vec{v}\|$	Cosine of angle with each axis
Distance	$d = \|\vec{B} - \vec{A}\|$	Distance between two points

Real-World Applications

🚀

Physics

Speed from velocity, force magnitude, field strength

🎮

Game Dev

Distance checks, collision detection, normalization

🤖

Machine Learning

Feature scaling, distance metrics, gradient norms

📡

Signal Processing

Signal energy, RMS calculations, noise amplitude

🗺

Navigation

Displacement, straight-line distance, bearing

📊

Data Science

Euclidean distance in clustering, similarity

How to Use the Vector Magnitude Calculator

Select a dimension — choose 2D, 3D, or Custom for higher dimensions. Or click a quick example to auto-fill a sample vector.
Enter components — type the vector components separated by commas (e.g., 3, 4 for 2D or 1, 2, 3 for 3D). Spaces, semicolons, and international number formats are also supported.
Click Calculate — press the "Calculate Magnitude" button to compute all results.
Review results — see the magnitude, unit vector, direction angles, step-by-step formulas, component contribution breakdown, and the interactive vector diagram with toggleable layers.

Frequently Asked Questions

What is the magnitude of a vector?

The magnitude (also called length, norm, or absolute value) of a vector is a scalar that represents the distance from the origin to the point defined by the vector. For a vector v = (v₁, v₂, ..., vₙ), the magnitude is √(v₁² + v₂² + ... + vₙ²), using the Euclidean distance formula derived from the Pythagorean theorem.

How do you find the magnitude of a 3D vector?

For a 3D vector v = (x, y, z), the magnitude is |v| = √(x² + y² + z²). Square each component, add them together, and take the square root of the sum. For example, for v = (3, 4, 0), |v| = √(9 + 16 + 0) = √25 = 5.

What is a unit vector and how do you find it?

A unit vector is a vector with magnitude 1 that points in the same direction as the original vector. To find the unit vector, divide each component of the vector by its magnitude: v̂ = v / |v|. Unit vectors are used to represent pure direction without magnitude. The standard unit vectors are î = (1,0,0), ĵ = (0,1,0), and k̂ = (0,0,1).

Can a vector have a magnitude of zero?

Yes. The only vector with magnitude zero is the zero vector, where all components are 0. The zero vector has no direction and is the additive identity in vector spaces. Its unit vector is undefined because you cannot divide by zero.

What is the difference between magnitude and distance?

The magnitude of a vector gives the distance from the origin to the tip of the vector. The distance between two points A and B equals the magnitude of the vector from A to B: d(A,B) = |B − A|. Both use the same Euclidean formula, but magnitude is a property of a single vector while distance relates two points.

Reference this content, page, or tool as:

"Vector Magnitude Calculator" at https://MiniWebtool.com/vector-magnitude-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-10

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Vector Magnitude Calculator

About Vector Magnitude Calculator

The Magnitude Formula

Magnitude by Dimension

2D Vector

3D Vector

Unit Vector

n-Dimensional

Key Formulas Reference

Real-World Applications

How to Use the Vector Magnitude Calculator

Frequently Asked Questions

Linear Algebra:

Top & Updated:

Formula	Expression	Description
Magnitude	\(\|\vec{v}\| = \sqrt{\sum v_i^2}\)	Length of the vector (L2 norm)
Unit Vector	\(\hat{v} = \vec{v} / \|\vec{v}\|\)	Normalized direction vector
Direction Cosine	\(\cos \alpha_i = v_i / \|\vec{v}\|\)	Cosine of angle with each axis
Distance	\(d = \|\vec{B} - \vec{A}\|\)	Distance between two points