Calculadora de Magnitud de Vector

Calcula la magnitud (longitud o norma euclidiana) de un vector en 2D, 3D o n dimensiones. Obtén el cálculo paso a paso mostrando cada componente al cuadrado, la suma bajo el radical y el resultado final de la raíz cuadrada, con un diagrama de vector interactivo.

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Calculadora de Magnitud de Vector

La Calculadora de Magnitud de Vectores calcula la longitud (norma euclidiana) de un vector en cualquier número de dimensiones. Introduce los componentes de tu vector y obtén al instante la magnitud, el vector unitario, los ángulos de dirección, un cálculo detallado paso a paso de la raíz cuadrada, un análisis de la contribución de cada componente y un diagrama interactivo que muestra el vector y sus proyecciones.

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.
Introduce los componentes: escribe los componentes del vector separados por comas (por ejemplo, 3, 4 para 2D o 1, 2, 3 para 3D). También se admiten espacios, puntos y comas, y formatos numéricos internacionales.
Click Calculate — press the "Calculate Magnitude" button to compute all results.
Revisa los resultados: consulta la magnitud, el vector unitario, los ángulos de dirección, las fórmulas paso a paso, el desglose de la contribución de cada componente y el diagrama vectorial interactivo con capas activables.

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.

Cite este contenido, página o herramienta como:

"Calculadora de Magnitud de Vector" en https://MiniWebtool.com/es/calculadora-de-magnitud-de-vector/ de MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-10

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Calculadora de Magnitud de Vector

Calculadora de Magnitud de Vector

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

Álgebra Lineal:

Herramientas destacadas:

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