向量大小計算機

計算 2D、3D 或 n 維向量的大小（長度或歐幾里得範數）。獲取顯示每個分量平方、根號下總和以及最終平方根結果的逐步計算過程，並附有互動式向量圖。

向量大小計算機

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.

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by miniwebtool team. Updated: 2026-04-10

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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

向量大小計算機