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

向量長度計算機（Vector Magnitude Calculator）可計算任意維度向量的長度（歐幾里得範數）。輸入向量分量，即可立即取得長度、單位向量、方向角、詳細的逐步平方根計算過程、各分量貢獻分析，以及顯示向量與其投影的互動式圖表。

The Magnitude Formula

對於向量 $\vec{v} = \langle v_1, v_2, \ldots, v_n \rangle$，其長度（或稱歐幾里得範數）為：

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

這是畢氏定理的直接推廣。在二維中，$|\vec{v}| = \sqrt{x^2 + y^2}$ 給出由各分量所構成直角三角形的斜邊長。在三維中，則延伸為 $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$。

Magnitude by Dimension

2D Vector

$|\vec{v}| = \sqrt{x^2 + y^2}$
xy 平面上向量的長度，等同於原點到點 (x, y) 的距離。

3D Vector

$|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$
用於物理、工程及 3D 繪圖中的空間計算。

Unit Vector

$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$
長度為 1、方向與原向量相同的向量，用來表示純粹的方向。

n-Dimensional

$|\vec{v}| = \sqrt{\sum_{i=1}^{n} v_i^2}$
同一公式適用於任意維度，常用於資料科學與機器學習領域。

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

選擇維度——選擇 2D、3D 或自訂（Custom）以輸入更高維度。或點擊快速範例以自動填入範例向量。
輸入分量——輸入以逗號分隔的向量分量（例如二維輸入 3, 4，三維輸入 1, 2, 3）。也支援空格、分號及各國數字格式。
點擊計算——按下「Calculate Magnitude」按鈕以計算所有結果。
檢視結果——查看長度、單位向量、方向角、逐步計算公式、各分量貢獻分解，以及可切換圖層的互動式向量圖。

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.

引用此內容、頁面或工具為：

"向量大小計算機" 於 https://MiniWebtool.com/zh-tw/向量大小計算機/，來自 MiniWebtool，https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-10

開發者 API 可用: 用一次 JSON HTTP 請求，把這個工具接入你的應用程式、自動化流程或代理。查看 API 文件

您還可以嘗試我們的 AI數學解題器 GPT，通過自然語言問答解決您的數學問題。

向量大小計算機