Tool to calculate the norm of a vector. The vector standard of a vector space represents the length (or distance) of the vector.
Vector Norm - dCode
Tag(s) : Matrix
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Vector Norm
Vector's Norm Calculator
Answers to Questions (FAQ)
What is the norm of a vector? (Definition)
The norm of a vector is its length. If $ A $ and $ B $ are two points (of a space of $ n $ dimensions) then the norm of the vector, noted with a double bar $ \|\overrightarrow{AB}\| $, is the distance between $ A $ and $ B $ (the length of the segment $ [AB] $).
The absolute value is the special case of the norm for a real number (one dimension).
How to calculate the norm of a vector?
In a vector space of dimension $ n $, a vector $ \vec{v} $ of components $ x_i $ : $ \vec{v} = (x_1, x_2, ..., x_n) $ is computed by the square root of the sum of the squares of the components: $$ \left\|\vec{v}\right\| = \sqrt{x_1^2 + x_2^2 + \cdots +x_n^2} $$
The norm of a vector can also be computed from the scalar product of the vector with itself: $ \| \vec{v} \| = \sqrt{ \vec{v} \cdot \vec{v} } $.
How to calculate the norm of a 2D vector?
In the 2D plane, for a vector $ \vec{v} = (x,y) $ the formula is simplified $$ \|\vec{v}\|= \sqrt{x^2+y^2} $$
Example: $ \vec{v} = \left( \begin{array}{c} 1 \ 2 \end{array} \right) $ so $ \|\vec{v}\| = \sqrt{1^2+2^2} = \sqrt{5} $
How to calculate the norm of a 3D vector?
In 3D space, for a vector $ \vec{u} = (x,y,z) $ the formula is simplified $$ \|{\vec{u}}\|= \sqrt{x^2+y^2+z^2} $$
How to calculate the components of a vector from the points?
From the coordinates of the points $ A (x_A,y_A) $ and $ B (x_B,y_B) $ of the vector $ \overrightarrow{AB} $, the components of the vector are $ {\overrightarrow {AB}} = \{ (x_B-x_A), (y_B-y_A) \} $ and therefore the norm is $ \|\overrightarrow {AB}\| = \sqrt{(x_B-x_A)^2+(y_B-y_A)^2} $
Source code
dCode retains ownership of the "Vector Norm" source code. Any algorithm for the "Vector Norm" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Vector Norm" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Vector Norm" or any other element are not public (except explicit open source licence like Creative Commons). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Vector Norm" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source.
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Vector Norm on dCode.fr [online website], retrieved on 2025-04-16,
