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Area Mathematics - General concepts and linear algebra / Vectors and tensors

IEV ref102-03-23

en
magnitude, <of a vector>
norm, <of a vector>
for any vector U, non-negative scalar, usually denoted by |U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aaaa@3CCD@ , equal to the non-negative square root of the scalar product or, in the case of a complex vector, of the Hermitian product of the vector by itself

Note 1 to entry: The magnitude of a vector U has the following properties:

  • U=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGH9aqpie qajugqbiaa=bdaaaa@3C1B@ if and only if |U|=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aiabg2da9iaaicdaaaa@3E8D@ ,
  • | αU|=|α||U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaeqySde MaaCyvaaGaay5bSlaawIa7aiabg2da9maaemaabaGaeqySdegacaGL hWUaayjcSdGaeyyXIC9aaqWaaeaacaWHvbaacaGLhWUaayjcSdaaaa@4A7D@ where α is a scalar,
  • | U+V||U|+|V| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvai abgUcaRiaahAfaaiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaahwfa aiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaahAfaaiaawEa7caGLiW oaaaa@4926@ where V is any other vector.

Note 2 to entry: For a vector U in the three-dimensional Euclidean or Hermitian space with orthonormal base, the magnitude is given by |U|= | U 1 | 2 + | U 2 | 2 + | U 3 | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aiabg2da9maakaaabaWaaqWaaeaacaWGvbWaaSba aSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdWaaWbaaSqabeaacaaIYa aaaOGaey4kaSYaaqWaaeaacaWGvbWaaSbaaSqaaiaaikdaaeqaaaGc caGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaqWaae aacaWGvbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdWaaWba aSqabeaacaaIYaaaaaqabaaaaa@5140@ .

Note 3 to entry: The terms "Euclidean norm" and "Hermitian norm" may be used for the real or the complex case, respectively.

Note 4 to entry: The magnitude of a vector U is represented by |U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaa4WaaqWaaOqaaKqzaf GaaCyvaaGccaGLhWUaayjcSdaaaa@3D9E@ or by U; U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaa4WaauWaaOqaaKqzaf GaaCyvaaGccaGLjWUaayPcSdaaaa@3DA3@ is also used.


fr
norme, <d'un vecteur> f
pour tout vecteur U, scalaire positif ou nul, noté généralement |U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aaaa@3CCD@ , égal à la racine carrée non négative du produit scalaire, ou du produit hermitien dans le cas d'un vecteur complexe, du vecteur par lui-même

Note 1 à l'article: La norme d'un vecteur U a les propriétés suivantes:

  • U=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGH9aqpie qajugqbiaa=bdaaaa@3C1B@ si et seulement si |U|=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aiabg2da9iaaicdaaaa@3E8D@ ,
  • | αU|=|α||U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaeqySde MaaCyvaaGaay5bSlaawIa7aiabg2da9maaemaabaGaeqySdegacaGL hWUaayjcSdGaeyyXIC9aaqWaaeaacaWHvbaacaGLhWUaayjcSdaaaa@4A7D@ α est un scalaire,
  • | U+V||U|+|V| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvai abgUcaRiaahAfaaiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaahwfa aiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaahAfaaiaawEa7caGLiW oaaaa@4926@ V est un autre vecteur quelconque.

Note 2 à l'article: Pour un vecteur U dans l'espace vectoriel euclidien ou hermitien à trois dimensions muni d'une base orthonormée, la norme est donnée par |U|= | U 1 | 2 + | U 2 | 2 + | U 3 | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aiabg2da9maakaaabaWaaqWaaeaacaWGvbWaaSba aSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdWaaWbaaSqabeaacaaIYa aaaOGaey4kaSYaaqWaaeaacaWGvbWaaSbaaSqaaiaaikdaaeqaaaGc caGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaqWaae aacaWGvbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdWaaWba aSqabeaacaaIYaaaaaqabaaaaa@5140@ .

Note 3 à l'article: Les termes «norme euclidienne» et «norme hermitienne» peuvent être employés dans les cas réel et complexe, respectivement.

Note 4 à l'article: La norme d'un vecteur U est notée |U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaa4WaaqWaaOqaaKqzaf GaaCyvaaGccaGLhWUaayjcSdaaaa@3D9E@ ou U; U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaa4WaauWaaOqaaKqzaf GaaCyvaaGccaGLjWUaayPcSdaaaa@3DA3@ est aussi utilisé.


de
Betrag (eines Vektors), m

es
norma (de un vector)
magnitud (de un vector)

ja
ベクトルの大きさ, <ベクトルの>
ノルム, <ベクトルの>

pl
norma (wektora)
moduł wektora (termin nie zalecany)

pt
norma (de um vector)

sr
магнитуда, <вектора> ж јд
норма, <вектора> ж јд

sv
belopp (av en vektor)

zh
长度, <向量的>
范数, <向量的>

Publication date: 2008-08
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