IEVref:102-03-23ID:
Language:enStatus: Obsolete
Term: magnitude (of a vector)
Synonym1: norm (of a vector)
[Preferred]
Synonym2:
Synonym3:
Symbol:
Definition: 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 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 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 The terms "Euclidean norm" and "Hermitian norm" may be used for the real or the complex case, respectively.

NOTE 4 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.


Publication date:2007-08
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Internal notes:2017-06-02: Cleanup - Remove Attached Image 102-03-23en.gif
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