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IEVref:102-03-36ID:
Language:enStatus: Standard
Term: vector product
Synonym1:
Synonym2:
Synonym3:
Symbol:
Definition: axial vector UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca WHwbaaaa@3CA1@ , orthogonal to two given vectors U and V, such that the three vectors U, V and UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca WHwbaaaa@3CA1@ form a right-handed trihedron or a left-handed trihedron according to the space orientation, with its magnitude equal to the product of the magnitudes of the given vectors and the sine of the angle ϑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaeqy0dOeaaa@3B24@ between them

| UV|=|U||V|sinϑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvai abgEna0kaahAfaaiaawEa7caGLiWoacqGH9aqpdaabdaqaaiaahwfa aiaawEa7caGLiWoacqGHflY1daabdaqaaiaahAfaaiaawEa7caGLiW oacqGHflY1jugqbiGacohacaGGPbGaaiOBaOGaeqy0dOeaaa@5297@

Note 1 to entry: In the three-dimensional Euclidean space with given space orientation, the vector product of two vectors U and V is the unique axial vector UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca WHwbaaaa@3CA1@ such that for any vector W in the same vector space the scalar triple product (U,V,W) is equal to the scalar product (UV)W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWHvbGaey 41aqRaaCOvaiaacMcacqGHflY1caWHxbaaaa@4124@ .

Note 2 to entry: For two vectors U= U x e x + U y e y + U z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabg2 da9iaadwfakmaaBaaaleaajug4aiaadIhaaSqabaqcLbuacaWHLbGc daWgaaWcbaqcLboacaWG4baaleqaaKqzafGaey4kaSIaamyvaOWaaS baaSqaaKqzGdGaamyEaaWcbeaajugqbiaahwgakmaaBaaaleaajug4 aiaadMhaaSqabaqcLbuacqGHRaWkcaWGvbGcdaWgaaWcbaqcLboaca WG6baaleqaaKqzafGaaCyzaOWaaSbaaSqaaKqzGdGaamOEaaWcbeaa aaa@5535@ and V= V x e x + V y e y + V z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCOvaiabg2 da9iaadAfakmaaBaaaleaajug4aiaadIhaaSqabaqcLbuacaWHLbGc daWgaaWcbaqcLboacaWG4baaleqaaKqzafGaey4kaSIaamOvaOWaaS baaSqaaKqzGdGaamyEaaWcbeaajugqbiaahwgakmaaBaaaleaajug4 aiaadMhaaSqabaqcLbuacqGHRaWkcaWGwbGcdaWgaaWcbaqcLboaca WG6baaleqaaKqzafGaaCyzaOWaaSbaaSqaaKqzGdGaamOEaaWcbeaa aaa@5539@ , where e x , e y , e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyzaOWaaS baaSqaaKqzGdGaamiEaaWcbeaajugqbiaabYcacaqGGaGaaCyzaOWa aSbaaSqaaKqzGdGaamyEaaWcbeaajugqbiaabYcacaqGGaGaaCyzaO WaaSbaaSqaaKqzGdGaamOEaaWcbeaaaaa@47EF@ is an orthonormal base, the vector product is expressed by UV=( U y V z U z V y ) e x +( U z V x U x V z ) e y +( U x V y U y V x ) e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgE na0kaahAfacqGH9aqpcaGGOaGaamyvaOWaaSbaaSqaaKqzGdGaamyE aaWcbeaajugqbiaadAfakmaaBaaaleaajug4aiaadQhaaSqabaqcLb uacqGHsislcaWGvbGcdaWgaaWcbaqcLboacaWG6baaleqaaKqzafGa amOvaOWaaSbaaSqaaKqzGdGaamyEaaWcbeaajugqbiaacMcacaWHLb GcdaWgaaWcbaqcLboacaWG4baaleqaaKqzafGaey4kaSIaaiikaiaa dwfakmaaBaaaleaajug4aiaadQhaaSqabaqcLbuacaWGwbGcdaWgaa WcbaqcLboacaWG4baaleqaaKqzafGaeyOeI0IaamyvaOWaaSbaaSqa aKqzGdGaamiEaaWcbeaajugqbiaadAfakmaaBaaaleaajug4aiaadQ haaSqabaqcLbuacaGGPaGaaCyzaOWaaSbaaSqaaKqzGdGaamyEaaWc beaajugqbiabgUcaRiaacIcacaWGvbGcdaWgaaWcbaqcLboacaWG4b aaleqaaKqzafGaamOvaOWaaSbaaSqaaKqzGdGaamyEaaWcbeaajugq biabgkHiTiaadwfakmaaBaaaleaajug4aiaadMhaaSqabaqcLbuaca WGwbGcdaWgaaWcbaqcLboacaWG4baaleqaaKqzafGaaiykaiaahwga kmaaBaaaleaajug4aiaadQhaaSqabaaaaa@83C9@ .

The vector product can also be expressed as U V=| e x e y e z U x U y U z V x V y V z | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca qGGaGaaCOvaiabg2da9maaemaabaqbaeqabmWaaaqaaiaahwgadaWg aaWcbaGaamiEaaqabaaakeaacaWHLbWaaSbaaSqaaiaadMhaaeqaaa GcbaGaaCyzamaaBaaaleaacaWG6baabeaaaOqaaiaadwfadaWgaaWc baGaamiEaaqabaaakeaacaWGvbWaaSbaaSqaaiaadMhaaeqaaaGcba GaamyvamaaBaaaleaacaWG6baabeaaaOqaaiaadAfadaWgaaWcbaGa amiEaaqabaaakeaacaWGwbWaaSbaaSqaaiaadMhaaeqaaaGcbaGaam OvamaaBaaaleaacaWG6baabeaaaaaakiaawEa7caGLiWoaaaa@5440@ using a sum similar to the sum used to obtain the determinant of a matrix. The vector product is therefore the axial vector associated with the antisymmetric tensor UVVU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgE PielaahAfacqGHsislcaWHwbGaey4LIqSaaCyvaaaa@41F5@ (see IEV 102-03-43).

Note 3 to entry: For two complex vectors U and V, either the vector product UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgE na0kaahAfaaaa@3D50@ or one of the vector products U * ×V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgw SixlaahAfakmaaCaaaleqabaGaaiOkaaaaaaa@3AEA@ or U× V * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgw SixlaahAfakmaaCaaaleqabaGaaiOkaaaaaaa@3AEA@ may be used depending on the application.

Note 4 to entry: A vector product can be similarly defined for a pair consisting of a polar vector and an axial vector and is then a polar vector, or for a pair of two axial vectors and is then an axial vector.

Note 5 to entry: In the usual three-dimensional space, the vector product of two vector quantities is the vector product of the associated unit vectors multiplied by the product of the scalar quantities.

Note 6 to entry: The vector product operation is denoted by a cross (×) between the two symbols representing the vectors. The use of the symbol ∧ is deprecated.


Publication date:2008-08
Source:
Replaces:
Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO
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