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

IEV ref102-03-36

en
vector product
axial vector U×V 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 U×V 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

| U×V|=|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 U×V 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 (U×V)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 U×V=( 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 U×V 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.


fr
produit vectoriel, m
produit extérieur, m
vecteur axial U×V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca WHwbaaaa@3CA1@ , orthogonal à deux vecteurs donnés U et V, tel que les trois vecteurs U, V et U×V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca WHwbaaaa@3CA1@ forment un trièdre direct ou un trièdre rétrograde selon l'orientation de l'espace, avec une norme égale au produit des normes des vecteurs donnés et du sinus de leur angle ϑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaeqy0dOeaaa@3B24@

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

Note 1 à l'article: Dans l'espace euclidien à trois dimensions d'orientation donnée, le produit vectoriel de deux vecteurs U et V est l'unique vecteur axial U×V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxdaTca WHwbaaaa@3CA1@ tel que, pour tout vecteur W du même espace, le produit mixte (U,V,W) soit égal au produit scalaire (U×V)W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWHvbGaey 41aqRaaCOvaiaacMcacqGHflY1caWHxbaaaa@4124@ .

Note 2 à l'article: Pour deux vecteurs 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@ et 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@ , où e x , e y , e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyzaOWaaS baaSqaaKqzGdGaamiEaaWcbeaajugqbiaabYcacaqGGaGaaCyzaOWa aSbaaSqaaKqzGdGaamyEaaWcbeaajugqbiaabYcacaqGGaGaaCyzaO WaaSbaaSqaaKqzGdGaamOEaaWcbeaaaaa@47EF@ est une base orthonormée, le produit vectoriel est exprimé par U×V=( 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@ .

On peut aussi exprimer le produit vectoriel sous la forme 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@ en utilisant une somme semblable à celle utilisée pour obtenir le déterminant d'une matrice. Le produit vectoriel est donc le vecteur axial associé au tenseur antisymétrique UVVU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgE PielaahAfacqGHsislcaWHwbGaey4LIqSaaCyvaaaa@41F5@ (voir IEV 102-03-43).

Note 3 à l'article: Pour deux vecteurs complexes U et V, on peut selon l'application utiliser soit le produit vectoriel U×V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgE na0kaahAfaaaa@3D50@ , soit l'un des produits U * ×V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgw SixlaahAfakmaaCaaaleqabaGaaiOkaaaaaaa@3AEA@ ou U× V * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaCyvaiabgw SixlaahAfakmaaCaaaleqabaGaaiOkaaaaaaa@3AEA@ .

Note 4 à l'article: On peut définir de la même manière, pour un couple constitué d'un vecteur polaire et d'un vecteur axial un produit vectoriel qui est un vecteur polaire, et pour un couple de deux vecteurs axiaux un produit vectoriel qui est un vecteur axial.

Note 5 à l'article: Dans l'espace usuel à trois dimensions, le produit vectoriel de deux grandeurs vectorielles est le produit vectoriel des vecteurs unitaires associés multiplié par le produit des grandeurs scalaires.

Note 6 à l'article: Le produit vectoriel est indiqué par une croix (×) entre les deux symboles représentant les vecteurs. L'emploi du symbole ∧ est déconseillé.


de
Vektorprodukt, n
vektorielles Produkt, n

es
producto vectorial
producto externo

ja
ベクトル積
外積

pl
iloczyn wektorowy

pt
produto vectorial

sr
векторски производ, м јд

sv
vektorprodukt, kryssprodukt

zh
向量积

Publication date: 2008-08
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