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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Seven-dimensional cross product ： ウィキペディア英語版 Seven-dimensional cross product In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in R7 a vector also in R7.〔 〕 Like the cross product in three dimensions, the seven-dimensional product is anticommutative and is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity. And while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to octonions as the three-dimensional product does to quaternions. The seven-dimensional cross product is one way of generalising the cross product to other than three dimensions, and it is the only other non-trivial bilinear product of two vectors that is vector valued, anticommutative and orthogonal.〔 In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results. ==Multiplication table== The product can be given by a multiplication table, such as the one here. This table, due to Cayley,〔 〕〔 〕 gives the product of basis vectors e''i'' and e''j'' for each ''i'', ''j'' from 1 to 7. For example from the table :$\backslash mathbf\_1\; \backslash times\; \backslash mathbf\_2\; =\; \backslash mathbf\_3\; =-\backslash mathbf\_2\; \backslash times\; \backslash mathbf\_1$ The table can be used to calculate the product of any two vectors. For example to calculate the e1 component of x × y the basis vectors that multiply to produce e1 can be picked out to give :$\backslash left(\; \backslash mathbf\backslash right)\_1\; =\; x\_2y\_3\; -\; x\_3y\_2\; +x\_4y\_5-x\_5y\_4\; +\; x\_7y\_6-x\_6y\_7.$ This can be repeated for the other six components. There are 480 such tables, one for each of the products satisfying the definition.〔 〕 This table can be summarized by the relation〔 :$\backslash mathbf\_i\; \backslash mathbf\; \backslash mathbf\_j\; =\; \backslash varepsilon\; \_\; \backslash mathbf\_k,$ where $\backslash varepsilon\; \_$ is a completely antisymmetric tensor with a positive value +1 when ''ijk'' = 123, 145, 176, 246, 257, 347, 365. The top left 3 × 3 corner of this table gives the cross product in three dimensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Seven-dimensional cross product」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.