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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 Binet–Cauchy identity ： ウィキペディア英語版 Binet–Cauchy identity In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that 〔 〕 : $$ \biggl(\sum_^n a_i c_i\biggr) \biggl(\sum_^n b_j d_j\biggr) = \biggl(\sum_^n a_i d_i\biggr) \biggl(\sum_^n b_j c_j\biggr) + \sum_ (a_i b_j - a_j b_i ) (c_i d_j - c_j d_i ) for every choice of real or complex numbers (or more generally, elements of a commutative ring). Setting ''ai'' = ''ci'' and ''bj'' = ''dj'', it gives the Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality for the Euclidean space $\backslash scriptstyle\backslash mathbb^n$. ==The Binet–Cauchy identity and exterior algebra== When ''n'' = 3 the first and second terms on the right hand side become the squared magnitudes of dot and cross products respectively; in ''n'' dimensions these become the magnitudes of the dot and wedge products. We may write it :$(a\; \backslash cdot\; c)(b\; \backslash cdot\; d)\; =\; (a\; \backslash cdot\; d)(b\; \backslash cdot\; c)\; +\; (a\; \backslash wedge\; b)\; \backslash cdot\; (c\; \backslash wedge\; d)\backslash ,$ where a, b, c, and d are vectors. It may also be written as a formula giving the dot product of two wedge products, as :$(a\; \backslash wedge\; b)\; \backslash cdot\; (c\; \backslash wedge\; d)\; =\; (a\; \backslash cdot\; c)(b\; \backslash cdot\; d)\; -\; (a\; \backslash cdot\; d)(b\; \backslash cdot\; c).\backslash ,$ In the special case of unit vectors ''a=c'' and ''b=d'', the formula yields :$|a\; \backslash wedge\; b|^2\; =\; |a|^2|b|^2\; -\; |a\; \backslash cdot\; b|^2.\; \backslash ,$ When both vectors are unit vectors, we obtain the usual relation :$1=\; \backslash cos^2(\backslash phi)+\backslash sin^2(\backslash phi)$ where φ is the angle between the vectors. 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「Binet–Cauchy identity」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.