सदिश बीजगणित के सूत्र

नीचे दी गयीं सर्वसमिकाएँ त्रिविमीय युक्लीडीय अवकाश के सदिशों से सम्बंधित हैंSee, for example, Albright's chemical engineering handbook Lyle Frederick Albright https://books.google.com/books?id=HYB3Udjx_FYC&pg=PA68 68 0-8247-5362-3 CRC Press §2.5.1 Vector algebra 2008 इनमें से कुछ (किन्तु सभी नहीं) अधिक वीमा वाले सदिशों के लिये भी सत्य हैं। उदाहरण के लिये, दो सदिशों का सदिश गुणनफल सभी विमाओं के लिये उपलब्ध नहीं है। परिमाण सदिश A का परिमाण, इस सदिश के तीन परस्पर लम्बवत दिशाओं में कम्पोनेन्ट के वर्गों के योग के वर्गमूल के बराबर होता है। यह परिमाण अदिश गुणनफल का प्रयोग करते हुए निम्नलिखित प्रकार से भी अभिव्यक्त किया जा सकता है- असमताएँ ; Cauchy–Schwarz inequality in three dimensions ; the triangle inequality in three dimensions ; the reverse triangle inequality Here the notation (A · B) denotes the dot product of vectors A and B. कोण The vector product and the scalar product of two vectors define the angle between them, say θ:Methods of applied mathematics Francis Begnaud Hildebrand 24 https://books.google.com/books?id=17EZkWPz_eQC&pg=PA24 0-486-67002-3 Reprint of Prentice-Hall 1965 2nd Courier Dover Publications 1992 3 जनवरी 2017 https://web.archive.org/web/20170817232929/https://books.google.com/books?id=17EZkWPz_eQC&pg=PA24 17 अगस्त 2017 live To satisfy the right-hand rule, for positive θ, vector B is counter-clockwise from A, and for negative θ it is clockwise. Here the notation A × B denotes the vector cross product of vectors A and B. The Pythagorean trigonometric identity then provides: If a vector A = (Ax, Ay, Az) makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then: and analogously for angles β, γ. Consequently: with unit vectors along the axis directions. क्षेत्रफल तथा आयतन The area Σ of a parallelogram with sides A and B containing the angle θ is: which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram. That is: The square of this expression is:Introduction to calculus and analysis, Volume II Richard Courant, Fritz John https://books.google.com/books?id=ngkQxS4eicgC&pg=PA191 190–195 Areas of parallelograms and volumes of parallelepipeds in higher dimensions 3-540-66569-2 2000 Springer Reprint of original 1974 Interscience where Γ(A, B) is the Gram determinant of A and B defined by: In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A, B and C is given by the Gram determinant of the three vectors: This process can be extended to n-dimensions. सदिशों का योग और गुणन Some of the following algebraic relations refer to the dot product and the cross product of vectors. These relations can be found in a variety of sources, for example, see Albright. ; distributivity of multiplication by a scalar and addition ; commutativity of addition ; associativity of addition ; commutativity of scalar (dot) product ; anticommutativity of vector cross product ; distributivity of addition wrt scalar product ; distributivity of addition wrt vector cross product ; scalar triple product ; vector triple product ; Binet–Cauchy identity in three dimensions In particular, when A = C and B = D, the above reduces to: ; Lagrange's identity in three dimensions A vector quadruple product, which is also a vector, can be defined, which satisfies the following identities:Mechanics and relativity Vidwan Singh Soni https://books.google.com/books?id=-3H5V0LGBOgC&pg=PA11 11–12 §1.10.2 Vector quadruple product PHI Learning Pvt. Ltd. 81-203-3713-1 2009This formula is applied to spherical trigonometry by Vector analysis: a text-book for the use of students of mathematics Edwin Bidwell Wilson, Josiah Willard Gibbs https://books.google.com/books?id=RC8PAAAAIAAJ&pg=PA77 §42 in Direct and skew products of vectors Scribner 1901 77 ff where [A, B, C] is the scalar triple product A · (B × C) or the determinant of the matrix {A, B, C} with the components of these vectors as columns . Given three arbitrary vectors not on the same line, A, B, C, any other vector D can be expressed in terms of these as:Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics Joseph George Coffin https://books.google.com/books?id=9mgGAQAAIAAJ&pg=PA56 56 1911 Wiley 2nd सन्दर्भ इन्हें भी देखें सदिश अवकाश (वेक्तर स्पेस) ज्यामितीय बीजगणित (Geometric algebra) सदिश कैलकुलस की सर्वसमिकाएँ श्रेणी:गणितीय सर्वसमिकाएँ