Applied Mathematics in Electrical and Computer Engineering
Opposition Product in mathematics 1
CLAUDE ZIAD BAYEH1, 2
Faculty of Engineering II, Lebanese University 2 EGRDI transaction on mathematics (2006) LEBANON Email:
[email protected]
NIKOS E.MASTORAKIS
WSEAS (Research and Development Department) http://www.worldses.org/research/index.html Agiou Ioannou Theologou 17-2315773, Zografou, Athens,GREECE
[email protected] Abstract: - The Opposition Product is an original study introduced by the author into the mathematical domain. The main goal of introducing this new study is to find applications and to facilitate many complicated issues that are difficult to resolve using the traditional Vectorial product (Cross Product). The idea of the Opposition Product is to multiply two vectors or more and the result will be a vector collinear to the second vector (or last vector) and will be in the opposite direction (inverse vector) and amplified by a scalar named k. in this paper a brief study is introduced with the definition of the Opposition Product and a simple application is developed as the Magnitude of the Opposition product of two vectors will give the Area of the two vectors. Many studies will follow this one in order to find more applications of this new Product of vectors. Key-words:- Opposition Product, Cross Product, mathematics.
1 Introduction
vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗ . Many studies will follow the Opposition Product in this paper in order to find other applications in many domains as in science, engineering and mathematics. The main goal of introducing this new study is to find applications and to facilitate many complicated issues that are difficult to resolve using the traditional Vectorial product (Cross Product). A definition of the Opposition product is defined in the section 2. In the third section, an application of the Opposition Product is presented in order to calculate the Area formed by two vectors. Finally a Conclusion is presented in the final section.
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space[1-10]. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them. The name is derived from the cross symbol "×" that is often used to designate this operation; the alternative name vector product emphasizes the vector (rather than scalar) nature of the result. It has many applications in mathematics, engineering and physics. In this paper, the author introduced a new and original product between vectors named Opposition Product. In mathematics, the Opposition Product is an Operation between two vectors or more in nDimension space; it is particularly used into 2-D space and 3-D Space. The Product of two vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗ gives as result a vector collinear with 𝑃𝑃�⃗ but with a negative sign (Opposed vector in the same Plan of the two vectors) and amplified by 𝑘𝑘(𝛼𝛼). The value of 𝑘𝑘(𝛼𝛼) is determined according to the application of the Opposition Product, for example, in the section 3 the value of 𝑘𝑘(𝛼𝛼) is developed in order to obtain the proportional Area of the two
ISBN: 978-1-61804-064-0
2
Definition
Product
of
the
Opposition
The Opposition Product of two or more vectors is denoted by “⊣”. The result of the multiplication of two vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗ is an inverse vector parallel to the second vector 𝑃𝑃�⃗ but amplified by 𝑘𝑘(𝛼𝛼) with 𝑘𝑘(𝛼𝛼) ∈ ℝ. The Opposition Product is anti-commutative 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ ≠ 𝑃𝑃�⃗ ⊣ 𝐹𝐹⃗ .
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3 Application Product
The Opposition Product is distributive over addition if 𝑘𝑘(𝛼𝛼) is a constant and it is not distributive over addition if 𝑘𝑘(𝛼𝛼)is variable.
of
the
Opposition
3.1 Area Opposition Product
Let’s Consider two vectors (as in figure 1) 𝐹𝐹⃗ = �����⃗ 𝐴𝐴𝐴𝐴, �����⃗ �⃗ �⃗ ������⃗ 𝑃𝑃 = 𝐵𝐵𝐵𝐵 and the result vector is 𝑅𝑅 = 𝐵𝐵𝐵𝐵 collinear with 𝑃𝑃�⃗ Therefore the equation is as the following: 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ = 𝑅𝑅�⃗ = −𝑘𝑘(𝛼𝛼) ∙ 𝑃𝑃�⃗ (1) with 𝑘𝑘 ∈ ℝ
The Area Opposition Product is a product between two vectors that gives the result in form of an Area of the two vectors multiplied by the inverse of the second vector. Let’s consider the following vectors in the three dimensions space (x, y, z) (as in figure 2) 𝐹𝐹⃗ = �����⃗ 𝐴𝐴𝐴𝐴, �����⃗ �⃗ �⃗ ������⃗ 𝑃𝑃 = 𝐵𝐵𝐵𝐵 and the result vector is 𝑅𝑅 = 𝐵𝐵𝐵𝐵 Therefore the equation is as the following: 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ = 𝑅𝑅�⃗ = −𝑘𝑘 ∙ 𝑃𝑃�⃗ (1) with 𝑘𝑘 ∈ ℝ �����⃗ = (𝑥𝑥𝐵𝐵 − 𝑥𝑥𝐴𝐴 ; 𝑦𝑦𝐵𝐵 − 𝑦𝑦𝐴𝐴 ; 𝑧𝑧𝐵𝐵 − 𝑧𝑧𝐴𝐴 ) = 𝐹𝐹⃗ = 𝐴𝐴𝐴𝐴 (𝑥𝑥𝐹𝐹 ; 𝑦𝑦𝐹𝐹 ; 𝑧𝑧𝐹𝐹 ) 𝑃𝑃�⃗ = �����⃗ 𝐵𝐵𝐵𝐵 = (𝑥𝑥𝐶𝐶 − 𝑥𝑥𝐵𝐵 ; 𝑦𝑦𝐶𝐶 − 𝑦𝑦𝐵𝐵 ; 𝑧𝑧𝐶𝐶 − 𝑧𝑧𝐵𝐵 ) = (𝑥𝑥𝑃𝑃 ; 𝑦𝑦𝑃𝑃 ; 𝑧𝑧𝑃𝑃 )
𝑅𝑅�⃗ = ������⃗ 𝐵𝐵𝐵𝐵 = (𝑥𝑥𝐷𝐷 − 𝑥𝑥𝐵𝐵 ; 𝑦𝑦𝐷𝐷 − 𝑦𝑦𝐵𝐵 ; 𝑧𝑧𝐷𝐷 − 𝑧𝑧𝐵𝐵 ) = (𝑥𝑥𝑅𝑅 ; 𝑦𝑦𝑅𝑅 ; 𝑧𝑧𝑅𝑅 )
Fig. 1: represents the result of the Opposition Product 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ = 𝑅𝑅�⃗ = −𝑘𝑘(𝛼𝛼) ∙ 𝑃𝑃�⃗ Theorem 1: 𝑎𝑎1 ⊣ ����⃗ ����⃗ 𝑎𝑎2 ⊣ ����⃗ 𝑎𝑎3 ⊣ ����⃗ 𝑎𝑎4 ⊣ ⋯ ⊣ ����⃗ 𝑎𝑎𝑛𝑛 𝑛𝑛−1 ∏𝑛𝑛−1 (𝑘𝑘 = (−1) 𝑎𝑎𝑛𝑛 𝑖𝑖 (𝛼𝛼𝑖𝑖 )) ∙ ����⃗ 𝑖𝑖=1
(2)
Demonstration: • ����⃗ 𝑎𝑎1 ⊣ ����⃗ 𝑎𝑎2 = −𝑘𝑘1 ∙ ����⃗ 𝑎𝑎2
• ����⃗ 𝑎𝑎1 ⊣ ����⃗ 𝑎𝑎2 ⊣ ����⃗ 𝑎𝑎3 = (−𝑘𝑘1 ∙ ����⃗) 𝑎𝑎2 ⊣ ����⃗ 𝑎𝑎3 = −𝑘𝑘1 ����⃗ 𝑎𝑎2 ⊣ ����⃗ 𝑎𝑎3 2 (−𝑘𝑘 )(−𝑘𝑘 )𝑎𝑎 = 𝑎𝑎3 1 2 ����⃗ 3 = (−1) 𝑘𝑘1 𝑘𝑘2 ����⃗ ⇒ ����⃗ 𝑎𝑎1 ⊣ ����⃗ 𝑎𝑎2 ⊣ ����⃗ 𝑎𝑎3 ⊣ ����⃗ 𝑎𝑎4 ⊣ ⋯ ⊣ ����⃗ 𝑎𝑎𝑛𝑛 = (−1)
𝑛𝑛−1
𝑛𝑛−1
�(𝑘𝑘𝑖𝑖 (𝛼𝛼𝑖𝑖 )) ∙ ����⃗ 𝑎𝑎𝑛𝑛
Fig. 2: represents the result of the Opposition Product 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ = 𝑅𝑅�⃗ = −𝑘𝑘 ∙ 𝑃𝑃�⃗ in 3D space
𝑖𝑖=1
Theorem 2: Let’s consider the Opposition Product of many vectors denoted by (⊣𝑛𝑛𝑖𝑖=1 ), therefore: (⊣𝑛𝑛𝑖𝑖=1 )𝑎𝑎 ����⃗ 𝑎𝑎1 ⊣ ����⃗ 𝑎𝑎2 ⊣ ����⃗ 𝑎𝑎3 ⊣ ����⃗ 𝑎𝑎4 ⊣ ⋯ ⊣ ����⃗ 𝑎𝑎𝑛𝑛 𝑛𝑛 = ����⃗ 𝑛𝑛−1 ∏𝑛𝑛−1 (𝑘𝑘 = (−1) 𝑎𝑎𝑛𝑛 (3) 𝑖𝑖 (𝛼𝛼𝑖𝑖 )) ∙ ����⃗ 𝑖𝑖=1
ISBN: 978-1-61804-064-0
Considering that 𝑘𝑘 = �
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(𝑌𝑌𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑌𝑌𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑌𝑌𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑌𝑌𝐹𝐹 )2 ((𝑋𝑋 𝑃𝑃 )2 +(𝑌𝑌𝑃𝑃 )2 +(𝑍𝑍𝑃𝑃 )2 )
(4)
Applied Mathematics in Electrical and Computer Engineering
Fig. 3: represents the parallelogram created with the points (A,B,C,D’) of the �𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗�
Therefore the Opposition Product will give the surface (Area) of the two vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗. 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ = 𝑅𝑅�⃗ = −�
(𝑌𝑌𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑌𝑌𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑌𝑌𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑌𝑌𝐹𝐹 )2 ((𝑋𝑋 𝑃𝑃 )2 +(𝑌𝑌𝑃𝑃 )2 +(𝑍𝑍𝑃𝑃 )2 )
4 Conclusion
∙ 𝑃𝑃�⃗
The magnitude product of two vectors will be as following: �𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗� = �𝑅𝑅�⃗� =
(𝑌𝑌𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑌𝑌𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑌𝑌𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑌𝑌𝐹𝐹 )2
�−�
((𝑋𝑋 𝑃𝑃 )2 +(𝑌𝑌𝑃𝑃 )2 +(𝑍𝑍𝑃𝑃 )2 )
�𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗� = �𝑅𝑅�⃗� = �
� ∙ �𝑃𝑃�⃗�
(𝑌𝑌𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑌𝑌𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑍𝑍𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑍𝑍𝐹𝐹 )2 +(𝑋𝑋 𝐹𝐹 ∙𝑌𝑌𝑃𝑃 −𝑋𝑋 𝑃𝑃 ∙𝑌𝑌𝐹𝐹 )2 ((𝑋𝑋 𝑃𝑃 )2 +(𝑌𝑌𝑃𝑃 )2 +(𝑍𝑍𝑃𝑃 )2 )
�((𝑋𝑋𝑃𝑃 )2 + (𝑌𝑌𝑃𝑃 )2 + (𝑍𝑍𝑃𝑃 )2 )
∙
�𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗� = �𝑅𝑅�⃗� = �(𝑌𝑌𝐹𝐹 ∙ 𝑍𝑍𝑃𝑃 − 𝑌𝑌𝑃𝑃 ∙ 𝑍𝑍𝐹𝐹 )2 + (𝑋𝑋𝐹𝐹 ∙ 𝑍𝑍𝑃𝑃 − 𝑋𝑋𝑃𝑃 ∙ 𝑍𝑍𝐹𝐹 )2 + (𝑋𝑋𝐹𝐹 ∙ 𝑌𝑌𝑃𝑃 − 𝑋𝑋𝑃𝑃 ∙ 𝑌𝑌𝐹𝐹 )2
(5) The equation (5) is the Surface (Area) of the parallelogram created with the points (A, B, C, D’) of the two vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗ as in the figure 3.
This is a simple application of the Opposition Product in which we can find the Area of the Parallelogram formed by two vectors by doing the magnitude value of two vectors; moreover the opposition product gives the negative inverse vector of the second vector amplified by k.
ISBN: 978-1-61804-064-0
In this paper, the author introduced a new and original product between vectors named Opposition Product. In mathematics, the Opposition Product is an Operation between two vectors or more in nDimension space; it is particularly used into 2-D space and 3-D Space. The Product of two vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗ gives as result a vector collinear with 𝑃𝑃�⃗ but with a negative sign (Opposed vector in the same Plan of the two vectors) and amplified by 𝑘𝑘(𝛼𝛼), the formula is written as following 𝐹𝐹⃗ ⊣ 𝑃𝑃�⃗ = 𝑅𝑅�⃗ = −𝑘𝑘(𝛼𝛼) ∙ 𝑃𝑃�⃗ . The value of 𝑘𝑘(𝛼𝛼) is determined according to the application of the Opposition Product, for example, in the section 3 the value of 𝑘𝑘(𝛼𝛼) is developed in order to obtain the proportional Area of the two vectors 𝐹𝐹⃗ and 𝑃𝑃�⃗ . As conclusion, many studies will follow the Opposition Product in this paper in order to find other applications in many domains as in science, engineering and mathematics. References: [1] Jeffreys H., Jeffreys B.S., “Methods of mathematical physics”. Cambridge University Press, (1999). [2] Dennis G. Zill, Michael R. Cullen "Definition 7.4: Cross product of two vectors", Advanced engineering mathematics (3rd ed.), Jones & Bartlett Learning, ISBN 076374591X, (2006), p. 324. [3] Dennis G. Zill, Michael R. Cullen, "Equation 7: a × b as sum of determinants", Jones & Bartlett Learning, ISBN 076374591X, (2006), p. 321. [4] Massey W.S., "Cross products of vectors in higher dimensional Euclidean spaces", The American Mathematical Monthly, Vol. 90, No. 10, (Dec. 1983), pp. 697–701. [5] Vladimir A. Boichenko, Gennadiĭ Alekseevich Leonov, Volker Reitmann, “Dimension theory for ordinary differential equations”, Vieweg+Teubner Verlag.. ISBN 3519004372, (2005), p. 26.
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[6] Pertti Lounesto, “Clifford algebras and spinors”, (2nd ed.), Cambridge University Press, ISBN 0521005515, (2001), p. 94. [7] Shuangzhe Liu, Gõtz Trenkler, "Hadamard, Khatri-Rao, Kronecker and other matrix products", International Journal of Information and systems sciences (Institute for scientific computing and education), (2008), pp. 160–177. [8] Eric W. Weisstein, "Binet-Cauchy identity", CRC concise encyclopedia of mathematics (2nd ed.), CRC Press, ISBN 1584883472, (2003), p. 228. [9] Lagrange J.L., "Solutions analytiques de quelques problèmes sur les pyramides triangulaires", Oeuvres, volume 3. (1773). [10] Wilson Edwin Bidwell, “Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs”, Yale University Press, (1901).
ISBN: 978-1-61804-064-0
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