Half Circular Vector in 2D space

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1 Half Circular Vector in D space CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (003) LEBANON claude_bayeh_cbegrdi@hotmail.com NIKOS E.MASTORAKIS WSEAS (Research and Development Department) Agiou Ioannou Theologou , Zografou, Athens,GREECE mastor@wseas.org Abstract: - The Half Circular Vector is an original study in mathematics introduced by the author in which the presented vector is not linear as the traditional vector, but the half circular vector takes the form of half circle with a lead at the head to distinguish it from the half circle. The half circular vector is a vector that can be studied similar to the linear vector with some difference. For example, the magnitude of this vector gives the perimeter of the half circle which is πd/ with D is the diameter of the vector, comparing to the linear vector in which the magnitude gives the length of the segment formed by the vector. The summation of many half circular vectors gives as result a half circular vector which is the sum of many half circular vectors. It has the same characteristics of the linear vector but the difference is that it is multiplied by π/ as it is a half circle. Many studies will follow this paper in order to find many applications in mathematics and scientific domains. As result, the Half Circular Vector is a vector that can be treated as the traditional linear vector, it has the same characteristics but multiplied by π/ as it is a half circle. Key-words:- Linear Vector, Half circular vector, magnitude, Euclidian space, Circular Space, mathematics. 1 Introduction In mathematics, a vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context [1] []. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field [3] [4]. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms [5] [6]. An example of a vector space is that of Euclidean vectors [7], which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector [8]. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces [9]. In this paper, a new and original study is introduced in the mathematical domain called half circular vector. In mathematics, the half circular vector is an original study in which it has similar characteristics of the linear vector (traditional vector) but it has the shape of a half circle and a lead on its head in order to distinguish it as a vector and not just a half circle. The half circular vector composed by two points similar to the linear vector, it has also a magnitude and a direction also similar to the linear vector but there are many different points that are not in common which are developed in this paper. The main goal of introducing the half circular vector is to create a circular space instead of the linear space such as the Euclidian space; many studies will follow this paper in order to find applications in mathematics or in any scientific domain. In this paper, the main concept of the half circular vector is developed. In the section, a definition of the half circular vector is presented with a complete study. In the section 3, some applications are presented. In section 4, a conclusion about the half circular vector is presented. ISBN:

2 Definition of the Half Circular vector The Half circular vector denoted by AAAA is not a linear vector such as the traditional one AAAA (figure ), but it is a vector that has a form of half circle (refer to figure 1). The direction of the half circular vector is counterclockwise. Fig. 1: represents the half circular vector AB which is the sum of many collinear intermediate vectors. Fig. : represents the traditional vector AAAA which is the sum of many collinear intermediate vectors. The half circular vector AAAA is equal to the sum of collinear intermediate vectors (figure 1). The magnitude of AAAA is equal to AAAA = dd AAAA With dd AAAA is the diameter of the half circle AB..1 Definition of the magnitude of the half circular vector Theorem 1: AAAA = AAAA + CCCC + DDDD + EEEE (1) With the points C, D, E are collinear intermediate points of the diameter AB (refer to figure 1) Demonstration: AAAA = dd AAAA AAAA = dd AAAA, CCCC = dd CCCC DDDD = dd DDDD, EEEE = dd EEEE The diameter dd AAAA is equal to the sum of the collinear intermediate points. Therefore, dd AAAA = dd AAAA + dd CCCC + dd DDDD + dd EEEE Thus, AAAA = dd AAAA = dd AAAA + dd CCCC = (dd AAAA +dd CCCC +dd DDDD +dd EEEE ) + dd DDEE + dd EEEE AAAA = AAAA + CCCC + DDDD + EEEE Theorem : The general formula of the half circular vector can be obtained as following with the sequence of AA nn are collinear intermediate points between the first point of the half circular vector and its end point. AA 0 ii = nn=ii 1 AA nn AA nn+1 nn=0 () With AA nn AA nn+1 is a half circular vector with diameter equal to dd AAnn AAnn +1. Demonstration: The same method of demonstration as the previous one can be used AA 0 ii = dd AA0AA ii AA 0 1 = dd AA0AA1 AA 1 = dd AA1AA AA ii 1 AA ii = dd AA ii 1 AA ii The diameter dd AA0 AA ii is equal to the sum of the collinear intermediate points. Therefore, dd AA0 AA ii = dd AA0 AA 1 + dd AA1 AA + + dd AAii 1 AA ii Thus, AA 0 ii = dd AA0AA ii = (dd AA0AA1 +dd AA1AA + +dd AA ii 1 AA ii ) = dd AA0AA1 + dd AA1AA + + dd AA ii 1 AA ii AA 0 ii = AA AA AA ii 1 AA ii AA 0 ii = nn=ii 1 AA nn AA nn+1 nn=0 Therefore the equality is verified. = dd AA 0 AA ii. Definition of the half circular vector Considering the two vectors AAAA and AAAA as in the figure 3. We can obtain the equation of the half circular vector from the linear vector as following: AAAA = (xx BB xx AA )ii + (yy BB yy AA )jj (3) (refer to figure 3) We have, cos(αα) = xx BB xx AA AAAA and sin(αα) = yy BB yy AA AAAA ISBN:

3 Then, AAAA = AAAA cos(αα) ii + AAAA sin(αα) jj (4) If we multiply the equation (4) by we obtain the following equation: AAAA = AAAA cos(αα) ii + AAAA sin(αα) jj (5) The magnitude of the half circular vector is denoted by AAAA and it is equal to: AAAA = AAAA = dd AAAA (6) So, AABB = AAAA cos(αα) ii + AAAA sin(αα) jj = AAAA cos(αα) ii + AAAA sin(αα) jj (7) AAAA If we find the module of the equation (7) we obtain the module of the half circular vector equation AAAA = AAAA cos(αα) + AAAA sin(αα) AAAA = AAAA cos(αα) + sin(αα) AAAA = AAAA.3 Other definition of the half circular vector Considering a half circular vector AAAA in the Cartesian coordinate system such as in the figure 3.1. The half circular vector AAAA is the sum of other half circular vectors YY AAAA and XX AAAA. XX AAAA = dd XX AAAA = (xx BB xx AA )ii (10) YY AAAA = dd YY AAAA = (yy BB yy AA )jj (11) Therefore, AAAA = XX AAAA + YY AAAA AABB = (xx BB xx AA )ii + (yy BB yy AA )jj (1) We remark that the equation (1) is the same as the equation (9), so this is a second method to demonstrate the equation of the half circular vector AAAA. Therefore we can deduce that the half circular vector is equal to: AAAA = AAAA = AAAA cos(αα) ii + AAAA sin(αα) jj (8) AAAA = (xx BB xx AA )ii + (yy BB yy AA )jj (9) Fig. 3.1: represents the half circular vector AAAA and other vectors such as YY AAAA and XX AAAA in the (xoy) coordinate system. Fig. 3: represents the half circular vector AAAA and the linear vector AAAA in the (xoy) coordinate system. As we can notice that the half circular vector has the same properties as the linear vector but multiplied by. Fig. 3.: represents the half circular vector AAAA and other vectors such as YY AAAA and XX AAAA in the (xoy) coordinate system. ISBN:

4 .4 Equation of the circle that contains the two points A and B The general equation of a circle with center O is equal to: (xx xx OO ) + (yy yy OO ) = RR (13) With RR is the radius of the circle And O is the center of the circle with OO(xx OO, yy OO ) And the two points A and B are included in the circle and O is the midpoint between them, therefore xx OO = xx BB +xx AA and yy OO = yy BB +yy AA So the equation is as following: (xx xx BB +xx AA ) + (yy yy BB +yy AA ) = RR (14) We can also know the equation of the radius RR With RR = (xx BB xx AA ) +(yy BB yy AA ) 4 Therefore the final equation is written as below: (xx xx BB +xx AA ) + (yy yy BB +yy AA ) = (xx BB xx AA ) + yy BB yy AA 4 (15) Fig. 4: represents the circle with diameter AB and a line that path through the two points A and B. By knowing the two points AA(xx AA, yy AA ) and BB(xx BB, yy BB ) we can deduce easily the equation of the circle that contains the two points A and B..5 Equation of the line that contains the two points A and B The general equation of a line that paths through two points A and B has the following equation: yy = aaaa + bb (16) With aa = yy BB yy AA xx BB xx AA And for a specific point in the line, we take AA(xx AA, yy AA ). Therefore the equation is as following: yy AA = yy BB yy AA xx xx BB xx AA + bb AA We have to calculate bb. So, bb = yy AA yy BB yy AA xx xx BB xx AA = yy AA xx BB yy BB xx AA AA xx BB xx AA Therefore the equation is equal to: yy = yy BB yy AA xx + yy AA xx BB yy BB xx AA (17) xx BB xx AA xx BB xx AA This line in equation (17) will divide the circle in equation (15) into two regions as shown in the figure 4. The region (I) is the region in which contains the half circular vector and the region (II) does not contains the half circular vector. Fig. 5: represents the circle with diameter AB and a line that path through the two points A and B..6 Magnitude of the half circular vector in the Cartesian coordinate system Considering the equation (9) AAAA = (xx BB xx AA )ii + (yy BB yy AA )jj (9) The magnitude of the equation (9) is denoted as following: AAAA = (xx BB xx AA ) + (yy BB yy AA ) AAAA = (xx BB xx AA ) + (yy BB yy AA ) (18) We can demonstrate that: AAAA = XX AAAA + YY AAAA (19) So the square of the half circle formed by AB is equal to the sum of the square half circle formed by XX AAAA and YY AAAA. Then we can deduce that the half ISBN:

5 circular vector has the same properties as the linear vector AAAA but multiplied be. The formed circles have the equations equals to the equation (15): (xx xx AA +xx OO With xx AA R ) + (yy yy AA +yy OO ) = (xx AA xx OO ) + yy AA yy OO 4 (1) 3. Half Circular Triangle in D space The half circular triangle in D space is a triangle formed by half circular vectors of three points (A, B and C) (refer to figure 8). The total summation of angles that form this triangle is equal to 70 0 comparing to the normal triangle that form Fig. 6: represents the half circular vector AAAA and other vectors such as YY AAAA and XX AAAA in the (xoy) coordinate system. 3 Application of the Half Circular Vector The equations of the three vectors that form this triangle are equals to: AAAA = (xx BB xx AA )ii + (yy BB yy AA )jj BBBB = (xx CC xx BB )ii + (yy CC yy BB )jj CCCC = (xx AA xx CC )ii + (yy AA yy CC )jj 3.1 Monopole circular D space The monopole circular D space is a space formed by a single point that has magnetic fields. All curves formed in this space are circular and have one directional vector. The formed vectors are similar to the half Circular Vector. So we can conclude that the Half Circular Vector form the basis of the Monopole circular D space. All vectors in this space have the equation: OOOO = (xx AA xx OO )ii + (yy AA yy OO )jj (0) With OO is the center of the space and AA is a point on the horizontal axis. Fig. 8: represents the half circular triangle in D space. Fig. 7: represents the monopole circular D space is a space formed by a single point that has magnetic field. 3.3 Half Circular Rectangle in D space The half circular Rectangle in D space is a rectangle formed by half circular vectors of four points (A, B, C and D) (refer to figure 9). ISBN:

6 The total summation of angles that form this rectangle is equal to comparing to the normal rectangle that form The equations of the four vectors that form this rectangle are equals to: AAAA = (xx BB xx AA )ii + (yy BB yy AA )jj BBBB = (xx CC xx BB )ii + (yy CC yy BB )jj CCDD = (xx DD xx CC )ii + (yy DD yy CC )jj DDAA = (xx AA xx DD )ii + (yy AA yy DD )jj Fig. 9: represents the half circular triangle in D space. 4 Conclusion In this paper, the author introduced a new and original study in the mathematical domain called half circular vector. In mathematics, the half circular vector is an original study in which it has similar characteristics of the linear vector (traditional vector) but it has the shape of a half circle and a lead on its head in order to distinguish it as a vector and not just a half circle. The half circular vector composed by two points similar to the linear vector, it has also a magnitude and a direction also similar to the linear vector but there are many different points that are not in common which are developed in this previous sections. The main goal of introducing the half circular vector is to create a circular space instead of the linear space such as the Euclidian space; some applications are developed in the previous sections such as the monopole circular D space which is the base of the magnetic field formed by a point in the space. Many studies will follow this paper in order to find applications in mathematics or in any other scientific domains. As conclusion, many studies will follow this paper in order to find other applications in many domains as in science, engineering and mathematics. The Circular Space is the result of the half circular vector and it will be studied in a separate paper. References: [1] Artin Michael, Algebra, Prentice Hall, ISBN , (1991). [] Blass Andreas, "Existence of bases implies the axiom of choice", Axiomatic set theory (Boulder, Colorado, 1983), Contemporary Mathematics, 31, Providence, R.I.: American Mathematical Society, (1984), pp [3] Brown William A., Matrices and vector spaces, New York: M. Dekker, ISBN , (1991). [4] Lang Serge, Linear algebra, Berlin, New York: Springer-Verlag, ISBN , (1987). [5] Lang Serge, Algebra, Graduate Texts in Mathematics, 11 (Revised third ed.), New York: Springer-Verlag, ISBN , (00). [6] Mac Lane Saunders, Algebra, (3rd ed.), ISBN , (1999), pp [7] Meyer Carl D., Matrix Analysis and Applied Linear Algebra, SIAM, ISBN , (000). [8] Roman Steven, Advanced Linear Algebra, Graduate Texts in Mathematics, 135 (nd ed.), Berlin, New York: Springer-Verlag, ISBN , (005). [9] Spindler Karlheinz, Abstract Algebra with Applications: Volume 1: Vector spaces and groups, CRC, ISBN , (1993). ISBN:

Introduction to the General operation of Matrices

Introduction to the General operation of Matrices CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (004) LEBANON Email: claude_bayeh_cbegrdi@hotmail.com