We will study some of the basic concepts about vectors various operations on vectors, their algebraic and geometric properties. These two types of properties, when considered to gather give full realizations to the concepts of vectors, and lead to their vital applicability in various area

M, MAHALAKSHMI

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A STUDY ON PLANE AND SPHERE (M.MAHALAKSHMI,Dr.S.SANGEETHA,R.RAMYA,P.PRIYA) (priyaperiyasamy13@gmail.com,sangeethasankar2016@gmailcom ) Department of Mathematics Dhanalakshmi Srinivasan College of Arts and Science for Women (Autonomous) Perambalur  ABSTRACT           We will study some of the basic concepts about vectors various operations on vectors, their algebraic and geometric properties. These two types of properties, when considered to gather give full realizations to the concepts of vectors, and lead to their vital applicability in various areas. KEYWORDS Scalar, Vector, Collinear vector, Like vector, Unlike vector, coplanar  INTRODUCTION                    Vector algebra is introduced in (1805-1865) by W.R. Hamilton.  However, an answer to the second query is a quantity (called force) which involves muscular strength (magnitude) and direction n (in which another player is positioned). Such quantities are called vectors. We frequently come across with both types of resistance etc... And vector quantities like displacement, velocity, acceleration, force, weight, and momentum electric field about vectors.  Vectors are main objects of study in multivariable calculus. They have the same role that numbers have in single-variable calculus. It is very important to gain a solid understanding of vectors before proceeding to multivariable calculus.  BASIC DEFINITIONS Multiplications of a vector by a scalar:  Let m be any scalar and    be any vector, then the product             of the vector    and the scalar  is a vector whose, i. Magnitude is     times that of      ii. Support is the same or parallel to that of     and iii. Sense is the same or opposite to that of    according as  is +ve or –ve. DEFINITION:   A vector   is said to be a linear combination of the vectors               if there exist scalars x, y, z...., such that                    DEFINITION:  A system of vector                         is said to be linearly dependent if there exist scalars        (not all zero)  such that                                  DEFINITION:  A system of vectors            is said to be linearly independent if there exist scalars            (all zero)  such that                     DEFINITION:   We take in space any arbitrary point 0, to be called as the origin of reference. Then the position vector of any point p, with respect to the origin 0 is the vector            We usually denoted the position vector of the points A, B, C by            respectively. CHARACTERISATION OF A VECTOR: 1. Length 2. Support 3. Sense Length:  The length of the directed line segment          is the distance between  P and Q. Support:  The line of an unlimited length of which a directed line segment is a part is called the Support. Sense:  The Sense of the directed line segment is from its initial point to its terminal point. DEFINITION:  DISPLACEMENT OF VECTOR:  If a point is displaced from position A to B then the displacement AB represents a vector           which is known as the displacement vector. PLANE:  A plane is a surface which is purely flat such that if any two points are considered on the surface and joined by a straight line lies wholly on the surface. The surface is known as plane surface or flat surface. Now we study the plane under various conditions using the tools of vector algebra.   Plane Through a Point and Parallel to Two Vectors    We assume that the plane passes through the point A and is parallel to the given vectors    and    . Let P be any point on the plane. Relative to a fixed point O chosen as origin, suppose that    and   are the position vector of A and P respectively. Then we have                         . Since          is parallel to    as well as    , so it is parallel to a vector which is the linear combination of    and     and hence we must have                  ,   and   are the scalar quantities. Thus from above we obtain                , i.e.,               ,  ..... (3.1) Which is that the required vector equation of the plane.   Corollary 1:  If the plane passes through the origin, then the equation of the plane (3.1) takes the form              .   Corollary 2:       If                                             and                   , then from (3.1) we obtain                                                        ,  Which yields                                          Which, on elimination of   and  , reduces to              , Where       and   are scalars. Thus we obtain the Cartesian equation of the plane in three dimensional coordinate geometry. Corollary 3: (Non-parametric form).  From the above discussion it follows that the vector      is parallel to the vector    and     and hence is perpendicular to the vector       . Hence as a result the vectors         and     are coplanar. Thus we obtain                          i.e.,                           .... (3.2) Which is the equation of the plane in non-parametric form. Plane Through Two Points and Parallel to a Vector   We consider A, B as the given points and     as the given vector. With respect to a fixed point O chosen as origin, we suppose that    and    are the position vectors of A and B respectively. Let P, with position vector  , be any point on the plane.   Then we have               and               . Since the required plane is parallel to     as well as        so it is parallel to a vector which is the linear combination of     and       and hence we must have                          and   being the scalar quantities. Thus we have                      i.e.,                   ,          ... (3.3) Which is that  the required vector equation of the plane. Corollary 4:  From the above discussion it is clear that the vector                    are coplanar. Hence we obtain                       , Which on simplification yields                               ,        ….. (3.4) Which is that the equation of the plane in non-parametric form.      SPHERE: DEFINITION:  A sphere is that the locus of a variable point in space whose distance from a hard and fast point within the space is usually a continuing . The fixed point is known as the centre of the sphere and the constant length of any point on the sphere from the centre is known as its radius. Now we will deduce the equations of sphere under different condition as follows: Sphere with Given Radius and Centre   we consider a sphere whose centre is at C with position vector    relative to a fixed point O chosen as origin. Let P be any point with position vector   on the surface of the sphere whose radius is of   units. Then we have              . Now               . Hence we have                                      , ..... (4.1)  Corollary 1:  If the centre of the sphere is at the origin then the equation of the sphere reduces to the following:             i.e.,         ..... (4.2) Corollary 2:  If the sphere passes through the origin O, then we have                        and hence the equation of the sphere reduces to the following:               .   ..... (4.3) Sphere with Given Extremities of a Diameter  Let P be any point with position vector   on the surface of the sphere. Further let A and B be the extremities of a diameter with position vector    and    respectively. Then we have               and              . Now for the sphere,          is perpendicular to         . Hence we have                               ,     Intersection of a Line and a Sphere  We consider a line through a given point A and parallel to a given unit vector   . Let    be the position vector of A. Further we consider a sphere whose centre is at C with position vector     and radius is of   units. Then the equation of the road and therefore the sphere are often respectively written as           and                       ,   Where   is a scalar. Eliminating   between the last two equations and using the fact that       , we obtain the required condition as follows:                                                    ,      ... (4.5) Which yields two values of   corresponding to each of which we have the points of  intersection as P and Q. Thus the position vectors of P and Q are         and         respectively, where       and       are the roots of the quadratic                      equation (4.5). Now the product of these roots is                        , which is independent of    and hence it is same for all lines drawn through A to intersect the sphere. If P and Q coincide at a point T on the sphere, then the line will be a tangent and we find the relation                                   .  NOTE:  The tangent drawn from A generates a cone having its vertex at A and enveloping the sphere is known as the enveloping cone. Also the quantity                             is known as the power of the point A.  Tangent Plane to a Sphere   We consider that the point A (under consideration in the subsection lies on the sphere, then the points A, P, Q coincide, which implies that         and hence the sum and product of the roots of the quadratic (4.5) are zero. As a result we have                         and                               ,   .... (4.6) Which are the required conditions of tangency. Now from the first equation of (4.6) it follows that the line          is perpendicular to the vector        and hence we get                        ,   .... (4.7) Which shows that all tangents through the point A lies on it and it is the tangent plane to the sphere at A. Further using the second equation of (4.6) and (4.7), we get the standard equation of the tangent plane at A to the sphere (4.1) as follows:                                       .... (4.8)  Corollary:3 (Condition of tangency)  If a plane         being the unit normal touches the sphere given in (4.1), then the perpendicular distance of the plane from the centre of the sphere equals to the radius of the sphere. Thus we have the following:                .  CONCLUSION       The vector algebra play fundamental role in the straight line, plane and sphere. In this applications to various fields such as geometry, mechanics, physics, and engineering, scientific research.  The concept of vector algebra is explained lucidly with the geometric notations and physical motivations. Further the geometrical and physical applications through this chapters help one in the advanced reading of the subject which is becoming more and more abstract.  REFERENCES 1. Absos Ali sheikh: “vector analysis with applications” narosa publishing house pvt. Ltd, new delhi-110 002. 2. E.B. Wilson: Vector Analysis, Dover publications, Inc., New York, 1960. 3. HARI KISHAN  “ Vector Algebra” and calculus Atlantic publications and distribution (P) Ltd, New delhi-110027.        

A STUDY ON PLANE AND SPHERE

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A STUDY ON PLANE AND SPHERE

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