Vectors, Matrices and Geometry

CHAPTER ONE: VECTORS AND GEOMETRY IN THE PLANE

CHAPTER ONE VECTORS AND GEOMETRY IN THE PLANE In school geometry, points on the plane are represented by pairs of real numbers which are called coordinates, and algebraic operations are carried out on the individual coordinates to discover properties of geometric configurations. For example, given two points P and Q represented by (a, b) and (c, d) respectively, the straight line passing through P and Q consists of points X whose coordinates (x, y) satisfy the polynomial equation (y - b)(c - a) = (x - a)(d - b) and the slope of the line is given by the algebraic expression (d - b)/ (c - a) in the coordinates of P and Q. We note that here algebraic operations are not carried out on the points P, Q and X but on their coordinates a, b, c, d, x, y. There are other ways in which algebra can be used in the study of plane geometry. For example, in an earlier book of the present series (see Section 7.14 of Fundamental Concepts of Mathematics) the correspondence between points of the plane and complex numbers is used. This correspondence enables us to use a single symbol for a complex number to refer a pair of coordinates. Moreover, it also allows us to operate directly on the complex numbers which now stand for points on the plane, so that the manifold properties of the complex number system are at our disposal in the study of plane geometry. Unfortunately this method cannot be extended to the study of geometry in space because we do not have a similar system of 3-dimensional numbers. In this chapter we shall explore yet another approach to plane geometry by using vectors in the plane. Vectors in the plane are another kind of algebraic entities and, like complex numbers, they have many properties that are useful for the study of plane geometry . With vectors we also have the same advantage of using a single symbol to refer a pair of components. Furthermore the notion of 1 Vectors, Matrices and Geometry vectors in the plane lends itself for easy generalization to the notion of vectors in spaces of higher dimensions. 1.1 Vectors in the plane A vector is usually described as a quantity that has a magnitude and a direction. For example in secondary school physics a displacement , a velocity and a force acting at a point are vectors while a mass and a temperature are not. Though it is possible to give an accurate description of magnitude and direction and use it to define vectors, it becomes a cumbersome task to generalize the notion of direction in higher dimensional spaces. Instead we shall lay down a simple definition of vector in terms of numbers and explain later the meaning of magnitude and direction of a vector. 1.1.1 DEFINITION A vector in the plane is an ordered pair ofreal numbers. The numbers of the ordered pair are called the components of the vector. Vectors shall be denoted by bold-faced letters and square brackets shall be used to enclose the two components. Accordingly the real numbers aI, a2 are the components of the vector a = [al' a2]. 1.1.2 REMARKS While bold-faced types are used for vectors in this book, they will be difficult in handwritten work. Students may find it easier to indicate a vector by placing an arrow over or a bar below the letter. Any other consistent usage is just as satisfactory. Being ordered pairs of real numbers, two vectors are equal if and only if they have identical components: for vectors a = [aI, a2] and b =[bl, b2 ], a =b if and only if al =bl and a2 =b2 • Vectors in the plane can be visualized as arrows in the cartesian plane. To present the vector a = [aI, a2] graphically, we draw a directed segment or an arrow from the origin 0 to the point A with coordinates (aI, a2). In this way every vector is represented by an arrow with initial point at the origin o. Conversely every arrow with initial point at 0 represents an vector in the plane. In Figure 1-1 the arrows representing the vectors b =[1,3]' c =[-2,4], d =[4,2] are drawn. 2 Vectors and Geometry in the Plane y c x Fig 1-1 1.1.3 REMARKS The reader will have no doubt noticed that an ordered pair of...