Vectors, Matrices and Geometry

CHAPTER TWO: VECTORS AND GEOMETRY IN SPACE

CHAPTER Two VECTORS AND GEOMETRY IN SPACE In this chapter we follow the pattern of last chapter to study algebra of vectors in space and solid geometry side by side. Readers will find a fairly complete treatment of the vector space R3 where most of the important topics are discussed. In spite of the extensive subject of geometry in space, we are only able to include some general algebraic methods in the treatment of lines and planes and a very small selection of classical theorems. 2.1 Cartesian coordinates in space The cartesian coordinates of a point in the plane are given in relation to a pair of mutually perpendicular axes; those of a point in space are given in relation to three mutually perpendicular lines intersecting at a point. Suppose that we have such a triad of lines. The point of intersection will be called the the origin and is denoted by o. Suppose that each line is given a fixed direction. Then we proceed to assign the directed lines as axes of coordinates: the xaxis , the y-axis and the z-axis. There are 6 different assignments which fall into two groups of three each. The first group is shown in Figure 2-1. Each assignment of the first group can be changed into any other of the same group by a rotation about o. Similarly each assignment of the second group (Figure 2-2) can be changed into any other of the same group. However it is impossible to change any assignment of one group into an assignment of the other group by a rotation about O. 39 Vectors, Matrices and Geometry Fig 2-1 z y Fig 2-2 To distinguish these two groups, we shall call each assignment in the first group a right-hand coordinate system because the x-axis, the y-axis and the z-axis correspond respectively to the thumb, the index finger and the middle finger of the right hand when extended in mutually perpendicular directions. Similarly the other three are called left-hand coordinate systems. Though it is immaterial as to which coordinate system is used, we shall henceforth adopt a right-hand coordinate system and proceed to set up a one-to-one correspondence between points in space and ordered triples of real numbers. To begin with we identify each coordinate axis with the real line in the usual manner. Given any ordered triple (ab a2, a3), each of the real numbers aI, a2, a3 individually corresponds to a point on the x-axis, y-axis, z-axis respectively. Through each of these points, we erect a plane perpendicular to the corresponding coordinate axis. The three planes so erected meet at a point A in space. In this way every ordered triple (aI, a2, a3) of real numbers is matched with a point A in space and we shall call the real numbers aI, a2, a3 the coordinates of the point A and write A = (ab a2, a3). In particular 0=(0,0,0). 40 x Vectors and Geometry in Space / ",,/ ",,// a3 ",'" "" z r:'------I I I I I I I I : O,,)---r--------:~___+y : /,,/"/ a2 I ,,'" : ",/ -------_____v" Fig 2-3 Conversely if B is a point in space, then through B we can pass three planes, each individually perpendicalur to a coordinate axis. These planes cut the x-axis, y-axis, z-axis at points corresponding to coordinates bl , b2 , b3 respectively. In this way the point B is matched with the ordered triple (b!, b2 , b3 ). Clearly the ordered triple (bl , b2 , b3 ) delivers back the point B by the previous process: B = (bl , b2 , b3 ). Therefore we have a one-to-one coorespondence between points in space and ordered triples of real numbers. By this correspondence the cartesian space of all ordered triples of real numbers becomes a mathematical model of the three-dimensional space. This model enables us to carry out algebraic works on coordinates of points in space. As an example we give a formula for the euclidean distance between two points in space. Let A = (al,a2,a3) and B = (bl ,b2 ,b3) be two points in space. Appropriate planes passing through A or B and perpendicular to the axes meet at a number of points (see Figure 2-3). Among them we find C =(b1 ,b2,a3), A' =(al,a2,O) and C' =(bl ,b2 ,O). A' and C...