1.1.  Vectors in space

In geometry, one is interested in the study of mutual position of points. This can be done best by specifying the distance between the points and the direction of one relative to the other. In other words, we want to study objects which have both 'magnitude' and 'direction' (such objects are also studied in Physics, e.g., velocity, force, etc.). To be more precise, let us make the following definition.   

Let , be two distinct points in space. The line segment from to with direction as that of to is called a  localized vector in space and is denoted by . The point is called the initial point and the point is called the final point of the localized vector . The length of the line segment is called the magnitude of  , denoted by ||.

In pictures,  a localized vector is depicted by an arrow as shown in figure below.

                   
                                                                      Figure 1.1

       We denote the set of all localized vectors by X. Suppose and are localized vectors such that the following hold:

      (i)   Line is parallel to line .
      (ii)  Lines  and have same direction.
      (iii) They have equal magnitudes, || = ||.

Since 'geometrical properties' are invariant under 'parallel transformations', we can say that the localized vectors  and   both represent the same arrow if properties (i), (ii) and (iii) above are satisfied. Also, physical quantities require only magnitude and direction. Thus, we shall not distinguish between such vectors. To be precise, we make the following definition. 

Two localized vectors and are said to be equivalent if one can be obtained from the other by a parallel transformation i.e., the properties (i), (ii) and (iii) above are satisfied. We write this as            .     

It is easy to check that ~ is an equivalence relation on X. The set of  equivalence classes of X under the equivalence relation ~ is denoted by V, and its elements are called vectors in space. We denote the elements of V by u, v, w, etc., and if a localized vector Î u, then is called a localization of u at. The vector having zero magnitude will be called the zero vector denoted by 0. This is also represented as , P being any point  in space. Pictorially, it is convenient to represent all vectors with the same initial point. This fixed initial point will be denoted by O, called the origin. Thus, every vector can be represented uniquely by a localized vector with initial point at O, and conversely. We shall call such representations as position vectors. Note that, every u ÎV determines a fixed direction in space.

         Points in space can be identified with elements of V. We choose a point O in space. For any u Î V, let be the localized vector with initial point at O such that Îu. This gives us a point P in space. The map u a P is a one-one map. For, if  P = Q in space then = and hence =  []. Also it is onto, for a given point P in space, consider u := . Then u will determine P. Thus points in space are elements of V, the point P being identified with .