Representation of a Vector

 Representation of a Vector

• In two dimensions, a vector can be represented by a straight line with an arrow in a plane. This is shown in the Fig. 1.3.1. The length of the segment is the magnitude of a vector while the arrow indicates the direction of the vector in a given co-ordinate system. The vector shown in the Fig. 1.3.1 is symbolically denoted as . The point O is its starting point while A is its terminating point. Its length is called its magnitude, which is R for the vector OA shown. It is represented as || = R. It is the distance between the starting point and terminating point of a vector.

Key Point : The vector hereafter will be indicated by bold letter with a bar over it.

1. Unit Vector

• A unit vector has a function to indicate the direction. Its magnitude is always unity, irrespective of the direction which it indicates and the co-ordinate system under consideration. Thus for any vector, to indicate its direction a unit vector can be used. Consider a unit vector  in the direction of  as shown in the Fig. 1.3.2. This vector indicates the direction of  but its magnitude is unity.

• So vector  can be represented completely as its magnitude R and the direction as indicated by unit vector along its direction.

where  = Unit vector along the direction OA and || =1

Key Point : Hereafter, letter  is used to indicate the unit vector and its suffix indicates the direction of the unit vector. Thus  indicates the unit vector along x axis direction.

• In case if a vector is known then the unit vector along that vector can be obtained by dividing the vector by its magnitude. Thus unit vector can be expressed as,

The idea and use unit vector will be more clear at the time of discussion of various co-ordinate systems, later in the chapter.

Review Question

1. What is a unit vector ? What is its significance in the vector representation ? How to find unit vector along a particular vector ?