the position of an object moving in a plane, we need to choose a convenient point, say O as origin. Let P and P ′ be the positions of the object at time t and t ′ , respectively [Fig. . (a)].
We join O and P by a straight line. Then, OP is the position vector of the object at time t . An arrow is marked at the head of this line. It is represented by a symbol r , i.e.
OP = r . Point P ′ is represented by another position vector, OP ′ denoted by r ′ . The length of the vector r represents the magnitude of the vector and its direction is the direction in which P lies as seen from O. If the object moves from P to P ′ , the vector PP ′ (with tail at P and tip at P ′ ) is called the displacement vector corresponding to motion from point P (at time t ) to point P ′ (at time t ′ ).
Fig. . (a) Position and displacement vectors. (b) Displacement vector P Q and different courses of motion.
It is important to note that displacement vector is the straight line joining the initial and final positions and does not depend on the actual path undertaken by the object between the two positions. For example, in Fig. . (b), given the initial and final positions as P and Q , the displacement vector is the same P Q for different paths of journey, say PABC Q , PD Q , and PBEF Q .
Therefore, the magnitude of displacement is either less or equal to the path length of an object between two points . This fact was emphasised in the previous chapter also while discussing motion along a straight line. . .
Equality of Vectors Two vectors A and B are said to be equal if, and only if, they have the same magnitude and the