6 



INTRODUCTORY. 



[CHAP. i. 



5. Change of position. Suppose that a point which, at 

 any particular instant, had a position P with reference to any 

 frame, has at some later instant a position Q relative to the same 

 frame. The point is said to have undergone a "change of posi- 

 tion " or a displacement. Let the line PQ be drawn. It is clear 

 that the displacement is precisely determined by this line; we 

 say that it is represented by this line. Suppose the line PQ 

 drawn through P to be produced indefinitely both ways, a parallel 

 line may be drawn through any other point, 

 for instance through 0, and then this line 

 determines a particular direction; this is 

 the direction of the displacement. Of the 

 two senses in which this line may be 

 described one, OR, is the sense from 

 towards the point R which is the fourth 

 corner of a parallelogram having OP, PQ 

 as adjacent sides; this is the sense of the 

 displacement. The measure of the length 

 of PQ is the number of units of length it 

 contains; this number is the magnitude 

 of the displacement. The subsequent position, Q, is entirely 

 determined by (1) the previous position, P, (2) the direction of 

 the displacement, (3) the sense of the displacement, (4) the 

 magnitude of the displacement. 



Further it is clear that exactly the same change of position 

 is effected in moving a point from P to K by 

 the straight line PK, and from K to Q by the 

 straight line KQ, as in moving the point from 

 P to Q directly by the straight line PQ. That 

 is to say, displacements represented by lines 

 PK, KQ are equivalent to the displacement 

 represented by the line PQ. 



Fig. 3. 



Displacement is a quantity, for one dis- 

 placement can be greater, equal to, or less than 

 another; but two displacements in different 

 directions, or in different senses, are clearly not 

 equivalent to each other; and thus displacement belongs to the 

 class of mathematical quantities known as vectors or directed 



Fig. 4. 



