'.8] 



POLYGON OF VECTORS. 



If AC, A'C' are 



Fig. 6. 



8. Examples of Equivalent Vectors. 



equal and parallel lines, 

 their ends can be joined 

 by two lines A A', CO' which 

 are equal and parallel; then 

 the vectors represented by 

 AC, A'C' are equivalent; 

 vectors represented by AC, 

 C'A are not equivalent. 



Again if A, B, C are any 

 three points, and a parallelogram 

 A, J5, C, D is constructed having 

 AB, BC as adjacent sides, AD and 

 BC are equivalent vectors. Also 

 the vector AC is equivalent to the 

 vectors AB, BC, or AD, DC, or 



AB, AD. 



t 

 Further if a polygon (plane or gauche) is constructed, having 



AC as one side, and having any points 

 P, Q, . . . T as corners, the vector repre- 

 sented by A C is equivalent to the vectors 

 represented by AP, PQ, ... TC. This is 

 clear because by definition the vectors 

 AP, PQ can be replaced by AQ, and so 

 on. The statement is independent of 

 the number of sides of the polygon, and 

 of the order in which its corners are 

 taken, no corner being taken more than 

 once, provided the points A, C are re- 

 garded as the first and last corners. [The 

 restriction that no corner is to be taken Fig. 7. 



more than once will be presently removed.] 



In particular if the polygon is a gauche quadrilateral ABDC a 

 parallelepiped can be constructed having its edges parallel to AB, 

 BD, DC, and having AC as one diagonal. Then the vector AC is 

 equivalent to the vectors represented by the edges AB, AP, AQ 

 which meet in A. (See Fig. 8.) 



The case of this which is generally most useful is the case 



