20 KEST AND MOTION 



can be replaced by three components parallel to the axes; these 

 vectors are of amount 



Of the 3n vectors thus obtained, the n vectors parallel to the 

 aj-axis can be compounded into the single vector 



X RI cos o^ + R z cos # 2 + + R n cos a n . (3) 



The whole system of vectors can thus be replaced by this vector 

 and two others parallel to the y and z axes respectively, namely 



Y=It l cos ft 4- R, cos j3 2 H ----- h R n cos /3 n . (4) 



Z = R^ cos 7 X + J2 2 cos 7 2 H ----- h R n cos 7 B . (5) 



Evidently the resultant of these three vectors, and consequently of 

 the original n vectors, is a diagonal of a rectangular parallelepiped 

 whose edges are of lengths X, F, Z. If the length of the resultant 

 be denoted by R, and the direction angles by a, /3, 7, we have 



R 2 = Z 2 + F 2 + ^ 2 , 



X. Y Z 



and cos a, = > cos ft = > cos, 7 = 



.K ^ a 



Hence the resultant is completely determined in magnitude and 

 direction. 



Centroids 



16. Let a system of vectors be represented in direction by OA l} 

 OA 2 , , OA n , and let their magnitudes be mf>A v - , m n OA n , where m lt 

 m z ,--,m n are any quantities. Denote by x r , y r , z r the coordinates of 

 A r with respect to axes through ; by a rt j3 r , j r the direction angles 

 of OA r with respect to these axes ; and by R r the magnitude of the 

 vector m r OA r . The components of this vector along these axes are 



y R r cos a r = m r OA r cos a r = 



JK r cos /3 r = m r OA r cos /3 r = 

 J2 r cos 7 r = m r OA r cos 7 r = 

 Hence equations (3), (4), (5) can be written thus : 



r y r , Z = m^ r . (6) 



