VECTOES IN A PLANE 17 



R sin e parallel to these axes, where e is the angle which R 

 makes with Ox. The components R cos e, R sin e are spoken of 

 as the components of R along Ox and Oy. 



There are two ways of compounding a number of vectors R l} 

 R 2 , -, R n . In the first place, we can construct a polygon 

 ABODE - N, such that the sides AB, BC, CD, , MN repre- 

 sent the vectors R I} R z , R z , :, R n . Then AN will represent the 

 resultant. For R I} R z can first be compounded into a vector R 1 

 represented by A C. Combining R 9 

 with this vector, we obtain a vec- 

 tor represented by AD, and so on 

 until finally AN is reached. 



As a second way, we can resolve 

 each vector,, such as R s , into its 

 two components 



R s cos e s , R s sin. s , 



along rectangular axes Ox, Oy. 



The n vectors are now resolved 



into 2 n vectors, of which n are parallel to Ox and n are parallel 



to Oy. The first set of n can be compounded into a single vector 



X = R l cos e t -f R 2 cos e 2 -f 



parallel to Ox, while the second set can be compounded into a 

 single vector Y = R^ sin l + R 2 sin e 2 + 



parallel to Oy. We now have two vectors X, Y parallel to Ox, Oy. 

 If their resultant is a vector R making an angle e with Ox, we 



have ^cose = X=^ 1 cos 1 + ^ 2 cos 2 + .... (1) 



R sin e = Y = R 1 sin e 1 + R 2 sin e 2 + . (2) 



To find the numerical value of R, we square and add (1) and 

 (2) and obtain 

 R 2 = X 2 + Y 2 



= (R l cos e l 4- R 2 cos e 2 H ) 2 + (R t sin Cj -f- R z sin e 2 -| ) 2 



= R\ + Rl-\ h 2^ 1 ^ 2 (cose 1 cose 2 + sin e x sin e a ) H 



= JBJ + R\ + ... + 2 R^R 2 cos ( l - e 2 ) + .... 



