RESOLUTION OF VECTORS 



427 



acts along the line of action from O to P, then the line OP repre- 

 sents the vector completely. 



If the sense is negative, that is, the vector acts towards the point 

 of application and the direction of the arrow must be reversed, 

 then the length OP' =/> must be measured from O along OA in 

 the opposite direction. 



It is obvious that OP and OP' represent two equal and opposite 

 vectors, which, if taken together, would neutralise one another. 

 Also if OP represents a positive vector, then OP' will represent 

 the corresponding negative vector, and therefore a positive vector 

 can be made negative by simply changing its sense ; this is very 

 important, and has to be used in all cases of subtraction of vectors. 



206. Resolution of Vectors. A vector can be resolved along any 

 two assigned directions ; that is, it can be replaced in effect by two 

 vectors, the first of which acts along one of these directions, and 

 the second along the other. 



M 



FIG. 139. 



Thus if OX and OY (Fig. 139) are the two given directions, the 

 angle XOY = a, and OP = p is the given vector whose line of 

 action makes an angle. with OX, then by making OP the diago- 

 nal of the parallelogram ONPM, ON represents the resolved part 

 of the vector in the direction OY, and OM represents the resolved 

 part in the direction OX. It is obvious that ON and OM should 

 be measured to the same scale as OP. 



Working with the triangle PMO and applying the sine rule. 

 OP PM OM 



Hence 



also 



sin (180 - a) sin sin (a -0) 



OP sin 



PM = ON -= 



OM = 



sin (180 - a) 

 p sin 

 sin a 

 OP sin (a - 0) 



sin a 



p sin (a 0) 

 sin a 



