428 PRACTICAL MATHEMATICS 



When x = 90, ON and OM become the rectangular components 

 of the given vector : 



and ON = p sin 



OM= p sm (90 -6) 

 = p cos 0. 



207. Addition and Subtraction of Vectors. In the previous para- 

 graph, since the vector OP can be replaced in effect by the vectors 

 OM and ON, it necessarily follows that the vector OP can replace 

 in effect the two vectors OM and ON. Thus OP can be taken as 

 the vector sum of OM and ON, and the sum of two vectors can be 

 obtained by making two adjacent sides of a parallelogram repre- 

 sent in every respect the two vectors, and the diagonal of the 

 parallelogram which passes through the point of intersection of 

 their lines of action will represent in every respect the sum of the 

 two vectors. 



Let A be a vector of magnitude p and direction X ; let B be 

 another vector of magnitude p. 2 and direction 2 . 



O 



FIG. 140. 



To find (A+B). 



Let OX (Fig. 140) be the reference line, and let OP X and 

 OP 2 make angles X and 2 respectively with OX. Let OP^ = /> x 

 and OP 2 = p 2 , and the parallelogram completed by drawing P 2 P 

 parallel to.OPj and P X P parallel to OP 2 . Then OP will re- 

 present the sum of the two vectors, or (A + B), 6 being the angle 

 its line of action makes with OX, and OP measured to the same 

 scale as OP X and OP 2 , the magnitude. The parallelogram law 

 can be used to find the difference of two vectors, since by altering 

 the sense of the vector which has to be subtracted the question 

 becomes one of addition of vectors. 



To /2nd (A -B). 



Let OPj and OP 2 make angles X and 2 respectively with the 

 reference line OX (Fig. 141), and let OPj = p 1 and OP 2 = /> 2 ; the 

 length OP a being now measured in the opposite direction. 



