TKEATISE ON ALTERNATING CURRENTS. 



34. Composition of Vectors. The sum of two vectors 

 OP and PQ (Fig. 12) is defined to be the vector OQ. The usual 

 meaning of the word sum is here extended, and if we write 



OP + PQ = OQ 



we should read OP together with PQ are equivalent to OQ. The 

 addition of vectors is thus the same as the combination of forces, 



for if OP and PQ represent 

 two forces, then OQ represents 

 their resultant ; in fact, as we 

 have already stated, a force is 

 a vector quantity. 



The difference of two vec- 

 tors OP and PQ is defined to 

 be the sum of the two vectors 

 OP and QP, and is, therefore, 

 the vector OQ (Fig. 12) where 

 PQ = PQ, that is PQ' equals 

 PQ in magnitude, but is drawn 

 in the opposite sense. It may 

 be well to emphasize here that 

 the conditions of equality of 



two vectors in no way lixes the vectors in space : they must merely 

 be of equal length parallel in direction and of the same sense ; so 

 long as these conditions are satisfied, the actual positions of the 

 vectors in space is immaterial. 



35. Algebraic Expression for a Vector Quan- 

 tity. Let OP (Fig. 13) represent any vector quantity. 



Through draw any two mutually perpendicular lines xOx', 

 yOy'. Draw PN perpendicular to xOx' ; the two vectors, ON and 

 NP t are then together equal to the vector OP. Any vector can 

 thus be resolved into two component vectors parallel respectively 

 to xOx' and yOy'. 



Now let us agree to represent unit vector along Ox by + 1 ; 

 unit vector along Ox' will then be represented by 1, since the 

 sense is exactly opposite. Let us further represent unit vector 

 along Oy by &; then unit vector along Oy' will be represented by /:. 



If, then, ON contains a units of length, and NP contains I 

 units, the vector OP is given by 



OP = ON + NP 

 = a 4- kb 



FIG. 12. 



