50 TREATISE ON ALTERNATING CURRENTS. 



Thus the scalar product of the two vectors 



a 4- kb 

 and 



a' + kb' 

 is 



aa' + bb' 



38. On A as an Operator. A vector whose length 

 is a along or parallel to Ox is represented by a ; whereas a vector 

 of the same length, whose direction is parallel to Oy, is represented 

 by ka. We may thus regard k as an operator which, when it 

 operates upon a vector along Ox, has the effect of turning the 

 vector through a right angle in the positive (counter-clockwise) 

 direction of rotation without altering its length. The effect of 

 similarly operating on ka must, therefore, to be consistent, be to 

 turn the vector ka through a right angle in the positive direction 

 without altering its length ; that is, it becomes a vector of length a 

 along Ox' ; that is, it becomes a. 



We thus have 



k . ka = k*a = a 

 or 



7i^ = 1 



that is, when using the symbol k in algebraical processes, we 

 must regard it as having the properties of the imaginary \/ 1. 



If we similarly operate with k on the vector - a, we get - ka, 

 a vector of length a along Oy\ ; and operating again, we get k*a, 



or -f a, a vector of length 

 a along Ox. 



Again, operating on any 

 vector 



a + kb 

 we get 



k(a + kb) = ka + ?>' 2 b 

 = ka b 



which (see Fig. 16) is the 

 vector 'a + kb turned through 



a right angle in the positive direction of rotation, its length being 



unaltered. 



That is, if OP is any vector, then k . OP is the 



