a + bi AS AN OPERATOR 



51 



n. -w position is b cos pt, and this is the real result of the 

 operation. 



If a complex quantity a+bi operates on sin pt, we have to 

 consider two ladial lines OA and OB each inclined at an angle pt 

 to the initial vertical line. 



OA = a OD = a sin pt 



OB = b OE = b sin pt 



OD and OE being the horizontal projections of OA and OB 

 respectively. The effect of the operation is to turn the radius 



FIG. 16. 



OB through a right angle, while the position of the radius OA 

 remains unchanged. 



CjD is the horizontal projection of the radial lines after the 

 operation has been performed. 



But CjD = a sin pt + b cos pt, and this is the real result of the 

 operation. 



If a bi operates on sin pt, the radius OB is turned through 

 a right angle in clockwise direction, and C 2 D is the horizontal 

 projection of the radial lines after the operation has been per- 

 formed. 



Thus the result of operating with a bi on sin pt is 

 a sin pt b cos pt 



I a- bi a - bi , 1 



Now r-. = r-pr = Then =-., operating on 



a + bt a 2 on 2 a 2 + b 2 a+ bt 



sin pt, will give a result which can be obtained by dividing the 

 result of operating with a bi on sin pt by a 2 + b 2 . 



1 a sin pt b cos pt 

 p operating on sin pt gives ^ ^ ~ 



1 a+bi a+ bi 



a- bi = a- - b 2 i- = a 2 + b 2 



Also, since 



