Example 1.2.14 (c): Multiplying geometrically
Find both geometrically and algebraically and confirm that both answers agree.
Working algebraically with rectangular coordinates we get:
To work geometrically we need to find length and angle of the vectors involved:
|1/(1+i)| = 1 and Arg(1+i) = arctan(1) = /4
since 1+i and c(1+i) have the same argument for any real number c > 0. Similarly
|1/2(1+i)| = 1 and Arg(1+i) = arctan() = /3
Thus, by our theorem about dividing geometrically, we have:
= 1/1*cis(/4-/3) = cis(-/12) = 0.9659258263 - 0.2588190451 i
Therefore, both ways give the same answer.