Rotation Theorem.
Rotational theorem i.e., angle between two intersecting lines. This is also known as coni method.
Let z 1 , z 2 and z 3 be the affixes of three points A, B and C respectively taken on argand plane.
Then we have AC  z 3  z 1 and AB = z 2  z 1
Y
C(z3)
and let arg AC  arg (z 3  z 1 )   and AB  arg (z 2  z 1 )  

Let CAB   ,  CAB      
B(z2)
A(z1)
 z  z1 

= arg AC  arg AB = arg (z 3  z 1 )  arg (z 2  z 1 ) = arg  3
z

z
2
1




X
O
 affix of C  affix of A 

angle between AC and AB = arg 
 affix of B  affix of A 
or
For any complex number z we have z | z | e i(argz )
 z  z1 
 
Similarly,  3
 z 2  z1 

z 3  z1 
 z 3  z 1  i  a rg z 2  z1 
z  z 1 | z 3  z 1 | i(CAB ) AC i

 e

e

e
or 3
z 2  z 1 | z 2  z1 |
AB
 z 2  z1 
Note: Here only principal values of the arguments are considered.
 z1  z 2 
 z  z2 
z  z2
   , if AB coincides with CD, then arg  1
  0 or   , so that 1
is real. It
z3  z4
 z3  z4 
 z3  z4 
arg 
follows that if
z1  z 2
is real, then the points A, B, C, D are collinear.
z3  z4
 z1  z 2 
   / 2 ,
 z3  z4 
If AB is perpendicular to CD, then arg 
so
z1  z 2
is
z3  z4
purely imaginary. It follows that if z 1  z 2 =  k z 3  z 4  , where k purely imaginary
D
P(z1)
S(z4)
A

R(z3)
C
Q(z2)
number, then AB and CD are perpendicular to each other.
(1) Complex number as a rotating arrow in the argand plane: Let z  rcos   i sin    re i
..…(i) r. e i be a complex number representing a point P in the argand plane.
Then OP | z |  r and POX  
Now consider complex number z1  ze i
or z1  re i .e i  re i  
{from (i)}
Q(zei)
Y
X'


O
Y'
P(z)
X
B
Clearly the complex number z1 represents a point Q in the argand plane, when OQ  r and
QOX     .
Clearly multiplication of z with e i rotates the vector OP through angle  in anticlockwise
sense.
Similarly multiplication of z with e  i will rotate the vector OP in clockwise sense.
Note: If z 1 , z 2 and z 3 are the affixes of the points A,B and C such that AC  AB and CAB   .
Therefore, AB  z 2  z1 , AC  z 3  z1 .
C(z3)
Then AC will be obtained by rotating AB through an angle  in anticlockwise
B(z2)

sense, and therefore,
AC  AB e i or (z 3  z1 )  (z 2  z1 )e i or
A(z1)
z 3  z1
 e i
z 2  z1
If A, B and C are three points in argand plane such that AC  AB and CAB   then use the
rotation about A to find e i , but if AC  AB use coni method.
Let z1 and z 2 be two complex numbers represented by point P and Q in the argand plane such
that  POQ   . Then, z1e i is a vector of magnitude | z 1 |  OP along OQ and
vector along OQ . Consequently, | z 2 | .
z2 
z1 e i
is a unit
| z1 |
z1 e i
is a vector of magnitude | z 2 |  OQ along OQ i.e.,
| z1 |
| z2 |
z
.z1e i  z 2  2 .
| z1 |
z1
(2) Condition for four points to be noncyclic:If points A,B,C and D are concyclic
 ABD   ACD
Using rotation theorem
(z  z 3 ) z 4  z 3 i
In  ACD 1

e
z1  z 3
z4  z3
From (i) and (ii)
D(z4)
A(z1)
(z  z 2 ) z 4  z 2 i
In  ABD 1

e
z1  z 2
z4  z2
.....(i)

.....(ii)
B(z2)

C(z3)
(z 1  z 2 ) (z 4  z 3 ) (z 1  z 2 ) (z 4  z 3 )

=Real
z1  z 3 z 4  z 2
(z 1  z 3 ) (z 4  z 2 )
So if z 1 , z 2 , z 3 and z 4 are such that
(z 1  z 2 ) (z 4  z 3 )
is real, then these four points are concyclic.
(z 1  z 3 ) (z 4  z 2 )