Rotation Principle: Complex Number

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 ROTATION PRINCIPLE

From the figure, 

 \arg {z_2} - \arg {z_1} = \theta

or, \arg \left( {\frac{{{z_2}}}{{{z_1}}}} \right) = \theta

Since z = |z|(\cos \theta + i\sin \theta ), where \theta  = arg.z

 Therefore, when z = \frac{{{z_2}}}{{{z_1}}},

 \Rightarrow \frac{{{z_2}}}{{{z_1}}} = \left| {\frac{{{z_2}}}{{{z_1}}}} \right|(\cos \theta + i\sin \theta ) 

 \Rightarrow \frac{{{z_2}}}{{{z_1}}} = \left| {\frac{{{z_2}}}{{{z_1}}}} \right|{e^{i\theta }}

 \Rightarrow \frac{{{z_2}}}{{{z_1}}} = \frac{{|{z_2}|}}{{|{z_1}|}}{e^{i\theta }}

 \Rightarrow \frac{{{z_2}}}{{|{z_2}|}} = \frac{{{z_1}}}{{|{z_1}|}}{e^{i\theta }}   (Important)

 unit vector along {{z_2}} = unit vector along {{z_1}} * rotation factor

 \theta  -> Anticlockwise direction

 When \theta  = 90 degree or pi/2

 \frac{{{z_2}}}{{|{z_2}|}} = \frac{{{z_1}}}{{|{z_1}|}}{e^{i\frac{\pi }{2}}}

 or, \frac{{{z_2}}}{{|{z_2}|}} = \frac{{{z_1}}}{{|{z_1}|}}(\cos \frac{\pi }{2} + i\sin \frac{\pi }{2})

 or, \frac{{{z_2}}}{{|{z_2}|}} = i\frac{{{z_1}}}{{|{z_1}|}}

 i.e. when any complex number is multiplied by 'i' it will rotate 90 degree in anticlockwise direction.
Ques: Find the Area of the triangle whose sides are z, iz and z + iz ?
Sol: Angle between z and iz is pi/2 because here the rotation factor is 'i' .

 And |z| = |iz|
 Therefore, 

 Area of Right angled triangle = 1/2 * Base * Height
                                           
                                             = \frac{1}{2}|z{|^2}  ( Answer )



Ques: From the given figure in which ABCD is a square find the relation between {z_1},{z_2} and {z_4} ?
Sol: unit vector along AD = unit vector along AB * rotating factor

  \frac{{{z_4} - {z_1}}}{{|{z_4} - {z_1}|}} = \frac{{{z_2} - {z_1}}}{{|{z_2} - {z_1}|}}{e^{i\frac{\pi }{2}}}

 or, \frac{{{z_4} - {z_1}}}{{|{z_4} - {z_1}|}} = \frac{{{z_2} - {z_1}}}{{|{z_2} - {z_1}|}}i

 Since, {|{z_2} - {z_1}| = |{z_4} - {z_1}|}  ( ABCD is a square)

 {z_4} - {z_1} = ({z_2} - {z_1})i

 or, {z_4} = {z_1} + i({z_2} - {z_1})  ( Answer )

Ques: If |z| = 1 find the location of 2z + 1 ?

 Sol : Let w = 2z + 1

 => z = (w - 1)/ 2 

 => |z| = \frac{{|w - 1|}}{2}

 or, |w - 1| = 2  ( Since |z| = 2 )

 Geometrically means the distance of 'w' from 1 is always equal to 2.

 it is an equation of circle ( see the graphical representation of complex number - click here )

 Centre at (1,0) and Radius = 2

Ques: If |z| = 2 find the location of z - 1 + i ?

Sol: Let z - 1 + i = w

 => z = w + 1 - i

 or, |z| = |w + 1 - i|

 or, |w + 1 - i| = 2

 or, |w - (-1 + i)| = 2

 it is an equation of circle having Centre at (-1,1) and Radius = 2.
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