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Geometrical Representation of Fundamental Operations

The various fundamental laws can be represented graphically. The graphical representation makes it easier to understand the laws. The four fundamental laws of geometry have been explained below:

(i) Geometrical representation of addition:

Geometrical Representation of addition of two complex numbersIf two points P and Q represent complex numbers z1 and z2 respectively in the Argand Plane, then the sum z1+z2 is represented by the extremity R of the diagonal OR of parallelogram OPRQ having OP and OQ as two adjacent sides. The addition of the complex numbers z1 and z2 can be assumed to be the addition of the vectors by utilizing the parallelogram law.

(ii) Geometrical representation of subtraction:

The representation of the difference of two complex numbers is slightly complicated than the addition of the complex numbers. The easiest way of presenting the subtraction is to think of addition of a negative vector. If we want to represent z1-z2, then the easiest way of representing it would be to think of adding a negative vector z1 + (-z2). The negative vector is the same as the positive one, the only difference being that the negative vector points in the opposite direction.
Representation of a negative vector
The difference vector z1 - z2 is represented below

Geometrical Representation of subtraction of two complex numbers
Remark: It is important to note here that the vector representing the difference of the vectors z1 - z2 may also be drawn joining the end point of z2 to the tip of z1 instead of the origin. This kind of representation does not alter the meaning or interpretation of the difference operator. The difference operator joining the tips of z1 and z2 is represented below:

Another representation of the difference operator

(iii) Modulus and argument of multiplication of two complex numbers: Modulus and argument are two extremely important concepts associated with complex numbers. 

Modulus of a complex number: The modulus of a complex number is defined to be the distance of the number from the origin. It is denoted by |z| and the value of the modulus of z = x+iy is √x2+y2.

Argument of a complex Number: Argument of a complex number is basically the angle that explains the direction of the complex number. It is measured in radians. If we have the complex number in polar form i.e. z = r(cos θ + i sin θ), then its argument is θ. 

Modulus of product of two complex numbers: Let z1 and z2 be two complex numbers in polar form

z1 = r1(cos θ1 + i sin θ1)

z2 = r2(cos θ2 + i sin θ2)

Hence their product is given by

z1z2 = r1r2 [(cos θ1 cos θ2 – sin θ1 sin θ2) + i(sin θ1 cos θ2 + cos θ1 sin θ2)].

This expression can be further simplified as

z1z2 = r1r2 [(cos (θ1 + θ2) + i sin (θ1 + θ2)].

The product z1z2 has the modulus r1r2 and the argument (θ1 + θ2).

Theorem: For any two complex numbers z1, z2

We have, |z1, z2| = |z1| |z2| and arg(z1, z2) = arg(z1) + arg(z2)

Proof: z1 = r1 eiθ1, z2 = r2 eiθ2

        z1z2 = r1 r2 ei(θ1+θ2)

        => |z1 z2|=|z1|=|z2|

arg (z1z2) = arg(z1) + arg(z2)

i.e., to multiply two complex numbers, we multiply their absolute values and add their arguments. 
View the following video for more on argument and modulus 

Some Key Points:

(i) P.V. arg(z1z2) ≠ P.V. arg(z1) + P.V. arg(z2)

(ii) |z1z2 ...... zn| = |z1| |z2| ......... |zn|

(iii) arg(z1.z2 ......... zn) = argz1+argz2+ ........ + argzn

(iv) Geometrical representation of multiplication of complex numbers -

Let P, Q be represented by z1= r1 eiθ1, z2 = r2 eiθ2 respectively. To find point R representing complex number z1 z2, we take a point L on real axis such that OL=1 and draw triangle OQR similar to triangle OLP. Therefore, 

Geometrical Representation of multiplication of complex numbersOR/OQ = OP/OL = OR = OP*OQ

i.e., OR = r1 r2 and ∠ QOR = θ,

R(z1z2) Q(z2) P(z1)

 ∠ LOR = ∠ LOP + ∠ POQ + ∠ QOR

           = Q1 + Q2 - Q1 + Q1

            = Q1 + Q2

Hence, R is represented by z1 z2 = r1 r2 ei(Q1+Q2)

(v) Modulus and argument of division of two complex numbers:

Theorem: If z1 and z2 ( ≠0 ) are two complex numbers, then |z1/z2| =

      |z1|/|z2|   and arg(z1/z2) = arg z1 - arg z2


       P.V. arg(z1/z2) ≠  P.V. arg (z1) - P.V. arg (z2)

 (vi) Geometrical representation of the division of complex numbers-

Let P, Q be represented by z1= r1 eiθ1, z2 = r2 eiθ2 respectively. To find point R representing complex number z1/z2, we tale a point L on real axis such that OL=1 and draw a triangle OPR similar to OQL.

Therefore, OP/OQ = OR/OL

                => OR = r1/r2

Representation of division of complex numbers
and ∠LOR = ∠LOP - ∠ROP

                = θ1 - θ2

Hence, R is represented by z1/z2 =r1/r2 ei(θ1 - θ2)

askIITians offers comprehensive study material which covers all the topics of IIT JEE Mathematics syllabus in detail. It is important to have a strong grip on the various algebraic operators as they lay the foundation of several other topics. The JEE aspirants must master the topic in order to remain competitive in the JEE.

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