IIT JEE , AIEEE Forum - Mathematics , Algebra

Anup

can someone please explain me the rotation of comlplex numbers and its applications in geometry.

thanks in advance

neeraj

Following explation includes the rotation of complex number and multiplication of complex number. Kindly go throuh it if you have any question you can ask me

Multiplying a complex number by i. In our goal toward finding a geometric interpretation of complex multiplication, let's consider next multiplying an arbitrary complex number z = x + yi by i.

z i = (x + yi) i = ?y + xi.

Let's interpret this statement geometrically. The point z in C is located x units to the right of the imaginary axis and y units above the real axis. The point z i is located y units to the left, and x units above. What has happened is that multiplying by i has rotated to point z 90° counterclockwise around the origin to the point z i. Stated more briefly, multiplication by i gives a 90° counterclockwise rotation about 0.

You can analyze what multiplication by ?i does in the same way. You'll find that multiplication by ?i gives a 90° clockwise rotation about 0. When we don't specify counterclockwise or clockwise when referring to rotations or angles, we'll follow the standard convention that counterclockwise is intended. Then we can say that multiplication by ?i gives a ?90° rotation about 0, or if you prefer, a 270° rotation about 0.

A geometric interpretation of multiplication. To completely justify what we're about to see, trigonometry is needed, and that is done in an optional section. For now, we'll see the results without the justification. We've seen two special cases of multiplication, one by reals which leads to scaling, the other by i which leads to rotation. The general case is a combination of scaling and rotation.

Let z and w be points in the complex plane C. Draw the lines from 0 to z, and 0 to w. The lengths of these lines are the absolute values |z| and |w|, respectively. We already know the length of the line from 0 to zw is going to be the absolute value |zw| which equals |z| |w|. (In the diagram, |z| is about 1.6, and |w| is about 2.1, so |zw| should be about 3.4. Note that the unit circle is shaded in.) What we don't know is the direction of the line from 0 to zw.

The answer is that "angles add". We'll determine the direction of the line from 0 to z by a certain angle, called the argument of z, sometimes denoted arg(z). This is the angle whose vertex is 0, the first side is the positive real axis, and the second side is the line from 0 to z. The other point w has angle arg(w). Then the product zw will have an angle which is the sum of the angles arg(z) + arg(w). (In the diagram, arg(z) is about 20°, and arg(w) is about 45°, so arg(zw) should be about 65°.)

In summary, we have two equations which determine where zw is located in C:

|zw| = |z| |w|

arg(zw) = arg(z) + arg(w)

v_gurucharan

PLS GIVE ALL THE STEPS

LET Z1,Z2 ARE COMPLEX NUMBERS REPRESENTED BY POINTS ON THE CIRCLE |Z1|=1 AND |Z2|=2, THEN:

a) max |2Z1+Z2|=4
b)|Z2+1/Z1|<4
c) min|Z1-Z2|=1
d) none of these

and also explain the representation of complex numbers in form of circles i.e. relation between a complex number and a circle and also parabola.

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