If O, z1 and z2 are vertices of an equilateral triangle, then (a) |z1|=|z2|=|z1-z2| (b) (z1+z2)^

If O, z1 and z2 are vertices of an equilateral triangle,...

Key Concepts

Rotation via Complex Multiplication

Multiplying a complex number by another complex number of unit modulus effects a rotation in the complex plane. This property is instrumental in analyzing problems where geometric figures, like triangles, are rotated, since a rotation by a specific angle can be achieved by multiplying by the corresponding exponential factor (in the form e^(i?)).

Complex Plane Representation

This concept involves interpreting complex numbers as points in a two-dimensional plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric view allows one to analyze relationships and distances between complex numbers using the familiar techniques of Euclidean geometry.

Modulus and Distance

The modulus (or absolute value) of a complex number represents its distance from the origin in the complex plane. It is analogous to the length of a vector and is fundamental in determining distances between points, comparing magnitudes, and applying the triangle inequality in geometric settings.

Argument of Complex Numbers

The argument of a complex number is the angle that the line joining the point to the origin makes with the positive real axis. It is crucial for understanding directional relationships and angular differences between different complex numbers, especially when considering rotations and orientation in the plane.

Geometric Properties of Equilateral Triangles

An equilateral triangle has all sides of equal length and each interior angle equal. In the context of complex numbers, these properties translate into equal distances (moduli) between vertices and specific relationships between their arguments, often involving fixed-angle rotations. This makes the equilateral triangle a useful model for exploring the interplay between algebraic and geometric properties in the complex plane.

Transcript

00:01 I have a giving here an equal triangle with vertices or origin z1 and z2.

00:08 Here and z2 are the two complex numbers.

00:11 So first i'll check the relation here, that is mod z1 equals mod z2 equals mod z1 negative z2.

00:25 If you observe here we have a mod z1 is more z1 negative 0.

00:30 That represents this is ab.

00:33 So that represents ab.

00:39 Next, mod z2 negative 0, that represents ac.

00:49 Next is mod z1 negative z2.

00:53 So z1 negative z2 with more, that represents bc.

00:57 Now, as given that, it is an equal to triangle.

00:59 So we have ab equals ac equals bc.

01:02 So therefore we can say more z1 equals, more g1 equals, mod z1 negative 0.

01:08 So we have this relation is next.

01:14 To check here z1 plus z2 whole square equals 3 z 1 z 2 before that let's understand here the concept here when we rotate a complex number by an angle so here we have this complex reactor here we have z 1 so this is rotated anticlockwise by 60 degrees because this is equal triangle so this is pi over 3 so we have z1 is rotated 60 degrees anticoquise and we reach z2 although we can write z2 as z2 equals z1 at the part iota pi over 3 and again repeating z1 is rotating anti -clockwise by 60 degrees and we get z2 so you can write z2 equals z1 8 to the part iota pi over now we just expand this left -hand side here we have z1 plus z2 whole square so we get it as left and side will be.

02:25 So we get here z1 square plus z2 square.

02:29 Z2 square is e to part iota 2, pi over 3, plus two times z1 times z2.

02:38 Z2 is z1, e to the power aorta, pi over 3.

02:44 Now we keep on simplifying this.

02:46 So we did as z1 square factor out, we get 1 plus not 8 to the par to pi over 3 that is given as cof 2 pi over 3 plus iota sign 2 pi over 3 then here we have z1 square effect off so here we have plus 2 times aorta pi over 3 that is cost 5 over 3 plus iota 5 over 3 so we obtain that let's simplify this so right now b1 square times now 1 now, cost 2 pi over 3 is cost pi negative pi over 3, there'll be negative cospy over 3.

03:31 So negative 1 over 2 plus we get aorta, root 3 over 2 plus we get 2 times post pi over 2 that is 1 over 2.

03:44 Then we have plus 2 times iota root 3 over 2...

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