Understanding Rotation in Complex Numbers

The concept of rotation in complex numbers is essential for JEE Main as it establishes a direct correspondence between algebraic operations and geometric transformations in the Argand plane. This article rigorously explores the rotation concepts, foundational theorems, derivations, and key examples required for a comprehensive understanding at the JEE level.

Complex Numbers in the Argand Plane

A complex number $z = x + iy$ is represented as a point $(x, y)$ in the Argand (complex) plane, where $x$ is the real part and $y$ is the imaginary part. The modulus $|z|$ gives the distance from the origin, and the argument $\arg(z)$ specifies its angle from the positive real axis.

The geometric interpretation supports algebraic operations: addition corresponds to vector addition, and multiplication encodes stretching and rotation. This duality is central to many higher-level results in [Understanding Complex Numbers](https://www.vedantu.com/jee-main/maths-complex-numbers).

Defining Rotation of a Complex Number

The rotation of a complex number involves turning its geometric representation by a certain angle about the origin. Algebraically, this is achieved by multiplying the number by a complex number of unit modulus (unimodular), commonly written as $e^{i\theta}$, which corresponds to a rotation by angle $\theta$ radians.

If $z$ is a complex number and $\alpha$ a real number, then the complex number $z' = z \cdot e^{i\alpha}$ represents the image of $z$ obtained by rotating it anti-clockwise by angle $\alpha$ about the origin. For clock-wise rotation, one uses $e^{-i\alpha}$ as the multiplier.

Algebraic Structure of Complex Rotation

Express $z$ in polar form as $z = r e^{i\theta}$, where $r = |z|$ and $\theta = \arg(z)$. Then, the rotated number is:

$$z' = z \cdot e^{i\alpha} = r e^{i(\theta + \alpha)}$$

Thus, multiplying a complex number by $e^{i\alpha}$ increases its argument by $\alpha$ but keeps its modulus unchanged.

Geometric Meaning and Properties

The transformation $z \mapsto z e^{i\alpha}$ rotates every point in the Argand plane by $\alpha$ radians about the origin. The modulus remains the same, confirming that pure rotation does not affect distance from the origin.

Multiplication by $i = e^{i\frac{\pi}{2}}$ rotates any complex number by $\dfrac{\pi}{2}$ radians (90 degrees) anti-clockwise. More generally, multiplication by $e^{in\pi/2}$ gives rotations by integral multiples of right angles.

Rotation Between Two Points: Ratio of Complex Numbers

Let $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$ be two non-zero complex numbers (not collinear with the origin). The quotient $\dfrac{z_1}{z_2}$ is given by:

$$\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$$

Here, $|\dfrac{z_1}{z_2}| = \dfrac{r_1}{r_2}$ gives the ratio of distances from the origin, and $\arg(\dfrac{z_1}{z_2}) = \theta_1 - \theta_2$ is the angle by which the vector $oz_2$ must be rotated to coincide with $oz_1$.

Rotation Theorem for Directed Segments

Given points $A(z_1)$ and $B(z_2)$, the directed segment $\overrightarrow{AB}$ is represented by $z_2 - z_1$. Let $C(z_3)$ and $D(z_4)$ define another segment $\overrightarrow{CD} = z_4 - z_3$. Then, the ratio:

$$\dfrac{z_4 - z_3}{z_2 - z_1} = \left|\dfrac{z_4 - z_3}{z_2 - z_1}\right| e^{i\theta}$$

Here, $|\cdots|$ gives the ratio of lengths, and $\theta = \arg\left(\dfrac{z_4-z_3}{z_2-z_1}\right)$ is the angle by which $\overrightarrow{AB}$ must be rotated and stretched to align with $\overrightarrow{CD}$. When only pure rotation is desired (no stretching), the modulus must be $1$.

Special Cases of Rotation

Multiplying any $z$ by a unimodular complex number ($|w|=1$) effects a pure rotation. For instance, multiplication by $i$ ($e^{i\pi/2}$) rotates by $90^\circ$ anti-clockwise. Multiplication by $-1$ ($e^{i\pi}$) rotates by $180^\circ$, and so forth.

When two complex numbers have the same modulus and their quotient is unimodular, they lie on a circle of radius $|z|$ centered at the origin, differing only by a rotation.

Stepwise Derivation: General Rotation Formula

Let $z$ be represented as $z = r e^{i\theta}$ and suppose we wish to rotate $z$ by an angle $\alpha$ about the origin. The rotated number is $z' = r e^{i(\theta + \alpha)}$.

This can also be written as $z' = z \cdot e^{i\alpha}$, reflecting the additive property of arguments in exponentials.

Alternatively, in rectangular form:

$$z' = (x + i y)(\cos\alpha + i \sin\alpha)$$

Expanding yields:

$$z' = (x\cos\alpha - y\sin\alpha) + i(x\sin\alpha + y\cos\alpha)$$

The real and imaginary parts transform as a standard planar rotation, confirming geometric intuition.

Rotation and Regular Polygons

For a regular $n$-gon centered at the origin, vertices are separated by equal angles $2\pi/n$. If $z_1$ represents one vertex, then after rotation by $2\pi/n$, the adjacent vertex is $z_2 = z_1 e^{i(2\pi/n)}$, the subsequent vertex by $z_k = z_1 e^{i 2\pi (k-1)/n}$.

This formulation is foundational for [Geometry of Complex Numbers](https://www.vedantu.com/jee-main/maths-geometry-of-complex-numbers) problems involving symmetries and rotational invariance.

Illustrative Examples

Example 1: Rotate the complex number $z = 5 + 6i$ by $90^\circ$ anti-clockwise about the origin.

To rotate by $90^\circ$, multiply by $i$:

$$(5 + 6i) \cdot i = 5i + 6i^2 = 5i + 6(-1) = -6 + 5i$$

Thus, the rotated complex number is $-6 + 5i$.

Example 2: Given vertices $z_1$, $z_2$, and $z_3$ of an isosceles right triangle with right angle at $z_3$ (i.e., at $R$), show that $(z_1 - z_2)^2 = 2(z_1 - z_3)(z_3 - z_2)$.

Since $PQ = QR$ and the angle at $R$ is $90^\circ$, the segment $z_2 - z_3$ must be a $90^\circ$ rotation of $z_1 - z_3$. Thus:

$$z_2 - z_3 = i(z_1 - z_3)$$

Squaring both sides:

$$(z_2 - z_3)^2 = i^2 (z_1 - z_3)^2 = - (z_1 - z_3)^2$$

Expanding $(z_1 - z_2)^2$ and substituting gives $(z_1 - z_2)^2 = 2(z_1 - z_3)(z_3 - z_2)$, as required.

Applications and Extended Observations

Rotation of complex numbers provides powerful tools for analyzing symmetries, geometric loci, and transformations within the complex plane. It is foundational for advanced results in [Complex Numbers and Quadratic Equations](https://www.vedantu.com/jee-main/maths-complex-numbers-and-quadratic-equations) and mathematical problem-solving strategies in coordinate geometry and trigonometric contexts.

Furthermore, the concept is central to the manipulation of complex loci, proof of results in cyclic quadrilaterals, and the geometric interpretation of complex inequalities.

Rotation and Unimodular Numbers

A unimodular complex number satisfies $|w| = 1$. Any multiplication by such a number results in pure rotation without change in magnitude. These numbers lie on the unit circle in the Argand plane and are of the form $e^{i\theta}$ for real $\theta$.

If $z_1$ and $z_2$ are complex numbers such that $|\dfrac{z_1}{z_2}|=1$, their arguments differ by a definite angle, and their magnitudes are equal.

Key Formulas and Summary Table

Conclusion: The Significance of Complex Number Rotation

The concept of rotation within the framework of complex numbers tightly couples algebraic operations with geometric intuition. Mastery of this correspondence is indispensable for mathematical reasoning in competitive exams and advanced geometry topics.

Further study on [Polar Form of Complex Numbers](https://www.vedantu.com/jee-main/maths-polar-and-eular-form-of-a-complex-number) and [Modulus of Complex Numbers](https://www.vedantu.com/jee-main/maths-complex-numbers-detailed-discussion-on-modulus) will strengthen foundational understanding and enable application of rotation methods to a wide class of problems.

Practice with [Practice Paper on Complex Numbers](https://www.vedantu.com/jee-main/maths-complex-numbers-and-quadratic-equations-practice-paper) to consolidate these principles through typical examination problems and deepen conceptual integration.