Problem 87 - A Complete Resource Book in Mathematics for JEE Main

Question

If for the complex numbers \(z_{1}\) and \(z_{2},\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|\), then \(\operatorname{amp} z_{1} \sim a m p z_{2}=\) (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{4}\) (D) None of these

Step-by-Step Solution

Answer

The amplitude difference is \(\frac{\pi}{2}\).

Step 1: Identify the Given Condition

We are given that the magnitude of the sum of two complex numbers is equal to the magnitude of their difference, i.e., \( |z_1 + z_2| = |z_1 - z_2| \).

Step 2: Geometric Interpretation of Condition

The condition \( |z_1 + z_2| = |z_1 - z_2| \) geometrically means that \( z_1 + z_2 \) and \( z_1 - z_2 \) are equidistant from the origin on the complex plane.

Step 3: Vector Analysis of Condition

Consider vectors \( z_1 \) and \( z_2 \). When \( z_1 + z_2 \) and \( z_1 - z_2 \) have equal magnitudes, \( z_2 \) must be perpendicular to \( z_1 \).

Step 4: Calculate the Argument

If \( z_2 \) is perpendicular to \( z_1 \), the angle between them, or the phase difference, is \( \frac{\pi}{2} \). This corresponds to a right angle.

Key Concepts

Magnitude of Complex Numbers

The magnitude of a complex number is often visualized as the distance from the origin to the point representing the complex number on the complex plane. This is similar to finding the length of a vector in 2D space. For a complex number in the form of \( z = a + bi \), its magnitude is calculated using the formula:\[|z| = \sqrt{a^2 + b^2}\]This equation computes the Euclidean distance of the point \((a, b)\) from the origin \((0, 0)\).
The magnitude is always a non-negative real number and gives an idea of how 'large' the complex number is, although this 'size' is not the same as a numerical value.
Knowing about the magnitude helps in many mathematical operations, such as dividing complex numbers or determining convergence in sequences of complex numbers.

Geometric Interpretation of Complex Numbers

Complex numbers can be interpreted geometrically as points or vectors in the complex plane, a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part of the complex number.
This visualization helps to understand operations with complex numbers, such as addition, subtraction, and the magnitude mentioned above.

• Addition: When you add two complex numbers, the result can be represented by translating the vector for one number by the vector for the other.
• Subtraction: This can be visualized as finding a vector from the endpoint of one number to the endpoint of the other.
• Magnitude: It is the length of the vector from the origin to the point or tip of the vector.

Understanding this geometric view also aids in easily grasping concepts like rotation and reflection of complex numbers, as well as their multiplication and division.

Vectors in Complex Plane

When we discuss complex numbers as vectors in the complex plane, we are referring to interpreting these numbers as arrows that have both direction and magnitude.
Each complex number \( z = a + bi \) can be thought of as a vector from the origin \((0, 0)\) to the point \((a, b)\). This vector representation is particularly useful in understanding operations like rotation.One key idea is that when you have two complex numbers, their vector addition forms a parallelogram. If the condition \(|z_1 + z_2| = |z_1 - z_2|\) holds, then the vectors \(z_1\) and \(z_2\) can be thought to form an "L-shape" or right angle.
This means that the angle between these vectors is \(\pi/2\) radians. With this understanding, it's evident that magnitude and directions are vital in visualizing complex number behavior on the plane, providing both algebraic and geometric perspectives.