Question

Let \(z_{1}\) and \(z_{2}\) be \(n^{\text {th }}\) roots of unity which subtends a right angle at the origin then 'n' must be of the form - (1) \(4 \mathrm{~K}+1, \mathrm{~K} \in \mathrm{I}\) (2) \(4 \mathrm{~K}+2, \mathrm{~K} \in \mathrm{I}\) (3) \(4 \mathrm{~K}+3, \mathrm{~K} \in \mathrm{I}\) (4) \(4 \mathrm{~K}, \mathrm{~K} \in \mathrm{I}\) (5) None of these

Short answer

Option (4) \(4K, K \in I\).

Step-by-step solution

Understand the roots of unity

The roots of unity are the complex numbers that satisfy the equation \(z^n = 1\). These can be expressed in the form \(z_k = e^{2\pi ik/n}\) for \(k = 0, 1, 2, \ldots, n-1\).

Calculate the angle between roots

Since \(z_1\) and \(z_2\) subtend a right angle at the origin, the angle between them should be \(\frac{\pi}{2}\) radians. The angle between any two \(n^{th}\) roots of unity can be expressed as \(\theta = \frac{2\pi (k_2 - k_1)}{n}\) where \(k_1, k_2\) are integers between 0 and n-1.

Set up the equation

Given that the angle between \(z_1\) and \(z_2\) is \(\frac{\pi}{2}\), set up the equation \(\frac{2\pi (k_2 - k_1)}{n} = \frac{\pi}{2}\).

Solve for n

Simplify \(\frac{2\pi (k_2 - k_1)}{n} = \frac{\pi}{2}\) to find that \(\frac{2 (k_2 - k_1)}{n} = \frac{1}{2}\). Multiplying both sides by 2n, we get \(4 (k_2 - k_1) = n\). Thus, \(n\) must be a multiple of 4, i.e., \(n = 4K\) for some integer \(K\).

Identify the form of n

From the options given, the correct form is \(4K, K \in I\). Hence the answer is option (4).

Key concepts

Complex Numbers

Complex numbers extend the idea of one-dimensional number lines (real numbers) into a plane consisting of two dimensions. They are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by \(i^2 = -1\).
A significant application of complex numbers is in the concept of roots of unity. The roots of unity are solutions to the equation \(z^n = 1\), where \(n\) is a positive integer. These roots can be expressed in the exponential form using Euler's formula: \(z_k = e^{2\pi ik/n}\) for \(k = 0, 1, ..., n-1\).
This means that the roots of unity are evenly spaced on the complex plane, forming a regular polygon centered at the origin with equal angles between consecutive roots.

Angle Calculation

In the complex plane, the angle between two points, also known as the argument, is crucial in understanding the behavior of complex numbers and their applications.
The angle between two \(n^{\text{th}}\) roots of unity can be determined by the difference in their arguments. If \(z_1 = e^{2\pi ik_1/n}\) and \(z_2 = e^{2\pi ik_2/n}\), the angle \(\theta\) between these two roots is given by:
\( \theta = \frac{2\pi (k_2 - k_1)}{n} \)
Given that \(z_1\) and \(z_2\) subtend a right angle at the origin, this angle \(\theta\) must be \(\frac{\pi}{2}\) radians. Setting up the equation, we get:
\( \frac{2\pi (k_2 - k_1)}{n} = \frac{\pi}{2} \)
Solving this equation involves simplifying and determining the value of \(n\) based on the problem's conditions.

Right Angle Subtension

A right-angle subtension in the context of roots of unity means that the vectors representing two roots form a 90-degree angle with each other at the origin.
Mathematically, if two complex numbers \(z_1\) and \(z_2\) subtend a right angle at the origin, then the angle between them must be \(\frac{\pi}{2}\) radians.
From the angle calculation section, we have derived that:
\( \frac{2 (k_2 - k_1)}{n} = \frac{1}{2} \)
Simplifying this further:
\( 4(k_2 - k_1) = n \)
Hence, \(n\) must be a multiple of 4. This means \(n = 4K\) where \(K\) is an integer. This result is derived from the geometrical properties of complex numbers and the periodic nature of exponential functions used to represent the roots of unity.