Problem 126 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the Argand diagram, if \(O, P\) and \(Q\) represent respectively the origin and the complex numbers \(z\) and \(z+i z\), then the \(\angle O P Q\) is (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

The angle \( \angle OPQ \) is \( \frac{\pi}{2} \).

Step 1: Identify Given Complex Numbers

The problem states that in the Argand diagram, point \( O \) is the origin, \( P \) represents the complex number \( z \), and \( Q \) represents the complex number \( z + i z \). We can express \( z \) as \( x + i y \) where \( x, y \) are real numbers.

Step 2: Determine Location of Q

Given \( z = x + iy \), the complex number \( z + iz \) becomes \( (x + iy) + i(x + iy) = x + iy + ix - y \), which simplifies to \( (x-y) + i(x+y) \). Thus, \( Q \) is located at \( (x-y, x+y) \).

Step 3: Express Vectors OP and PQ

Vector \( \overrightarrow{OP} \) is \( x + iy \) and vector \( \overrightarrow{PQ} \) is obtained by subtracting \( P \) from \( Q \), which is \( ((x-y) + i(x+y)) - (x + iy) = -y + ix \).

Step 4: Calculate Angle OPQ using Dot Product

The dot product of vectors \( \overrightarrow{OP} \) and \( \overrightarrow{PQ} \) is \(-yx + i(x^2 - y^2)\). Since these vectors are perpendicular when their dot product is zero, let's calculate the angle. The imaginary part, which does not affect the real angle calculation, is calculated by dividing the vectors into real and imaginary components: \( \overrightarrow{OP} = x \hat{i} + y \hat{j} \) and \( \overrightarrow{PQ} = -y \hat{i} + x \hat{j} \). The dot product is zero: \( x(-y) + y(x) = -xy + xy = 0 \). Thus, the angle between them is \( \frac{\pi}{2} \).

Step 5: Determine the Answer

Since the calculated angle \( \angle O P Q \) is \( \frac{\pi}{2} \), the correct answer is option (C) \( \frac{\pi}{2} \).

Key Concepts

Argand Diagram

The Argand Diagram is a powerful tool used to visualize complex numbers. Imagine it like a coordinate plane, where each complex number gets its own unique spot. This diagram helps us understand the location of complex numbers as points in the plane.

In the Argand diagram:

• The horizontal axis represents the real part of the complex number.
• The vertical axis represents the imaginary part.

For our exercise, point \( O \) is the origin, representing the complex number \( 0 + i0 \). Point \( P \) represents the complex number \( z = x + iy \). Point \( Q \) represents \( z + iz \) which simplifies to \( (x-y) + i(x+y) \). This shows how the imaginary part and real part combine to give a new location on the diagram. These geometric representations help us see transformations and combinations of complex numbers.

Vector Representation

Vector Representation is like giving directions between points on a map. In mathematics, it helps us understand points as arrow-like structures with direction and magnitude.

In the given exercise, we have:

• Vector \( \overrightarrow{OP} \), going from the origin \( O \) to point \( P \), represented as \( x + iy \).
• Vector \( \overrightarrow{PQ} \), which connects points \( P \) and \( Q \). It is represented as \( -y + ix \), obtained by the difference of points \( Q \) and \( P \).

These vector representations allow us to apply linear algebra techniques, such as the dot product, to calculate angles or check perpendicularity between vectors. They transform abstract positions on the Argand diagram into understandable, relatable line segments.

Dot Product

The Dot Product is a mathematical operation that multiplies two vectors and gives a scalar result. It is a fundamental tool in vector calculus and helps to determine the angle between two vectors.

The dot product of two vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \) is calculated by:\[\overrightarrow{a} \cdot \overrightarrow{b} = a_1b_1 + a_2b_2\]Where \( a_1, a_2 \) and \( b_1, b_2 \) are components of the vectors. In the exercise, we have:

• \( \overrightarrow{OP} \) with components \( x \) and \( y \).
• \( \overrightarrow{PQ} \) with components \( -y \) and \( x \).

The dot product is computed as \(-xy + xy = 0\), indicating that the vectors are perpendicular. This result is pivotal in determining the exact angle between the vectors.

Angle Calculation

Calculating the angle between vectors is very useful in understanding their relationship. When performing angle calculation, particularly between vectors in the Argand diagram, we often rely on the dot product.

If the dot product of two vectors is zero, it means they are perpendicular to each other. This is because:\[\cos(\theta) = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{\|\overrightarrow{a}\| \|\overrightarrow{b}\|}\]where \( \theta \) is the angle between them.

In the exercise, because the dot product of \( \overrightarrow{OP} \) and \( \overrightarrow{PQ} \) equals zero, the angle \( \angle OPQ \) is \( \frac{\pi}{2} \) radians, or 90 degrees. This demonstrates a perfect right angle, proving the perpendicular nature of the vectors on the Argand diagram.