Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}, z_{2}, z_{3}, z_{4}\) are the four complex numbers represented by the vertices of a quadrilateral taken in order such that \(z_{1}-z_{4}=z_{2}-z_{3}\) and amp \(\frac{z_{4}-z_{1}}{z_{2}-z_{1}}=\frac{\pi}{2}\) then the quadrilateral is a (A) square (B) rhombus (C) rectangle (D) a cyclic quadrilateral

Step-by-Step Solution

Answer

The quadrilateral is a rectangle (C).

Step 1: Understanding the Given Conditions

The problem gives us two conditions: 1. The complex number difference condition: \(z_1 - z_4 = z_2 - z_3\) which means the vectors representing these complex numbers are equal.2. The amplitude condition: \(\text{amp}\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \frac{\pi}{2}\). This implies the angle between the vectors \(z_4 - z_1\) and \(z_2 - z_1\) is 90 degrees.

Step 2: Interpreting the Complex Difference

Given the condition \(z_1 - z_4 = z_2 - z_3\), we interpret this to mean that the vectors from \(z_4\) to \(z_1\) and from \(z_3\) to \(z_2\) are equal in magnitude and direction. This implies parallelogram properties because opposite sides in a parallelogram are equal.

Step 3: Understanding Vector Angles

The amplitude condition \(\text{amp}\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \frac{\pi}{2}\) tells us that the vectors \(z_4 - z_1\) and \(z_2 - z_1\) are perpendicular. This implies that the angle between two consecutive sides is 90 degrees, a property of rectangles and squares.

Step 4: Combining Conditions: Parallelogram with Right Angles

Since we have both the condition of a parallelogram (from equal opposite sides) and a 90-degree angle between consecutive sides (perpendicular vectors), this shape is more specifically a rectangle, because all rectangles meet these criteria.

Key Concepts

Complex Vectors

Complex vectors are a fascinating aspect of complex numbers that can be visualized on the complex plane. In this context, each complex number can be considered a point or a vector originating from the origin. The expression \( z = a + bi \) can be represented as a vector \((a, b)\) in a 2D coordinate system, where \( a \) is the real part and \( b \) is the imaginary part.
When we subtract two complex numbers, for example, \( z_1 - z_2 \), we are essentially finding the vector that points from \( z_2 \) to \( z_1 \). This operation is analogous to finding the difference between two vectors in geometry.

In this exercise, one of the conditions given is \( z_1 - z_4 = z_2 - z_3 \). This means the vector pointing from \( z_4 \) to \( z_1 \) is equal to the vector pointing from \( z_3 \) to \( z_2 \). Such an equality of vectors simplifies some geometric problems as it reflects the idea of congruent vectors, which have the same magnitude and direction.

• This concept heavily relies on vector arithmetic.
• It helps to visualize and solve problems involving geometric shapes using algebraic operations of complex numbers.

Parallelograms

A parallelogram is a four-sided shape where opposite sides are parallel and equal in length. In the language of vectors, if we consider vectors forming the sides of a quadrilateral, it becomes a parallelogram if the pairs of opposite vectors are equal.
From the given problem, the condition \( z_1 - z_4 = z_2 - z_3 \) directly hints that the quadrilateral can be a parallelogram. This is because opposite sides in a parallelogram are represented by equal vectors.

Parallelograms possess several useful properties:

• Opposite sides are equal and parallel.
• Opposite angles are equal, and adjacent angles are supplementary.
• The diagonals bisect each other.

These properties not only apply to traditional geometric shapes but also easily extend to scenarios involving complex vectors, as shown in this exercise.

Angles in Geometry

Angles are a fundamental part of geometry, creating the rules that determine the shapes and their properties. In the context of complex numbers and vectors, calculating the angle between two vectors can be achieved through the usage of complex conjugates and amplitudes.
An important concept here is orthogonality; vectors are orthogonal (or perpendicular) if the angle between them is \( \frac{\pi}{2} \) radians or 90 degrees. This is captured in complex terms using the amplitude notation.

For example, in our exercise, the condition \( \text{amp}\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \frac{\pi}{2} \) implies that the vectors \( z_4 - z_1 \) and \( z_2 - z_1 \) are perpendicular. This immediately directs us to characteristics of geometrical shapes like rectangles or squares, which have right angles between their sides.
Understanding angles helps to:

• Determine the relationships between sides of polygons.
• Classify shapes based on angle properties.
• Ensure dimensional accuracy in shapes when using vector algebra.

This foundational concept of angles is crucial when exploring complex shapes in geometry, including quadrilaterals formed by complex numbers.