Problem 16 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) is possible if (A) \(z_{2}=\bar{z}_{1}\) (B) \(z_{2}=\frac{1}{z_{1}}\) (C) \(\arg z_{1}=\arg z_{2}\) (D) \(\left|z_{1}\right|=\left|z_{2}\right|\)

Step-by-Step Solution

Answer

Option C: \(\arg z_1 = \arg z_2\) ensures the equality holds.

Step 1: Recognize the Formula of Equality

The given equation \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) indicates that the magnitudes or moduli of the complex numbers \(z_1\) and \(z_2\) add up linearly. This is true if and only if the two complex numbers are non-negative real multiples of each other, meaning they are on the same line through the origin in the complex plane.

Step 2: Analyze Each Option

Review each option to determine which one meets the condition that the real parts of \(z_1\) and \(z_2\) are proportional and in the same direction:- **Option A**: \(z_2 = \bar{z_1}\) could potentially make \(z_1\) and \(z_2\) directions opposite unless purely real.- **Option B**: \(z_2 = \frac{1}{z_1}\) might not ensure real number multiplication as \(z_1\) and \(\frac{1}{z_1}\) would typically not be linearly related.- **Option C**: \(\arg z_{1} = \arg z_{2}\) implies the arguments are equal, thus showing they are in the same direction.- **Option D**: \(|z_1| = |z_2|\) indicates equal lengths, but does not assure directionality.

Step 3: Select the Correct Option

Option C, \(\arg z_{1} = \arg z_{2}\), ensures both the direction and proportionality necessary for \( \left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right| \). This holds true as two vectors being in the same direction implies they are positively scaled, which meets the condition required by the triangle inequality condition being an equality.

Key Concepts

Complex Plane

The complex plane is a fundamental concept in mathematics, particularly when dealing with complex numbers. Imagine a regular graph with an x-axis and y-axis, but instead, you're plotting complex numbers. Here, the x-axis represents the real part of the complex number, while the y-axis represents the imaginary part. This way of representing complex numbers allows us to see them as points or vectors in a two-dimensional plane.
A complex number such as \( z = a + bi \) can be plotted, with \( a \) as the horizontal component and \( b \) as the vertical component.

• The distance from the origin to the point representing the complex number is called the magnitude.
• The angle from the positive x-axis to the line connecting the origin to the point is called the argument.

Understanding the complex plane helps us visualize complex number addition and multiplication. When you add complex numbers, you are just adding their respective components, similar to vector addition in physics. When dealing with an equation like \( \left|z_1+z_2\right|\), it becomes easier to see why this is affected by the direction and magnitude of the complex numbers involved.

Magnitude Equality

Magnitude equality in the context of complex numbers refers to a situation where two complex numbers have the same length or distance from the origin. The magnitude (or modulus) of a complex number \( z = a + bi \) is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \).
For the equality \( \left|z_1+z_2\right| = \left|z_1\right| + \left|z_2\right| \) to hold true, specific conditions must be met. This relation means that the magnitudes add up linearly, which is a special case of the triangle inequality.

• This equality occurs only if the complex numbers \( z_1 \) and \( z_2 \) align in the same direction.
• Essentially, they should be non-negative real multiples of one another.

By ensuring that the arguments (angles) of the complex numbers are the same, you assure they're pointing the same way. This makes the line along which you measure their effect simple and clear, ensuring straightforward addition of their magnitudes.

Triangle Inequality

The triangle inequality is a crucial geometric concept that holds true in any metric space, including the complex plane. In simple terms, it states that the length of the sum of two sides of a triangle is always greater than or equal to the length of the third side. For complex numbers, this is mathematically expressed as:

• \( \left|z_1 + z_2\right| \leq \left|z_1\right| + \left|z_2\right| \)

This means the magnitude of the sum of two complex numbers can never exceed the sum of their individual magnitudes. However, when the equality \( \left|z_1 + z_2\right| = \left|z_1\right| + \left|z_2\right| \) is true, it indicates a special scenario:

• The inequality becomes an equality if and only if \( z_1 \) and \( z_2 \) are in the same direction (i.e., aligned).
• Think of the triangle as being reduced to a straight line where the two vectors are added without any angular deviation.

This straight-line scenario demonstrates the importance of direction in vector addition, as only when the vectors are perfectly aligned does the sum of their lengths become their combined magnitude.