Question

If \(z_{1}, z_{2}\) are complex numbers and \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) then (a) \(\arg \left(\mathrm{z}_{1}\right)+\arg \left(\mathrm{z}_{2}\right)=0\) (b) \(\arg \left(\mathrm{z}_{1} \mathrm{z}_{2}\right)=0\) (c) \(\arg \left(\mathrm{z}_{1}\right)=\arg \left(\mathrm{z}_{2}\right)\) (d) None of these

Short answer

The correct option is (c), which states that \(\arg \left(\mathrm{z}_{1}\right)=\arg \left(\mathrm{z}_{2}\right)\). This implies that the two complex numbers lie along the same line in the complex plane originating from the origin, satisfying the given condition \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\).

Step-by-step solution

Interpret the Given Information

Given the relation \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\), this is true if and only if \( z_1 \) and \( z_2 \) are non-negative real numbers or are complex numbers that lie on the same line in the Argand plane.

Analyze Options

(a) Option (a) states that the sum of the arguments of the two complex numbers is equal to 0 (\(\arg (z_{1}) + \arg (z_{2}) = 0\)). This isn't necessarily true because it implies that the arguments of \( z_1 \) and \( z_2 \) must be additive inverses of each other, which is not always the case for complex numbers that lie on the same line. (b) Option (b) says that the argument of the product of the two complex numbers is equal to 0 (\(\arg (z_{1}z_{2}) = 0\)). This indicates that the product of \( z_1 \) and \( z_2 \) is a positive real number. However, this statement is only correct if both \( z_1 \) and \( z_2 \) are positive real numbers, which would mean they lie on the positive real axis. This is possible but not always the case. (c) Option (c) states that the arguments of the two complex numbers are equal (\(\arg (z_{1}) = \arg (z_{2})\). It implies that the two complex numbers lie on the same ray originating from the origin. This condition is true for all pairs of complex numbers satisfying the given relation and is thus, the correct option. (d) Option (d) says 'None of these'. It would be applicable only if none of the previous three options were correct.

Conclusion

From the evaluation of options, it is established that option (c) is the only correct choice. Therefore, if \(z_{1}, z_{2}\) are complex numbers and \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\), then the conditions imply that \(\arg \left(\mathrm{z}_{1}\right)=\arg \left(\mathrm{z}_{2}\right)\), meaning that the two complex numbers lie along the same line in the complex plane originating from the origin. The other options are too restrictive or incorrect, so option (c) is the answer.

Key concepts

Argand Plane

The Argand Plane is like a flat map where every complex number can be plotted. In mathematics, the horizontal axis is called the real axis, and the vertical axis is the imaginary axis. Imagine placing any complex number, say \( z = a + bi \), on this plane:

• The real part \( a \) corresponds to how far you move along the horizontal axis.
• The imaginary part \( b \) determines your position along the vertical axis.

This helps in visualizing how complex numbers interact.
For instance, if two complex numbers lie on the same line in the Argand Plane, they're either on the same line from the origin or on the same straight path.
This idea is crucial for understanding the solution to our exercise, where the condition \( \left|z_{1}+z_{2}\right| = \left|z_{1}\right| + \left|z_{2}\right| \) implies they are dependent in a linear manner. Being on the same line means their arguments, which define their direction relative to the positive real axis, are equal.

Arguments of Complex Numbers

When we talk about the argument of a complex number, we're referring to the angle it forms with the positive real axis.
Consider a complex number \( z = a + bi \), the argument \( \text{arg}(z) \) is the angle \( \theta \) measured counterclockwise from the positive real axis to the line representing the complex number.

• For a number on the positive real axis, the argument is 0.
• If it's on the negative real axis, the argument is \( \pi \) (or \(-\pi\)).
• Numbers above the axis have positive arguments and those below have negative arguments.

In our exercise, the condition where \( \left|z_{1} + z_{2}\right| = \left|z_{1}\right| + \left|z_{2}\right| \) implies that \( \text{arg}(z_1) = \text{arg}(z_2) \). This means that the two numbers not only lie on a straight line from the origin but also have precise angles that match perfectly, reinforcing their straight-line relationship.

Triangle Inequality

The triangle inequality is a fundamental concept in both geometry and complex numbers. For complex numbers \( z_1 \) and \( z_2 \), it states:

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

This principle mirrors the rule in geometry: the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side.
If the equality holds, as in \( \left| z_1 + z_2 \right| = \left| z_1 \right| + \left| z_2 \right| \), both vectors representing the complex numbers must be collinear and in the same direction, forming a straight line rather than a triangle. This situation allowed us to conclude about the arguments of the complex numbers in the original exercise.
So, it is this equality condition that gave us the solution stating \( \text{arg}(z_1) = \text{arg}(z_2) \), which meant they share the same direction, fundamentally altering the typical triangle shape into a single, unfolded line.