If z1 and z2 are two non-zero complex numbers such that |z1+z2|=|z1|+|z2| then z1-z2 is equal to

step 1 We are given that z1 plus z2 mod is equal to mod z1 plus mod z2 Mod z2 now if we square both the

If z1 and z2 are two non-zero complex numbers such that...

If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$ then $\arg z_{1}-\arg z_{2}$ is equal to
[2005]
(A) $\frac{\pi}{2}$
(B) $-\pi$
(C) 0
(D) $-\frac{\pi}{2}$

Key Concepts

Equality Condition in the Triangle Inequality

The equality condition in the triangle inequality occurs if and only if the two complex numbers have the same or exactly opposite direction in the complex plane. In contexts where the equality |z1 + z2| = |z1| + |z2| holds, it implies that the complex numbers are aligned in the same direction, leading to specific conclusions about the difference of their arguments.

Argument of a Complex Number

The argument of a complex number is the angle that the line connecting the origin to the complex number makes with the positive real axis. It provides information about the direction of the complex number in the plane and is essential when expressing the complex number in polar form.

Triangle Inequality

The triangle inequality states that for any two complex numbers, the modulus of their sum is less than or equal to the sum of their individual moduli. This inequality reflects the geometric idea that the shortest distance between two points is a straight line, and it helps in understanding conditions under which equality holds.

Complex Numbers

Complex numbers extend the real numbers by incorporating the imaginary unit, often represented in the form a + bi. They allow for a geometrical interpretation in the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Modulus (Magnitude) of a Complex Number

The modulus of a complex number is a measure of its magnitude or distance from the origin in the complex plane. It is calculated as the square root of the sum of the squares of its real and imaginary parts, and plays a crucial role in understanding the size and relative distances between complex numbers.

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