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

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
(A) $-\pi$
(B) $-\frac{\pi}{2}$
(C) $\pi$
(D) $\frac{\pi}{2}$

Key Concepts

Triangle Inequality in Complex Numbers

The triangle inequality for complex numbers states that for any complex numbers z1 and z2, the modulus of their sum is less than or equal to the sum of their moduli: |z1 + z2| ? |z1| + |z2|. This property is analogous to the triangle inequality in Euclidean geometry.

Argument of a Complex Number

The argument of a complex number is the angle formed with the positive real axis by the line segment joining the origin to the point representing the complex number in the complex plane. It is denoted by arg z and is typically measured in radians.

Condition for Equality in the Triangle Inequality

Equality in the triangle inequality, |z1 + z2| = |z1| + |z2|, holds if and only if the two complex numbers have the same direction in the complex plane, meaning that their arguments are the same (or differ by an integer multiple of 2?). This situation implies that the numbers are positive real multiples of each other.

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the square root of -1. They can be represented in the complex plane with a as the horizontal (real) component and b as the vertical (imaginary) component, which is essential for discussing notions such as modulus and argument.

Modulus of a Complex Number

The modulus (or absolute value) of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It is denoted by |z| and calculated using the formula ?(a² + b²) for a complex number z = a + bi.

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