# Solve the following

Question:

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg (z1) – arg (z2) = ____________.

Solution:

Given for complex number $z_{1}$ and $z_{2}$

$\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$

i. e $\left|z_{1}+z_{2}\right|^{2}=\left(\left|z_{1}\right|+\left|z_{2}\right|\right)^{2}$

i. e $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+2\left|z_{1}\right|\left|z_{2}\right| \cos \theta$

$=\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+2\left|z_{1}\right|\left|z_{2}\right| \quad\left(\because z_{1} \cdot z_{2}=\left|z_{1}\right|\left|z_{2}\right| \cos \theta\right.$ where $\theta$ is angle between $z_{1}$ and $\left.z_{2}\right)$

i. e $2\left|z_{1}\right|\left|z_{2}\right| \cos \theta=2\left|z_{1}\right|\left|z_{2}\right|$

i. e $\cos \theta=1$

i. e $\theta=0$

i.e angle between $z_{1}$ and $z_{2}$ is 0

i.e $\arg \left(z_{1}\right)-\arg z_{2}=0$