Question

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

Answer 1

Given for complex number z₁ and z_{2
$$\begin{gathered} \left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right| \\ \mathrm{i}.\mathrm{e}{\left|z_1+z_2\right|}^2={\left(\left|z_1\right|+\left|z_2\right|\right)}^2 \\ \mathrm{i}.\mathrm{e}{\left|z_1\right|}^2+{\left|z_2\right|}^2+2\left|z_1\right|\left|z_2\right|\cos \theta \\ ={\left|z_1\right|}^2+{\left|z_2\right|}^2+2\left|z_1\right|\left|z_2\right|\left(\because z_1.z_2=\left|z_1\right|\left|z_2\right|\cos \theta \mathrm{where}\theta \mathrm{is}\mathrm{angle}\mathrm{between}{z}_1\mathrm{and}{z}_2\right) \\ \mathrm{i}.\mathrm{e}2\left|z_1\right|\left|z_2\right|\cos \theta =2\left|z_1\right|\left|z_2\right| \\ \mathrm{i}.\mathrm{e}\cos \theta =1 \\ \mathrm{i}.\mathrm{e}\theta =0 \end{gathered}$$}
i.e angle between z₁ and z₂ is 0
i.e arg (z₁) – arg z₂ = 0