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202212311329 Problem 1.2

Show that |z_1+z_2|=|z_1|+|z_2| iff z_1 and z_2 lie on a common ray from 0 iff one of z_1 or z_2 is a nonnegative multiple of the other.

Extracted from R. B. Ash & W. P. Novinger. (2004). Complex Variables.

Roughwork.

Let z_1=r_1e^{i\theta_{1}} and z_2=r_2e^{i\theta_{2}}, rewrite

\begin{aligned} |r_1e^{i\theta_{1}}+r_2e^{i\theta_{2}}| & = |r_1e^{i\theta_{1}}|+|r_2e^{i\theta_{2}}| \\ ? = r_3 & = r_1 + r_2 \\ \end{aligned}

by Lemma.

\begin{aligned} r_3e^{i\theta_{3}} & = r_1e^{i\theta_{1}}+r_2e^{i\theta_{2}} \\ & = r_1(\cos\theta_1+i\sin\theta_1) + r_2(\cos\theta_2+i\sin\theta_2) \\ & = (r_1\cos\theta_1+r_2\cos\theta_2) + i(r_1\sin\theta_1+r_2\sin\theta_2) \\ r_3^2 & = r_1^2+r_2^2+2r_1r_2\cos (\theta_1-\theta_2) \\ r_3 & = \sqrt{r_1^2+r_2^2+2r_1r_2\cos (\theta_1-\theta_2)} \\ \end{aligned}

we have

\begin{aligned} \cos (\theta_1-\theta_2) & = 1 \\ \theta_1 - \theta_2 & = 0 \\ \theta_1 & = \theta_2 \\ \end{aligned}

The rest is left an exercise for the reader.