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202212311359 Problem 1.3

Let z_1 and z_2 be nonzero complex numbers, and let \theta (0\leqslant \theta\leqslant \pi) be the angle between them. Show that

(a) \mathrm{Re}z_1\overline{z}_2=|z_1||z_2|\cos\theta, \mathrm{Im}z_1\overline{z}_2=\pm |z_1||z_2|\sin\theta, and consequently
(b) The area of the triangle formed by z_1, z_2, and z_2-z_1 is |\mathrm{Im}z_1\overline{z}_2|/2.

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

Roughwork.

The area of a triangle \triangle ABC, constructed by any two sides \mathbf{AB} and \mathbf{AC} with an included angle \theta, is

\begin{aligned} \textrm{Area} & = \frac{1}{2}\mathbf{AB}\times \mathbf{AC} \\ \bigg( & =\frac{1}{2}(AB)(AC)\sin\theta\bigg) \\ \end{aligned}

i.e., half the area of the parallelogram spanned by these two vectors.

Hence, \mathtt{(a)} \Longrightarrow \mathtt{(b)}.

This problem is not to be attempted.