September 29, 2022 – Physicspupil

202209291831 Problem 1.1

Prove the parallelogram law

$|z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2]$

and give a geometric interpretation.

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

Draw a picture.

Apply the Law of Cosines,

$\begin{aligned} |z_1+z_2|^2 & =|z_1|^2+|z_2|^2-2(-z_1)\cdot z_2 \\ |z_1-z_2|^2 & =|z_1|^2+|z_2|^2-2z_1\cdot z_2 \\ \end{aligned}$

and that is all I could interpret.

物理後生﹑掣數學之肘.
小子算數﹑指物理之月.