Let $$M=\{(x_1,x_2,x_3) \in \mathbb{C}^3\;\hbox{such that}\;2\Re e(x_1\overline{x_2})=0,\;\;|x_1|^2-|x_2|^2=0,\;\hbox{and}\;|x_1|^2+|x_2|^2=1\},$$ Is this set empty? And thank you
2026-04-03 21:18:27.1775251107
Is this set empty?
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It is not empty, since $\left(\frac1{\sqrt2},\frac i{\sqrt2},0\right)\in M$.