Coplex numbers

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Hari Shankar (9109)

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We have $|z_1+z_2|^2 = |z_1|^2 + |z_2|^2 + 2 Re(z_1 \overline{z_2})$

Hence $|i \sqrt 2 z_1+\sqrt 3z_2|^2 = 2|z_1|^2 + 3|z_2|^2 + 2 Re(i \sqrt 6 z_1 \overline{z_2})$ and

$|\sqrt 3 z_1+i\sqrt 2z_2|^2 = 3|z_1|^2 + 2|z_2|^2 + 2 Re(-i \sqrt 6 z_1 \overline{z_2}) \\ \\ = 3|z_1|^2 + 2|z_2|^2 - 2 Re(i \sqrt 6 z_1 \overline{z_2})$

Adding the two we get 5(|z_1|^2 + |z_2|^2)

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