Complex Arithmetic/Examples/(Modulus of 2 z 2 - 3 z 1)^2

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Example of Complex Arithmetic

Let $z_1 = 1 - i$ and $z_2 = -2 + 4 i$.

Then:

$\cmod {2 z_2 - 3 z_1}^2 = 170$


Proof

\(\ds \cmod {2 z_2 - 3 z_1}^2\) \(=\) \(\ds \cmod {2 \paren {-2 + 4 i} - 3 \paren {1 - i} }^2\)
\(\ds \) \(=\) \(\ds \cmod {\paren {-4 + 8 i} - \paren {3 - 3 i} }^2\)
\(\ds \) \(=\) \(\ds \cmod {-7 + 11 i}^2\)
\(\ds \) \(=\) \(\ds 7^2 + 11^2\)
\(\ds \) \(=\) \(\ds 49 + 121\)
\(\ds \) \(=\) \(\ds 170\)

$\blacksquare$


Sources