Distance between Points in Complex Plane

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Let $A$ and $B$ be points in the complex plane such that:

$A = \tuple {x_1, y_1}$
$B = \tuple {x_2, y_2}$

Then the distance between $A$ and $B$ is given by:

\(\ds \size {AB}\) \(=\) \(\ds \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}\)
\(\ds \) \(=\) \(\ds \cmod {z_1 - z_2}\)

where $z_1$ and $z_2$ are represented by the complex numbers $z_1$ and $z_2$ respectively.


We have:

\(\ds AB\) \(=\) \(\ds z_2 - z_1\) Geometrical Interpretation of Complex Subtraction
\(\ds \) \(=\) \(\ds \paren {x_2 + i y_2} - \paren {x_1 + i y_1}\)
\(\ds \) \(=\) \(\ds \paren {x_2 - x_1} + \paren {y_2 - y_1} i\) Definition of Complex Subtraction
\(\ds \leadsto \ \ \) \(\ds \size {AB}\) \(=\) \(\ds \cmod {\paren {x_2 - x_1} + \paren {y_2 - y_1} i}\)
\(\ds \) \(=\) \(\ds \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}\) Definition of Complex Modulus