Equation for Line through Two Points in Complex Plane/Parametric Form 2

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Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $L$ be a straight line through $z_1$ and $z_2$ in the complex plane.


$L$ can be expressed by the equations:

\(\ds x - x_1\) \(=\) \(\ds t \paren {x_2 - x_1}\)
\(\ds y - y_1\) \(=\) \(\ds t \paren {y_2 - y_1}\)


These are the parametric equations of $L$, where $t$ is the parameter.


Proof

From Equation for Line through Two Points in Complex Plane: Parametric Form 1:

$z = z_1 + t \paren {z_2 - z_1}$


Letting:

\(\ds z\) \(=\) \(\ds x + i y\)
\(\ds z_1\) \(=\) \(\ds x_1 + i y_1\)
\(\ds z_2\) \(=\) \(\ds x_2 + i y_2\)

the parametric equations follow by equating real parts and imaginary parts.

$\blacksquare$


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