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.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $11$