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

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

or:
 * $z = \paren {1 - t} z_1 + t z_2$

This form of $L$ is known as the parametric form, where $t$ is the parameter.

Proof
Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.

Let $z$ be an arbitrary point on $L$ represented by the point $P$.


 * Line-in-Complex-Plane-through-Two-Points.png

From Geometrical Interpretation of Complex Addition:

As $AP$ and $AB$ are collinear:
 * $AP = t AB$

and so:
 * $z - z_1 = t \paren {z_2 - z_1}$

The given expressions follow after algebra.