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.

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 $P$.

$\ds OA + AP$

$=$

$\ds OP$

$\ds \leadsto \ \ $

$\ds z_1 + AP$

$=$

$\ds z$

$\ds \leadsto \ \ $

$\ds AP$

$=$

$\ds z - z_1$

$\ds OA + AB$

$=$

$\ds OB$

$\ds \leadsto \ \ $

$\ds z_1 + AB$

$=$

$\ds z_2$

$\ds \leadsto \ \ $

$\ds AB$

$=$

$\ds z_2 - z_1$

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.

$\blacksquare$

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