Equation for Line through Two Points in Complex Plane

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.

Formulation 1

$L$ can be expressed by the equation:

$\map \arg {\dfrac {z - z_1} {z_2 - z_1} } = 0$

Parametric Form $1$

$L$ 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.

Parametric Form $2$

$L$ 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.

Symmetric Form

$L$ can be expressed by the equation:

$z = \dfrac {m z_1 + n z_2} {m + n}$

This form of $L$ is known as the symmetric form.

Examples

Line through $2 + i$ and $3 - 2 i$

Let $L$ be a straight line through $2 + i$ and $3 - 2 i$ in the complex plane.

Then $L$ can be expressed as an equation in the following ways:

Parametric Form: $1$

$z - \paren {2 + i} = t \paren {1 - 3 i}$

Parametric Form: $2$

$x = 2 + t, y = 1 - 3 t$

Standard Form

$3 x + y = 7$