Condition For Collinearity Of Points In Complex Plane - ProofWiki

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Theorem

Formulation 1

Let $z_1$, $z_2$ and $z_3$ be points in the complex plane.

Then $z_1$, $z_2$ and $z_3$ are collinear if and only if:

$\dfrac {z_1 - z_3} {z_3 - z_2} = \lambda$

where $\lambda \in \R$ is a real number.

If this is the case, then $z_3$ divides the line segment in the ratio $\lambda$.

If $\lambda > 0$ then $z_3$ is between $z_1$ and $z_2$, and if $\lambda < 0$ then $z_3$ is outside the line segment joining $z_1$ to $z_2$.

Formulation 2

Let $z_1, z_2, z_3$ be distinct complex numbers.

Then:

$z_1, z_2, z_3$ are collinear in the complex plane

if and only if:

$\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 = 0$ where: $\alpha + \beta + \gamma = 0$ not all of $\alpha, \beta, \gamma$ are zero. Retrieved from "https://proofwiki.org/w/index.php?title=Condition_for_Collinearity_of_Points_in_Complex_Plane&oldid=512104" Categories:
  • Geometry of Complex Plane
  • Analytic Geometry
  • Condition for Collinearity of Points in Complex Plane
  • Collinear Points

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Tag » Collinear Points Complex Numbers