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Formulas/maths/M4a Geometry/Condition for Collinearity

Condition for Collinearity

Three points are collinear iff the ratio (z₃−z₁)/(z₂−z₁) is real — equivalently, arg((z₃−z₁)/(z₂−z₁)) = 0 or π. The determinant form: the 3×3 determinant with rows (zₖ, z̄ₖ, 1) equals zero. Both forms are used in JEE problems.
Derivation

Three points are collinear when one lies on the line through the other two. In the complex plane, this becomes a condition on arguments — which in turn becomes a condition on a ratio being real.

The Argument Condition

z1z_1, z2z_2, z3z_3 are collinear iff the vector from z1z_1 to z3z_3 points in the same or opposite direction as the vector from z1z_1 to z2z_2.

Same direction: arg⁡(z3−z1)=arg⁡(z2−z1)\arg(z_3 - z_1) = \arg(z_2 - z_1)

Opposite direction: arg⁡(z3−z1)=arg⁡(z2−z1)+π\arg(z_3 - z_1) = \arg(z_2 - z_1) + \pi

Both cases are captured by:

arg⁡ ⁣(z3−z1z2−z1)=0 or π\arg\!\left(\frac{z_3 - z_1}{z_2 - z_1}\right) = 0 \text{ or } \pi

A complex number has argument 00 or π\pi iff it is real. Therefore:

z1,z2,z3 collinear  ⟺  z3−z1z2−z1∈Rz_1, z_2, z_3 \text{ collinear} \iff \frac{z_3 - z_1}{z_2 - z_1} \in \mathbb{R}

Algebraic Form

A complex number ww is real iff w=wˉw = \bar{w}. Applying this to the ratio:

z3−z1z2−z1=zˉ3−zˉ1zˉ2−zˉ1\frac{z_3 - z_1}{z_2 - z_1} = \frac{\bar{z}_3 - \bar{z}_1}{\bar{z}_2 - \bar{z}_1}

Cross-multiplying:

(z3−z1)(zˉ2−zˉ1)=(zˉ3−zˉ1)(z2−z1)(z_3 - z_1)(\bar{z}_2 - \bar{z}_1) = (\bar{z}_3 - \bar{z}_1)(z_2 - z_1)

Expanding and rearranging:

(z3−z1)(zˉ2−zˉ1)−(zˉ3−zˉ1)(z2−z1)=0(z_3 - z_1)(\bar{z}_2 - \bar{z}_1) - (\bar{z}_3 - \bar{z}_1)(z_2 - z_1) = 0

Determinant Form

The condition above is equivalent to:

∣z1zˉ11z2zˉ21z3zˉ31∣=0\begin{vmatrix} z_1 & \bar{z}_1 & 1 \\ z_2 & \bar{z}_2 & 1 \\ z_3 & \bar{z}_3 & 1 \end{vmatrix} = 0

This is the complex analogue of the Cartesian collinearity determinant ∣x1y11x2y21x3y31∣=0\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0. Writing z=x+iyz = x + iy and zˉ=x−iy\bar{z} = x - iy in each entry, the two determinants differ only by column operations and a constant factor — they contain the same information.

Which Form to Use

  • Ratio form is faster for problems where you can read off the ratio directly.
  • Determinant form is better when the three points are given parametrically and you need to eliminate a parameter.