How to tell whether a point is to the right or lef

Question

I have a set of points. I want to separate them into 2 distinct sets. To do this, I choose two points (a and b) and draw an imaginary line between them. Now I want to have all points that are left from this line in one set and those that are right from this line in the other set.

How can I tell for any given point z whether it is in the left or in the right set? I tried to calculate the angle between a-z-b – angles smaller than 180 are on the right hand side, greater than 180 on the left hand side – but because of the definition of ArcCos, the calculated angles are always smaller than 180°. Is there a formula to calculate angles greater than 180° (or any other formula to chose right or left side)?

Answer 1

@AVB's answer in ruby

det = Matrix[
  [(x2 - x1), (x3 - x1)],
  [(y2 - y1), (y3 - y1)]
].determinant

If det is positive its above, if negative its below. If 0, its on the line.