How can I tell if a point belongs to a certain line?

Question

How can I tell if a point belongs to a certain line?

Examples are appreciated, if possible.

Answer 1

Given two points on the line L0 and L1 and the point to test P.

               (L1 - L0) * (P - L0)
n = (P - L0) - --------------------- (L1 - L0)
               (L1 - L0) * (L1 - L0)

The norm of the vector n is the distance of the point P from the line through L0 and L1. If this distance is zero or small enough (in the case of rounding errors), the point lies on the line.

The symbol * represents the dot product.

Example

P = (5, 5)

L0 = (0, 10)
L1 = (20, -10)

L1 - L0 = (20, -20)
P  - L0 = (5, -5)

              (20, -20) * (5, -5)
n = (5, -5) - --------------------- (20, -20)
              (20, -20) * (20, -20)

              200
  = (5, -5) - --- (20, -20)
              800

  = (5, -5) - (5, -5)

  = (0, 0)