How to determine if 3 points on a 3-D graph are collinear? Let the points $A, B$ and $C$ be $(x_1, y_1, z_1), (x_2, y_2, z_2)$ and $(x_3, y_3, z_3)$ respectively. How do I prove that the 3 points are collinear? What is the formula?
From $A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)$ we can get their position vectors.

$\vec{AB}=(x_2-x_1,y_2-y_1,z_2-z_1)$ and $\vec{AC}=(x_3-x_1,y_3-y_1,z_3-z_1)$.

Then $||\vec{AB}\times\vec{AC}||=0\implies A,B,C$ collinear.
If the distance between |AB|+|BC|=|AC| then A,B,C are collinear.