[Math] How to determine if 3 points on a 3-D graph are collinear

Question

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?

Answer 1

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.