Three points with position vectors $\vec a, \vec b, \vec c$ will be collinear, if there exist scalars $x, y, z$ such that

$x\vec a+y\vec b+z\vec c=\vec 0$, where $x + y + z=0$

Let, A, B, C be the points with position vectors $\vec a, \vec b, \vec c$ respectively. These points will be collinear, iff

$\vec {AB}=λ \vec{AC}$

$⇒\vec b-\vec a=λ(\vec a-\vec a)$

$⇒(λ-1)\vec a+\vec b+(-λ)\vec c=\vec 0$

$⇒x\vec a+y\vec b+z\vec c=\vec 0$, where $x = λ-1, y=1, z=-λ$

$⇒x\vec a+y\vec b+z\vec c=\vec 0$, where $x + y + z=0$