I want to confirm two things :
In space $(\mathbb{R}^3)$ if line $\ell_1$ is perpendicular to line $\ell_2$ and line $\ell_2$ is perpendicular to line $\ell_3$, then $\ell_1$ is parallel to $\ell_3$.
The points $A,B,C$ (pairwise different) are collinear iff the vectors $\vec{AB}, \vec{CB}$ are parallel.
I think the first statement is correct due to the direction vectors and the cross product.
I think the second statement is wrong, since we have that these 3 points are collinear iff $AB + BC = AC$, right?
The first statement is false
$x$ axis is perpendicular to $y$ axis which is perpendicular to $z$ axis. $x$ and $z$ are not parallel.
The second looks correct.
If $A,B,C$ are collinear, then they lie on the same line and $\vec{AB},\vec{CB}$ are parallel.
If $\vec{AB}$ and $\vec{CB}$ are parallel, then there is a scalar $\lambda$ such that
$\vec{AB}=\lambda\vec{CB}$ that is $A=B+\lambda\vec{CB}$ therefore $A,B,C$ are collinear.