Let us understand that, two more points are said to be collinear if they all lie on a single straight line

Let the points be A, B and C having position vectors such that,

Position vector of A = \(\vec i+2\vec j+3\vec k\)

Position vector of B = \(3\vec i+4\vec j+7\vec k\)

Position vector of C = \(-3\vec i-2\vec j-7\vec k\)

So, in this case if we prove that \(\vec{AB}\) and \(\vec{BC}\) are parallel to each other, then we can easily show that A, B and C are collinear.

**Therefore**, \(\vec{AB}\) is given by

\(\vec{AB}\) = Position vector of C - Position vector of B

Let us note the relation between \(\vec{AB}\) and \(\vec{BC}\).

**We know,**

This relation shows that \(\vec{AB}\) and \(\vec{BC}\) are parallel to each other.

But also, \(\vec B\) is the common vector in \(\vec{AB}\) and \(\vec{BC}\).

⇒ \(\vec{AB}\) and \(\vec{BC}\) are not parallel but lies on a straight line.

**Thus**, A, B and C are collinear.