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.
