Using vector method, prove that the following points are collinear.

A(–3, –2, –5), B(1, 2, 3) and C(3, 4, 7)

Given: A (–3, –2, –5), B (1, 2, 3) and C (3, 4, 7).

To Prove: A, B and C are collinear.

Proof:

Let us define position vectors. So,

So, in this case if we prove that and are parallel to each other, then we can easily show that A, B and C are collinear.

Therefore, is given by

And is given by

Let us note the relation between and .

We know,

Or

Or [∵, ]

This relation shows that and are parallel to each other.

But also, is the common vector in and .

⇒ and are not parallel but lies on a straight line.

Thus, proved that A, B and C are collinear.