If are two non-collinear vectors, prove that the points with position vectors and are collinear for all real values of

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

Given that, and are two non-collinear vectors.

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

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

…(i)

And is given by

Let us note the relation between and .

We know,

Or

Or [∵, from (i)]

Or …(ii)

If λ is any real value, then is also a real value.

Then, for any real value , we can write

From (ii) equation, we can write

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, A, B and C are collinear.