If are non-coplanar vectors, prove that the points having the following position vectors are collinear:

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 non-coplanar vectors.

And we know that, vectors that do not lie on the same plane or line are called non-coplanar vectors.

To Prove: , and are collinear.

Proof: Let the points be A, B and C.

Then,

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, A, B and C are collinear.