Using vectors show that the points
A(–2, 3, 5), B(7, 0, 1) C(–3, –2, –5) and D(3, 4, 7) are such that AB and CD intersect at the point P(1, 2, 3).

We have been given the points A(–2, 3, 5), B(7, 0, 1), C(–3, –2, –5), D(3, 4, 7) and P(1, 2, 3).

Let us define it position vectors.

So,

Now, we need to show that AB and CD intersect at the point P.

For this, if we prove that A, B and P are collinear & C, D and P are collinear so that P is the common point between them and we can show that AB and CD intersect at P.

Let us find position vector of AP and PB.

And

Now, we can draw out a relation between and .

We know,

This relation clearly shows that and are parallel.

And since, P is the common point between them, we can say that these vectors and are actually not parallel but lie on a straight line.

⇒ Points A, P, B are collinear

[∵, Two more points are said to be collinear if they all lie on a single straight line.]

Now let us find the position vector of CP and PD.

And

Now, we can draw out a relation between and .

We know,

This relation clearly shows that and are parallel.

And since, P is the common point between them, we can say that these vectors and are actually not parallel but lie on a straight line.

⇒ Points C, P and D are collinear.

[∵, Two more points are said to be collinear if they all lie on a single straight line.]

Since, we know that A, P, B and C, P, D are collinear separately.

Note that, P is the common point between the two pairs of collinear points.

Thus, AB and CD intersect each other at a point P.