Show that the points A (1, –2, –8), B (5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

We have been given the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7).

We need to show that A, B and C are collinear.

Let us define the position vector.

So, in this case if we find a relation between , and , then we can easily show that A, B and C are collinear.

Therefore, is given by

And is given by

And is given by

Let us add and , we get

Thus, clearly A, B and C are collinear.

We need to find the ratio in which B divides AC.

Let the ratio at which B divides AC be λ : 1. Then, position vector of B is:

But the position vector of B is .

So, by comparing the position vectors of B, we can write

Solving these equations separately, we get

⇒ 11λ + 1 = 5(λ + 1)

⇒ 11λ + 1 = 5λ + 5

⇒ 11λ – 5λ = 5 – 1

⇒ 6λ = 4

The ratio at which B divides AC is λ : 1.

Since,

We can say

Solving it further, multiply the ratio by 3.

⇒ λ : 1 = 2 : 3

Thus, the ratio in which B divides AC is 2 : 3.