Using vectors, find the value of such that the points (1, –1, 3) and (3, 5, 3) are collinear.

Let the points be A (λ, –10, 3), B (1, –1, 3), C (3, 5, 3).

Let us define the position vectors of A, B and C.

Then,

And

And since, A, B and C are collinear.

Then, it has a relation as such

, where k is scalar quantity.

Comparing the coefficients of and . We get

1 – λ = k (3 – λ)

And 9 = 15k

First, we need to find the value of k.

So take 9 = 15k

Substitute the value of k in (1 – λ) = k (3 – λ)

⇒ 5(1 – λ) = 3(3 – λ)

⇒ 5 – 5λ = 9 – 3λ

⇒ 5λ – 3λ = 5 – 9

⇒ 2λ = –4

⇒ λ = –2

Hence, the value of λ is –2.