Let us indicate given points with A, B and C.

⇒ A = (2,3,4)

⇒ B = (–1,–2,1)

⇒ C = (5,8,7)

We know that for points D, E, F to be collinear the direction ratios of any two lines from DE, DF, EF are to be proportional;

We know that direction ratios for a line passing through points (x₁, y₁, z₁) and (x₂, y₂, z₂) is (x₂–x₁, y₂–y₁, z₂–z₁).

Let us assume direction ratios for AB is (r₁, r₂, r₃) and BC is (r₄, r₅, r₆).

The proportional condition can be stated as .

Let us find the direction ratios of AB

⇒ (r₁, r₂, r₃) = (–1–2, –2–3, 1–4)

⇒ (r₁, r₂, r₃) = (–3,–5,–3)

Let us find the direction ratios of BC

⇒ (r₄, r₅, r₆) = (5–(–1), 8–(–2), 7–1)

⇒ (r₄, r₅, r₆) = (5+1, 8+2, 7–1)

⇒ (r₄, r₅, r₆) = (6, 10, 6)

Now

⇒

⇒ ……(1)

⇒

⇒ ……(2)

⇒

⇒ ……(3)

From (1),(2),(3) we get,

⇒

So, from the above relational we can say that points (2,3,4), (–1,–2,1) , (5,8,7) are collinear.