Let us denote the points as follows:

⇒ A = (4,7,8)

⇒ B = (2,3,4)

⇒ C = (–1,–2,1)

⇒ D = (1,2,5)

If two lines are said to be parallel the directional ratios of two lines need to be proportional.

Let us assume the direction ratios for line AB be (r₁,r₂,r₃) and CD be (r₄,r₅,r₆)

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’s find the direction ratios for the line AB

⇒ (r₁,r₂,r₃) = (2–4, 3–7, 4–8)

⇒ (r₁,r₂,r₃) = (–2,–4,–4)

Let’s find the direction ratios for the line CD

⇒ (r₄,r₅,r₆) = (1–(–1), 2–(–2), 5–1)

⇒ (r₄,r₅,r₆) = (1+1, 2+2, 5–1)

⇒ (r₄,r₅,r₆) = (2,4,4)

The proportional condition can be stated as .

Let check whether the directional ratios are proportional or not,

⇒

⇒ ……(1)

⇒

⇒ ……(2)

⇒

⇒ ……(3)

From (1),(2),(3) we can say that the direction ratios of the lines are proportional. So, the lines are parallel to each other.