Show that the line through points (1,–1,2) and (3,4,–2) is perpendicular to the line through the points (0,3,2) and (3,5,6).

Let us denote the points as follows:

⇒ A = (1,–1,2)

⇒ B = (3,4,–2)

⇒ C = (0,3,2)

⇒ D = (3,5,6)

If two lines of direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂) are said to be perpendicular to each other. Then the following condition is need to be satisfied:

⇒ a₁.a₂+b₁.b₂+c₁.c₂=0 ……(1)

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) = (3–1, 4–(–1), –2–2)

⇒ (r₁,r₂,r₃) = (3–1, 4+1, –2–2)

⇒ (r₁,r₂,r₃) = (2,5,–4)

Let’s find the direction ratios for the line CD

⇒ (r₄,r₅,r₆) = (3–0, 5–3, 6–2)

⇒ (r₄,r₅,r₆) = (3,2,4)

Let us check whether the lines are perpendicular or not using (1)

⇒ r₁.r₄+r₂.r₅+r₃.r₆ = (2×3)+(5×2)+(–4×4)

⇒ r₁.r₄+r₂.r₅+r₃.r₆ = 6+10–16

⇒ r₁.r₄+r₂.r₅+r_3.r₆ = 0

Since the condition is clearly satisfied, we can say that the given lines are perpendicular to each other.