Show that the line joining the origin to the point (2,1,1) is perpendicular to the line determined by the points (3,5,–1) and (4,3,–1).

Let us denote the points as follows:

⇒ O = (0,0,0)

⇒ A = (2,1,1)

⇒ B = (3,5,–1)

⇒ C = (4,3,–1)

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 OA be (r₁,r₂,r₃) and BC 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 OA

⇒ (r₁,r₂,r₃) = (2–0, 1–0, 1–0)

⇒ (r₁,r₂,r₃) = (2,1,1)

Let’s find the direction ratios for the line BC

⇒ (r₄,r₅,r₆) = (4–3, 3–5, –1–(–1))

⇒ (r₄,r₅,r₆) = (4–3, 3–5, –1+1)

⇒ (r₄,r₅,r₆) = (1,–2,0)

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

⇒ r₁.r₄+r₂.r₅+r₃.r₆ = (2×1)+(1×–2)+(1×0)

⇒ r₁.r₄+r₂.r₅+r₃.r₆ = 2–2+0

⇒ 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.