Let OA be the line joining the origin (0,0,0) and the point A(2,1,1).

Let BC be the line joining the points B(3,5,−1) and C(4,3,−1)

Direction ratios of OA = (a₁, b₁, c₁) ≡ [(2 - 0), (1 - 0), (1 - 0)] ≡ (2,1,1)

Direction ratios of BC = (a₂, b₂, c₂) ≡ [(4 - 3), (3 - 5), (-1 + 1)]

≡ (1, -2, 0)

Given-

OA is ⊥ to BC

∴ we have to prove that -

a₁a₂ + b₁b₂ + c₁c₂ = 0

L.H.S = a₁a₂ + b₁b₂ + c₁c₂ = 2 × 1 + 1 × (−2) + 1 × 0 = 2 - 2 = 0

R.H.S = 0

Thus, L.H.S = R.H.S ….PROVED

Hence OA is ⊥ to BC.