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), (4, 3, –1).

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 = (a_{1}, b_{1}, c_{1}) ≡ [(2 - 0), (1 - 0), (1 - 0)] ≡ (2,1,1)

Direction ratios of BC = (a_{2}, b_{2}, c_{2}) ≡ [(4 - 3), (3 - 5), (-1 + 1)]

≡ (1, -2, 0)

Given-

OA is ⊥ to BC

∴ we have to prove that -

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

L.H.S = a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 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.

1