Question

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).

Answer 1

Let OA be the line joining the origin, O (0, 0, 0), and the point, A (2, 1, 1).
Also, let BC be the line joining the points, B (3, 5, −1) and C (4, 3, −1).
The direction ratios of OA are 2, 1, and 1 and of BC are (4 − 3) = 1, (3 − 5) = −2, and
(−1 + 1) = 0
OA is perpendicular to BC, if a₁a₂ + b₁b₂ + c₁c₂ = 0
a₁a₂ + b₁b₂ + c₁c₂ = 2 × 1 + 1 (−2) + 1 ×0 = 2 − 2 = 0
Thus, OA is perpendicular to BC.