Show that the line joining the origin to the point (2, 1, 1) is perpen

1.	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» 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.

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»

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.

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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).

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