Show that the line through the points (1, −1, 2) (3, 4, −2) is per

1.	Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Answer» Let AB be the line joining the points, (1, −1, 2) and (3, 4, − 2), and CD be the line joining the points, (0, 3, 2) and (3, 5, 6). The direction ratios, a₁, b₁, c₁, of AB are (3 − 1), (4 − (−1)), and (−2 − 2) i.e., 2, 5, and −4. The direction ratios, a₂, b₂, c₂, of CD are (3 − 0), (5 − 3), and (6 −2) i.e., 3, 2, and 4. AB and CD will be perpendicular to each other, if a₁a₂+ b₁b₂+ c₁c₂ = 0 a₁a₂ + b₁b₂+ c₁c₂= 2 × 3 + 5 × 2 + (− 4) × 4 = 6 + 10 − 16 = 0 Therefore, AB and CD are perpendicular to each other.

Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Answer»

Let AB be the line joining the points, (1, −1, 2) and (3, 4, − 2), and CD be the line joining the points, (0, 3, 2) and (3, 5, 6).
The direction ratios, a₁, b₁, c₁, of AB are (3 − 1), (4 − (−1)), and (−2 − 2) i.e., 2, 5, and −4.

The direction ratios, a₂, b₂, c₂, of CD are (3 − 0), (5 − 3),

and (6 −2) i.e., 3, 2, and 4. AB and CD will be perpendicular to each other,

if a₁a₂+ b₁b₂+ c₁c₂ = 0 a₁a₂ + b₁b₂+ c₁c₂= 2 × 3 + 5 × 2 + (− 4) × 4 = 6 + 10 − 16 = 0

Therefore, AB and CD are perpendicular to each other.

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