Book: Mathematics Part-II
Chapter: 11. Three Dimensional Geometry
Subject: Maths - Class 12th
Q. No. 2 of Exercise 11.2
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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).
We know that
Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are perpendicular if the angle between them is θ = 90°, i.e. a1a2 + b1b2 + c1c2 = 0
Also, we know that the direction ratios of the line segment joining (x1, y1, z1) and (x2, y2, z2) is taken as x2 – x1, y2 – y1, z2 – z1 (or x1 – x2, y1 – y2, z1 – z2).
⇒ The direction ratios of the line through the points (1, –1, 2) and (3, 4, –2) is:
a1 = 3 – 1 = 2, b1 = 4 – (-1) = 4 + 1 = 5, c1 = -2 –2 = -4
and the direction ratios of the line through the points (0, 3, 2) and (3, 5, 6) is:
a2 = 3 – 0 = 3, b2 = 5 – 3 = 2, c2 = 6 – 2 = 4
Now, consider
a1a2 + b1b2 + c1c2 = 2 × 3 + 5 × 2 + (-4) × 4 = 6 + 10 + (-16) = 16 + (-16) = 0
⇒ 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).
