Mathematics Part-II

Book: Mathematics Part-II

Chapter: 11. Three Dimensional Geometry

Subject: Maths - Class 12th

Q. No. 2 of Exercise 11.2

Listen NCERT Audio Books - Kitabein Ab Bolengi

2

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).


2

Chapter Exercises

More Exercise Questions