If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.

Mathematics Part-II

Book: Mathematics Part-II

Chapter: 11. Three Dimensional Geometry

Subject: Maths - Class 12^th

Q. No. 1 of Exercise 11.1

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1

If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.

Let the direction cosines of the line making ∠ α with x-axis, β – with y axis and γ- with z axis are l, m and n

⇒ l = cos α, m = cos β and n = cos γ

Here α = 90°, β = 135° and γ = 45°

So direction cosines are

l = cos 90° = 0

m = cos 135°= cos (180° - 45°) = -cos 45° =

n = cos 45° =

⇒ Direction cosines of the line

1

Chapter Exercises

Exercise 11.1

Exercise 11.2

Exercise 11.3

Miscellaneous Exercise

1

If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.

2

Find the direction cosines of a line which makes equal angles with the coordinate axes.

3

If a line has the direction ratios –18, 12, –4, then what are its direction cosines?

4

Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

5

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (-1, 1, 2) and (–5, –5, –2).