These NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.1 Questions and Answers are prepared by our highly skilled subject experts.

## NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.1

Question 1.

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

Solution:

Let the direction angles be α, ß, γ.

i.e., α = 90°, ß = 135° and γ = 45°

The direction cosines are cos α, cos ß, cos γ = cos 90°, cos 135°, cos45°

= 0, \(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\)

Question 2.

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

Solution:

Let the direction angles be α, ß, γ. Angles of the line. Since this makes equal angles with the axes, we get

Question 3.

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

Solution:

The direction ratios are – 18, 12, – 4

\(\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}=\sqrt{384}\) = 22

The direction cosines are

\(\frac{-18}{22}, \frac{12}{22}, \frac{-4}{22}=\frac{-9}{11}, \frac{6}{11}, \frac{-2}{11}\)

Question 4.

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

Solution:

Let A (2, 3, 4), B (- 1, – 2, 1) and C (5, 8, 7) be the points.

Direction ratios of AB

= – 1 – 2, – 2 – 3, 1 – 4

= – 3, – 5, – 3

The direction ratios of BC

= 5 + 1, 8 + 2, 7 – 1 = 6, 10, 6

The direction ratios of AB and BC are proportional

∴ AB and BC are parallel.

Hence A, B, C are collinear.

Question 5.

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

Solution:

Let A (3, 5, – 4) B (- 1, 1, 2) and C (- 5, – 5, – 2) are the vertices of ∆ABC