RBSE Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.1

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.1 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 11 Three Dimensional Geometry Ex 11.1

Question 1.
If a line makes angles 90°, 135°, 45° with the x, y and z-axis respectively, find its direction cosines.
Answer:
A line makes angles 90°, 135°, 45° with the x, y and z-axis respectively.
∴ cos α = cos 90° = 0,
cos β = cos 135° = - \(\frac{1}{\sqrt{2}}\),
cos γ = cos 45° = \(\frac{1}{\sqrt{2}}\)
∴ Direction cosines are 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 axis.
Answer:
Let α be the angle made by the line with the coordinate axis.
∴ Direction cosines of the line are cos α, cos α, cos α
But l² + m² + n² = 1
cos² α + cos² α + cos² α = 1
3 cos²α = 1
cos² α = \(\frac{1}{3}\)
∴ cos α = ± \(\frac{1}{\sqrt{3}}\)
Hence, the direction cosines of the line are:
±\(\frac{1}{\sqrt{3}}\), ±\(\frac{1}{\sqrt{3}}\), ±\(\frac{1}{\sqrt{3}}\).

Question 3.
If a line has the direction ratios - 18, 12, - 4 then what are its direction cosines?
Answer:
Let a, b, c be the direction ratios, then direction cosines of the line are as follows
(where a = - 18, b = 12, c = - 4)

Question 4.
Show that the points (2, 3, 4), (- 1, - 2, 1) (5, 8, 7) are collinear.
Answer:
Let A(2, 3, 4), B(- 1, - 2, 1) and C(5, 8, 7) be the given points.

Hence A, B, C are collinear.
Hence proved.

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).
Answer:
Let A(3, 5, - 4), B(- 1, 1, 2) and C(- 5, - 5, - 2) be the vertices of ∆ABC.
(i) Direction ratios of AB:
x₂ - x₁, y₂ - y₁, z₂ - z₁

(ii) Direction ratios of BC:

(iii) Direction ratios of CA: