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. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-11-ex-11-1/
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