NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 – Free PDF Download
The Exercise 11.1 of NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry is based on the following topics:
- Introduction to Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- The relation between the direction cosines of a line
- Direction cosines of a line passing through two points
Solving the problems of this exercise of Chapter 11 in Class 12 will help the students understand the topics mentioned above in a better way.
NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry Exercise 11.1
Access Other Exercises of Class 12 Maths Chapter 11
Exercise 11.1 Solutions 5 Questions
Exercise 11.2 Solutions 17 Questions
Exercise 11.3 Solutions 14 Questions
Miscellaneous Exercise On Chapter 11 Solutions 23 Questions
Access Answers to NCERT Class 12 Maths Chapter 11 Exercise 11.1
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 cosines of the line be l, m and n.
Here let α = 90°, β = 135° and γ = 45°
So,
l = cos α, m = cos β and n = cos γ
So direction cosines are
l = cos 90° = 0
m = cos 135°= cos (180° – 45°) = -cos 45° = -1/√2
n = cos 45° = 1/√2
∴ The direction cosines of the line are 0, -1/√2, 1/√2
2. Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution:
Given:
Angles are equal.
So let the angles be α, β, γ
Let the direction cosines of the line be l, m and n
l = cos α, m = cos β and n = cos γ
Here given α = β = γ (Since the line makes equal angles with the coordinate axes) … (1)
The direction cosines are
l = cos α, m = cos β and n = cos γ
We have,
l2 + m 2 + n2 = 1
cos2 α + cos2β + cos2γ = 1
From (1), we have,
cos2 α + cos2 α + cos2 α = 1
3 cos2 α = 1
Cos α = ± √(1/3)
∴ The direction cosines are
l = ± √(1/3), m = ± √(1/3), n = ± √(1/3)
3. If a line has the direction ratios –18, 12, –4, then what are its direction cosines?
Solution:
Given
Direction ratios as -18, 12, -4
Where, a = -18, b = 12, c = -4
Let us consider the direction ratios of the line as a, b and c
Then the direction cosines are
∴ The direction cosines are
-18/22, 12/22, -4/22 => -9/11, 6/11, -2/11
4. Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.
Solution:
If the direction ratios of two line segments are proportional, then the lines are collinear.
Given:
A(2, 3, 4), B(−1, −2, 1), C(5, 8, 7)
The direction ratio of the line joining A (2, 3, 4) and B (−1, −2, 1), is
(−1−2), (−2−3), (1−4) = (−3, −5, −3)
Where, a1Â = -3, b1Â = -5, c1Â = -3
Direction ratio of line joining B (−1, −2, 1) and C (5, 8, 7) is
(5− (−1)), (8− (−2)), (7−1) = (6, 10, 6)
Where, a2Â = 6, b2Â = 10 and c2Â =6
Hence it is clear that the direction ratios of AB and BC are of the same proportions
By
∴ A, B, and C 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).
Solution:
Given:
The vertices are (3, 5, –4), (-1, 1, 2) and (–5, –5, –2).
The direction cosines of the two points passing through A(x1, y1, z1) and B(x2, y2, z2) is given by (x2 – x1), (y2-y1), (z2-z1)
Firstly let us find the direction ratios of AB
Where, A = (3, 5, -4) and B = (-1, 1, 2)
Ratio of AB = [(x2 – x1)2, (y2 – y1)2, (z2 – z1)2]
= (-1-3), (1-5), (2-(-4)) = -4, -4, 6
Then by using the formula,
√[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]
√[(-4)2 + (-4)2 + (6)2] = √(16+16+36)
= √68
= 2√17
Now let us find the direction cosines of the line AB
By using the formula,
-4/2√17 , -4/2√17, 6/2√17
Or -2/√17, -2/√17, 3/√17
Similarly,
Let us find the direction ratios of BC
Where, B = (-1, 1, 2) and C = (-5, -5, -2)
Ratio of AB = [(x2 – x1)2, (y2 – y1)2, (z2 – z1)2]
= (-5+1), (-5-1), (-2-2) = -4, -6, -4
Then by using the formula,
√[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]
√[(-4)2 + (-6)2 + (-4)2] = √(16+36+16)
= √68
= 2√17
Now let us find the direction cosines of the line AB
By using the formula,
-4/2√17, -6/2√17, -4/2√17
Or -2/√17, -3/√17, -2/√17
Similarly,
Let us find the direction ratios of CA
Where, C = (-5, -5, -2) and A = (3, 5, -4)
Ratio of AB = [(x2 – x1)2, (y2 – y1)2, (z2 – z1)2]
= (3+5), (5+5), (-4+2) = 8, 10, -2
Then by using the formula,
√[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]
√[(8)2 + (10)2 + (-2)2] = √(64+100+4)
= √168
= 2√42
Now let us find the direction cosines of the line AB
By using the formula,
8/2√42, 10/2√42, -2/2√42
Or 4/√42, 5/√42, -1/√42
Also, explore –Â
Good