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Exercise - 11.1 - Three Dimensional Geometry NCERT Solutions | Mathematics (Maths) Class 12 - JEE PDF Download

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 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
EXERCISE 11.1                                                                        PAGE NO: 467 
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 a = 90°, ß = 135° and ? = 45° 
So, 
l = cos a, m = cos ß and n = cos ? 
So, the direction cosines are 
l = cos 90° = 0 
m = cos 135°= cos (180° – 45°) = -cos 45° = -1/v2 
n = cos 45° = 1/v2 
? The direction cosines of the line are 0, -1/v2, 1/v2 
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 a, ß, ? 
Let the direction cosines of the line be l, m and n. 
l = cos a, m = cos ß and n = cos ? 
Here, given a = ß = ? (Since, line makes equal angles with the coordinate axes) … (1) 
The direction cosines are 
l = cos a, m = cos ß and n = cos ? 
We have, 
l
2
 + m 
2
 + n
2
 = 1 
cos
2
 a + cos
2
ß + cos
2
? = 1 
From (1) we have, 
Page 2


 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
EXERCISE 11.1                                                                        PAGE NO: 467 
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 a = 90°, ß = 135° and ? = 45° 
So, 
l = cos a, m = cos ß and n = cos ? 
So, the direction cosines are 
l = cos 90° = 0 
m = cos 135°= cos (180° – 45°) = -cos 45° = -1/v2 
n = cos 45° = 1/v2 
? The direction cosines of the line are 0, -1/v2, 1/v2 
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 a, ß, ? 
Let the direction cosines of the line be l, m and n. 
l = cos a, m = cos ß and n = cos ? 
Here, given a = ß = ? (Since, line makes equal angles with the coordinate axes) … (1) 
The direction cosines are 
l = cos a, m = cos ß and n = cos ? 
We have, 
l
2
 + m 
2
 + n
2
 = 1 
cos
2
 a + cos
2
ß + cos
2
? = 1 
From (1) we have, 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
cos
2
 a + cos
2
 a + cos
2
 a = 1 
3 cos
2
 a = 1 
Cos a = ± v(1/3) 
? The direction cosines are 
l = ± v(1/3), m = ± v(1/3), n = ± v(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 lines segments are proportional, then the lines are collinear. 
Given: 
A(2, 3, 4), B(-1, -2, 1), C(5, 8, 7) 
Direction ratio of line joining A (2, 3, 4) and B (-1, -2, 1), are 
(-1-2), (-2-3), (1-4) = (-3, -5, -3) 
Page 3


 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
EXERCISE 11.1                                                                        PAGE NO: 467 
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 a = 90°, ß = 135° and ? = 45° 
So, 
l = cos a, m = cos ß and n = cos ? 
So, the direction cosines are 
l = cos 90° = 0 
m = cos 135°= cos (180° – 45°) = -cos 45° = -1/v2 
n = cos 45° = 1/v2 
? The direction cosines of the line are 0, -1/v2, 1/v2 
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 a, ß, ? 
Let the direction cosines of the line be l, m and n. 
l = cos a, m = cos ß and n = cos ? 
Here, given a = ß = ? (Since, line makes equal angles with the coordinate axes) … (1) 
The direction cosines are 
l = cos a, m = cos ß and n = cos ? 
We have, 
l
2
 + m 
2
 + n
2
 = 1 
cos
2
 a + cos
2
ß + cos
2
? = 1 
From (1) we have, 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
cos
2
 a + cos
2
 a + cos
2
 a = 1 
3 cos
2
 a = 1 
Cos a = ± v(1/3) 
? The direction cosines are 
l = ± v(1/3), m = ± v(1/3), n = ± v(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 lines segments are proportional, then the lines are collinear. 
Given: 
A(2, 3, 4), B(-1, -2, 1), C(5, 8, 7) 
Direction ratio of line joining A (2, 3, 4) and B (-1, -2, 1), are 
(-1-2), (-2-3), (1-4) = (-3, -5, -3) 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
Where, a 1 = -3, b 1 = -5, c 1 = -3 
Direction ratio of line joining B (-1, -2, 1) and C (5, 8, 7) are 
(5- (-1)), (8- (-2)), (7-1) = (6, 10, 6) 
Where, a 2 = 6, b 2 = 10 and c 2 =6 
Now, 
 
? A, B, 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(x 1, y 1, z 1) and B(x 2, y 2, z 2) is given by (x 2 – x 1), (y 2-y 1), (z 2-z 1) 
Page 4


 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
EXERCISE 11.1                                                                        PAGE NO: 467 
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 a = 90°, ß = 135° and ? = 45° 
So, 
l = cos a, m = cos ß and n = cos ? 
So, the direction cosines are 
l = cos 90° = 0 
m = cos 135°= cos (180° – 45°) = -cos 45° = -1/v2 
n = cos 45° = 1/v2 
? The direction cosines of the line are 0, -1/v2, 1/v2 
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 a, ß, ? 
Let the direction cosines of the line be l, m and n. 
l = cos a, m = cos ß and n = cos ? 
Here, given a = ß = ? (Since, line makes equal angles with the coordinate axes) … (1) 
The direction cosines are 
l = cos a, m = cos ß and n = cos ? 
We have, 
l
2
 + m 
2
 + n
2
 = 1 
cos
2
 a + cos
2
ß + cos
2
? = 1 
From (1) we have, 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
cos
2
 a + cos
2
 a + cos
2
 a = 1 
3 cos
2
 a = 1 
Cos a = ± v(1/3) 
? The direction cosines are 
l = ± v(1/3), m = ± v(1/3), n = ± v(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 lines segments are proportional, then the lines are collinear. 
Given: 
A(2, 3, 4), B(-1, -2, 1), C(5, 8, 7) 
Direction ratio of line joining A (2, 3, 4) and B (-1, -2, 1), are 
(-1-2), (-2-3), (1-4) = (-3, -5, -3) 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
Where, a 1 = -3, b 1 = -5, c 1 = -3 
Direction ratio of line joining B (-1, -2, 1) and C (5, 8, 7) are 
(5- (-1)), (8- (-2)), (7-1) = (6, 10, 6) 
Where, a 2 = 6, b 2 = 10 and c 2 =6 
Now, 
 
? A, B, 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(x 1, y 1, z 1) and B(x 2, y 2, z 2) is given by (x 2 – x 1), (y 2-y 1), (z 2-z 1) 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
Firstly let us find the direction ratios of AB 
Where, A = (3, 5, -4) and B = (-1, 1, 2) 
Ratio of AB = [(x 2 – x 1)
2
, (y 2 – y 1)
2
, (z 2 – z 1)
2
] 
= (-1-3), (1-5), (2-(-4)) = -4, -4, 6 
Then by using the formula, 
v[(x 2 – x 1)
2
 + (y 2 – y 1)
2
 + (z 2 – z 1)
2
] 
v[(-4)
2
 + (-4)
2
 + (6)
2
] = v(16+16+36) 
= v68 
= 2v17 
Now let us find the direction cosines of the line AB 
By using the formula, 
 
-4/2v17 , -4/2v17, 6/2v17 
Or -2/v17, -2/v17, 3/v17 
Similarly, 
Let us find the direction ratios of BC 
Where, B = (-1, 1, 2) and C = (-5, -5, -2) 
Ratio of AB = [(x 2 – x 1)
2
, (y 2 – y 1)
2
, (z 2 – z 1)
2
] 
= (-5+1), (-5-1), (-2-2) = -4, -6, -4 
Then by using the formula, 
v[(x 2 – x 1)
2
 + (y 2 – y 1)
2
 + (z 2 – z 1)
2
] 
v[(-4)
2
 + (-6)
2
 + (-4)
2
] = v(16+36+16) 
= v68 
= 2v17 
Now, let us find the direction cosines of the line AB 
Page 5


 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
EXERCISE 11.1                                                                        PAGE NO: 467 
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 a = 90°, ß = 135° and ? = 45° 
So, 
l = cos a, m = cos ß and n = cos ? 
So, the direction cosines are 
l = cos 90° = 0 
m = cos 135°= cos (180° – 45°) = -cos 45° = -1/v2 
n = cos 45° = 1/v2 
? The direction cosines of the line are 0, -1/v2, 1/v2 
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 a, ß, ? 
Let the direction cosines of the line be l, m and n. 
l = cos a, m = cos ß and n = cos ? 
Here, given a = ß = ? (Since, line makes equal angles with the coordinate axes) … (1) 
The direction cosines are 
l = cos a, m = cos ß and n = cos ? 
We have, 
l
2
 + m 
2
 + n
2
 = 1 
cos
2
 a + cos
2
ß + cos
2
? = 1 
From (1) we have, 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
cos
2
 a + cos
2
 a + cos
2
 a = 1 
3 cos
2
 a = 1 
Cos a = ± v(1/3) 
? The direction cosines are 
l = ± v(1/3), m = ± v(1/3), n = ± v(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 lines segments are proportional, then the lines are collinear. 
Given: 
A(2, 3, 4), B(-1, -2, 1), C(5, 8, 7) 
Direction ratio of line joining A (2, 3, 4) and B (-1, -2, 1), are 
(-1-2), (-2-3), (1-4) = (-3, -5, -3) 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
Where, a 1 = -3, b 1 = -5, c 1 = -3 
Direction ratio of line joining B (-1, -2, 1) and C (5, 8, 7) are 
(5- (-1)), (8- (-2)), (7-1) = (6, 10, 6) 
Where, a 2 = 6, b 2 = 10 and c 2 =6 
Now, 
 
? A, B, 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(x 1, y 1, z 1) and B(x 2, y 2, z 2) is given by (x 2 – x 1), (y 2-y 1), (z 2-z 1) 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
Firstly let us find the direction ratios of AB 
Where, A = (3, 5, -4) and B = (-1, 1, 2) 
Ratio of AB = [(x 2 – x 1)
2
, (y 2 – y 1)
2
, (z 2 – z 1)
2
] 
= (-1-3), (1-5), (2-(-4)) = -4, -4, 6 
Then by using the formula, 
v[(x 2 – x 1)
2
 + (y 2 – y 1)
2
 + (z 2 – z 1)
2
] 
v[(-4)
2
 + (-4)
2
 + (6)
2
] = v(16+16+36) 
= v68 
= 2v17 
Now let us find the direction cosines of the line AB 
By using the formula, 
 
-4/2v17 , -4/2v17, 6/2v17 
Or -2/v17, -2/v17, 3/v17 
Similarly, 
Let us find the direction ratios of BC 
Where, B = (-1, 1, 2) and C = (-5, -5, -2) 
Ratio of AB = [(x 2 – x 1)
2
, (y 2 – y 1)
2
, (z 2 – z 1)
2
] 
= (-5+1), (-5-1), (-2-2) = -4, -6, -4 
Then by using the formula, 
v[(x 2 – x 1)
2
 + (y 2 – y 1)
2
 + (z 2 – z 1)
2
] 
v[(-4)
2
 + (-6)
2
 + (-4)
2
] = v(16+36+16) 
= v68 
= 2v17 
Now, let us find the direction cosines of the line AB 
 
 
 
 
 NCERT Solutions for Class 12 Maths Chapter 11 – 
Three Dimensional Geometry  
By using the formula, 
 
-4/2v17, -6/2v17, -4/2v17 
Or -2/v17, -3/v17, -2/v17 
Similarly, 
Let us find the direction ratios of CA 
Where, C = (-5, -5, -2) and A = (3, 5, -4) 
Ratio of AB = [(x 2 – x 1)
2
, (y 2 – y 1)
2
, (z 2 – z 1)
2
] 
= (3+5), (5+5), (-4+2) = 8, 10, -2 
Then, by using the formula, 
v[(x 2 – x 1)
2
 + (y 2 – y 1)
2
 + (z 2 – z 1)
2
] 
v[(8)
2
 + (10)
2
 + (-2)
2
] = v(64+100+4) 
= v168 
= 2v42 
Now, let us find the direction cosines of the line AB 
By using the formula, 
 
8/2v42, 10/2v42, -2/2v42 
Or 4/v42, 5/v42, -1/v42
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Document Description: NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The notes and questions for NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry have been prepared according to the JEE exam syllabus. Information about NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry covers topics like and NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry.
Introduction of NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry in English is available as part of our Mathematics (Maths) Class 12 for JEE & NCERT Solutions: Exercise - 11.1 - Three Dimensional Geometry in Hindi for Mathematics (Maths) Class 12 course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Exercise - 11.1 - Three Dimensional Geometry NCERT Solutions | Mathematics (Maths) Class 12 - JEE
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