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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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