Page 1
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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