NCERT Textbook: Three Dimensional Geometry | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


THREE DIMENSIONAL GEOMETRY 377
v The moving power of mathematical invention is not
reasoning but imagination. – A.DEMORGAN v
11.1  Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional geometry. The purpose of this
approach to 3-dimensional geometry is that it makes the
study simple and elegant*.
In this chapter, we shall study the direction cosines
and direction ratios of a line joining two points and also
discuss about the equations of lines and planes in space
under different conditions, angle between two lines, two
planes, a line and a plane, shortest distance between two
skew lines and distance of a point from a plane. Most of
the above results are obtained in vector form. Nevertheless, we shall also translate
these results in the Cartesian form which, at times, presents a more clear geometric
and analytic picture of the situation.
11.2  Direction Cosines and Direction Ratios of a Line
From Chapter 10, recall that if a directed line L passing through the origin makes
angles a, ß and ? with x, y and z-axes, respectively, called direction angles, then cosine
of these angles, namely, cos a, cos ß and cos ? are called direction cosines of the
directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements,
i.e., ,  and . Thus, the signs of the direction cosines are reversed.
Chapter 11
THREE DIMENSIONAL GEOMETRY
* For various activities in three dimensional geometry, one may refer to the Book
“A Hand Book for designing Mathematics Laboratory in Schools”, NCERT, 2005
Leonhard Euler
(1707-1783)
Rationalised 2023-24
Page 2


THREE DIMENSIONAL GEOMETRY 377
v The moving power of mathematical invention is not
reasoning but imagination. – A.DEMORGAN v
11.1  Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional geometry. The purpose of this
approach to 3-dimensional geometry is that it makes the
study simple and elegant*.
In this chapter, we shall study the direction cosines
and direction ratios of a line joining two points and also
discuss about the equations of lines and planes in space
under different conditions, angle between two lines, two
planes, a line and a plane, shortest distance between two
skew lines and distance of a point from a plane. Most of
the above results are obtained in vector form. Nevertheless, we shall also translate
these results in the Cartesian form which, at times, presents a more clear geometric
and analytic picture of the situation.
11.2  Direction Cosines and Direction Ratios of a Line
From Chapter 10, recall that if a directed line L passing through the origin makes
angles a, ß and ? with x, y and z-axes, respectively, called direction angles, then cosine
of these angles, namely, cos a, cos ß and cos ? are called direction cosines of the
directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements,
i.e., ,  and . Thus, the signs of the direction cosines are reversed.
Chapter 11
THREE DIMENSIONAL GEOMETRY
* For various activities in three dimensional geometry, one may refer to the Book
“A Hand Book for designing Mathematics Laboratory in Schools”, NCERT, 2005
Leonhard Euler
(1707-1783)
Rationalised 2023-24
 MATHEMATICS 378
Note that a given line in space can be extended in two opposite directions and so it
has two sets of direction cosines. In order to have a unique set of direction cosines for
a given line in space, we must take the given line as a directed line. These unique
direction cosines are denoted by l, m and n.
Remark If the given line in space does not pass through the origin, then, in order to find
its direction cosines, we draw a line through the origin and parallel to the given line.
Now take one of the directed lines from the origin and find its direction cosines as two
parallel line have same set of direction cosines.
Any three numbers which are proportional to the direction cosines of a line are
called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are
direction ratios of a line, then a = ?l, b=?m and c = ?n, for any nonzero ? ? R.
A
Note   Some authors also call direction ratios as direction numbers.
Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines
(d.c’s) of the line. Then
l
a
 = 
m
b
 =
n
k
c
=
 (say), k being a constant.
Therefore l = ak, m = bk, n = ck ... (1)
But l
2
 + m
2
 + n
2
 = 1
Therefore k
2
 (a
2
 + b
2
 + c
2
) = 1
or k =
2 2 2
1
a b c
±
+ +
Fig 11.1
Rationalised 2023-24
Page 3


THREE DIMENSIONAL GEOMETRY 377
v The moving power of mathematical invention is not
reasoning but imagination. – A.DEMORGAN v
11.1  Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional geometry. The purpose of this
approach to 3-dimensional geometry is that it makes the
study simple and elegant*.
In this chapter, we shall study the direction cosines
and direction ratios of a line joining two points and also
discuss about the equations of lines and planes in space
under different conditions, angle between two lines, two
planes, a line and a plane, shortest distance between two
skew lines and distance of a point from a plane. Most of
the above results are obtained in vector form. Nevertheless, we shall also translate
these results in the Cartesian form which, at times, presents a more clear geometric
and analytic picture of the situation.
11.2  Direction Cosines and Direction Ratios of a Line
From Chapter 10, recall that if a directed line L passing through the origin makes
angles a, ß and ? with x, y and z-axes, respectively, called direction angles, then cosine
of these angles, namely, cos a, cos ß and cos ? are called direction cosines of the
directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements,
i.e., ,  and . Thus, the signs of the direction cosines are reversed.
Chapter 11
THREE DIMENSIONAL GEOMETRY
* For various activities in three dimensional geometry, one may refer to the Book
“A Hand Book for designing Mathematics Laboratory in Schools”, NCERT, 2005
Leonhard Euler
(1707-1783)
Rationalised 2023-24
 MATHEMATICS 378
Note that a given line in space can be extended in two opposite directions and so it
has two sets of direction cosines. In order to have a unique set of direction cosines for
a given line in space, we must take the given line as a directed line. These unique
direction cosines are denoted by l, m and n.
Remark If the given line in space does not pass through the origin, then, in order to find
its direction cosines, we draw a line through the origin and parallel to the given line.
Now take one of the directed lines from the origin and find its direction cosines as two
parallel line have same set of direction cosines.
Any three numbers which are proportional to the direction cosines of a line are
called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are
direction ratios of a line, then a = ?l, b=?m and c = ?n, for any nonzero ? ? R.
A
Note   Some authors also call direction ratios as direction numbers.
Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines
(d.c’s) of the line. Then
l
a
 = 
m
b
 =
n
k
c
=
 (say), k being a constant.
Therefore l = ak, m = bk, n = ck ... (1)
But l
2
 + m
2
 + n
2
 = 1
Therefore k
2
 (a
2
 + b
2
 + c
2
) = 1
or k =
2 2 2
1
a b c
±
+ +
Fig 11.1
Rationalised 2023-24
THREE DIMENSIONAL GEOMETRY 379
Hence, from (1), the d.c.’s of the line are
2 2 2 2 2 2 2 2 2
, ,
a b c
l m n
a b c a b c a b c
= ± = ± = ±
+ + + + + +
where, depending on the desired sign of k, either a positive or a negative sign is to be
taken for l, m and n.
For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k ? 0 is also a
set of direction ratios. So, any two sets of direction ratios of a line are also proportional.
Also, for any line there are infinitely many sets of direction ratios.
11.2.1  Direction cosines of a line passing through two points
Since one and only one line passes through two given points, we can determine the
direction cosines of a line passing through the given points P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
)
as follows (Fig 11.2 (a)).
Fig 11.2
Let l, m, n be the direction cosines of the line PQ and let it makes angles a, ß and  ?
with the x, y and z-axis, respectively.
Draw perpendiculars from P and Q to XY-plane to meet at R and S. Draw a
perpendicular from P to QS to meet at N. Now, in right angle triangle PNQ, ?PQN= ?
(Fig 11.2 (b).
Therefore, cos ? =
2 1
NQ
PQ PQ
z z -
=
Similarly cosa =
2 1 2 1
and cos
PQ PQ
x x y y - -
ß=
Hence, the direction cosines of the line segment joining the points P(x
1
, y
1
, z
1
) and
Q(x
2
, y
2
, z
2
) are
Rationalised 2023-24
Page 4


THREE DIMENSIONAL GEOMETRY 377
v The moving power of mathematical invention is not
reasoning but imagination. – A.DEMORGAN v
11.1  Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional geometry. The purpose of this
approach to 3-dimensional geometry is that it makes the
study simple and elegant*.
In this chapter, we shall study the direction cosines
and direction ratios of a line joining two points and also
discuss about the equations of lines and planes in space
under different conditions, angle between two lines, two
planes, a line and a plane, shortest distance between two
skew lines and distance of a point from a plane. Most of
the above results are obtained in vector form. Nevertheless, we shall also translate
these results in the Cartesian form which, at times, presents a more clear geometric
and analytic picture of the situation.
11.2  Direction Cosines and Direction Ratios of a Line
From Chapter 10, recall that if a directed line L passing through the origin makes
angles a, ß and ? with x, y and z-axes, respectively, called direction angles, then cosine
of these angles, namely, cos a, cos ß and cos ? are called direction cosines of the
directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements,
i.e., ,  and . Thus, the signs of the direction cosines are reversed.
Chapter 11
THREE DIMENSIONAL GEOMETRY
* For various activities in three dimensional geometry, one may refer to the Book
“A Hand Book for designing Mathematics Laboratory in Schools”, NCERT, 2005
Leonhard Euler
(1707-1783)
Rationalised 2023-24
 MATHEMATICS 378
Note that a given line in space can be extended in two opposite directions and so it
has two sets of direction cosines. In order to have a unique set of direction cosines for
a given line in space, we must take the given line as a directed line. These unique
direction cosines are denoted by l, m and n.
Remark If the given line in space does not pass through the origin, then, in order to find
its direction cosines, we draw a line through the origin and parallel to the given line.
Now take one of the directed lines from the origin and find its direction cosines as two
parallel line have same set of direction cosines.
Any three numbers which are proportional to the direction cosines of a line are
called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are
direction ratios of a line, then a = ?l, b=?m and c = ?n, for any nonzero ? ? R.
A
Note   Some authors also call direction ratios as direction numbers.
Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines
(d.c’s) of the line. Then
l
a
 = 
m
b
 =
n
k
c
=
 (say), k being a constant.
Therefore l = ak, m = bk, n = ck ... (1)
But l
2
 + m
2
 + n
2
 = 1
Therefore k
2
 (a
2
 + b
2
 + c
2
) = 1
or k =
2 2 2
1
a b c
±
+ +
Fig 11.1
Rationalised 2023-24
THREE DIMENSIONAL GEOMETRY 379
Hence, from (1), the d.c.’s of the line are
2 2 2 2 2 2 2 2 2
, ,
a b c
l m n
a b c a b c a b c
= ± = ± = ±
+ + + + + +
where, depending on the desired sign of k, either a positive or a negative sign is to be
taken for l, m and n.
For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k ? 0 is also a
set of direction ratios. So, any two sets of direction ratios of a line are also proportional.
Also, for any line there are infinitely many sets of direction ratios.
11.2.1  Direction cosines of a line passing through two points
Since one and only one line passes through two given points, we can determine the
direction cosines of a line passing through the given points P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
)
as follows (Fig 11.2 (a)).
Fig 11.2
Let l, m, n be the direction cosines of the line PQ and let it makes angles a, ß and  ?
with the x, y and z-axis, respectively.
Draw perpendiculars from P and Q to XY-plane to meet at R and S. Draw a
perpendicular from P to QS to meet at N. Now, in right angle triangle PNQ, ?PQN= ?
(Fig 11.2 (b).
Therefore, cos ? =
2 1
NQ
PQ PQ
z z -
=
Similarly cosa =
2 1 2 1
and cos
PQ PQ
x x y y - -
ß=
Hence, the direction cosines of the line segment joining the points P(x
1
, y
1
, z
1
) and
Q(x
2
, y
2
, z
2
) are
Rationalised 2023-24
 MATHEMATICS 380
2 1
PQ
x x -
, 
2 1
PQ
y y -
, 
2 1
PQ
z z -
where PQ =
( )
2
2 2
2 1 2 1 2 1
( ) ( ) x x y y z z - + - + -
A
Note   The direction ratios of the line segment joining P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
)
may be taken as
x
2 
– x
1
, y
2 
– y
1
, z
2 
– z
1
 or x
1 
– x
2
, y
1 
– y
2
, z
1 
– z
2
Example 1 If a line makes angle 90°, 60° and 30° with the positive direction of x, y and
z-axis respectively, find its direction cosines.
Solution Let the d.c.'s of the lines be l , m, n. Then l = cos 90
0
 = 0, m = cos 60
0
 = 
1
2
,
n = cos 30
0
 = 
2
3
.
Example 2 If a line has direction ratios 2, – 1, – 2, determine its direction cosines.
Solution Direction cosines are
2 2 2
) 2 ( ) 1 ( 2
2
- + - +
,  
2 2 2
) 2 ( ) 1 ( 2
1
- + - +
-
,  
( )
2 2 2
) 2 ( 1 2
2
- + - +
-
or 
2 1 2
, ,
3 3 3
- -
Example 3 Find the direction cosines of the line passing through  the two points
(– 2, 4, – 5) and (1, 2, 3).
Solution We know the direction cosines of the line passing through  two points
P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) are given by
2 1 2 1 2 1
, ,
PQ PQ PQ
x x y y z z - - -
where PQ =
( )
2
1 2
2
1 2
2
1 2
) ( ) ( z z y y x x - + - + -
Here P is (– 2, 4, – 5) and Q is (1, 2, 3).
So PQ =
2 2 2
(1 ( 2)) (2 4) (3 ( 5)) - - + - + - - = 
77
Thus, the direction cosines of the line joining two points is
3 2 8
, ,
77 77 77
-
Rationalised 2023-24
Page 5


THREE DIMENSIONAL GEOMETRY 377
v The moving power of mathematical invention is not
reasoning but imagination. – A.DEMORGAN v
11.1  Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional geometry. The purpose of this
approach to 3-dimensional geometry is that it makes the
study simple and elegant*.
In this chapter, we shall study the direction cosines
and direction ratios of a line joining two points and also
discuss about the equations of lines and planes in space
under different conditions, angle between two lines, two
planes, a line and a plane, shortest distance between two
skew lines and distance of a point from a plane. Most of
the above results are obtained in vector form. Nevertheless, we shall also translate
these results in the Cartesian form which, at times, presents a more clear geometric
and analytic picture of the situation.
11.2  Direction Cosines and Direction Ratios of a Line
From Chapter 10, recall that if a directed line L passing through the origin makes
angles a, ß and ? with x, y and z-axes, respectively, called direction angles, then cosine
of these angles, namely, cos a, cos ß and cos ? are called direction cosines of the
directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements,
i.e., ,  and . Thus, the signs of the direction cosines are reversed.
Chapter 11
THREE DIMENSIONAL GEOMETRY
* For various activities in three dimensional geometry, one may refer to the Book
“A Hand Book for designing Mathematics Laboratory in Schools”, NCERT, 2005
Leonhard Euler
(1707-1783)
Rationalised 2023-24
 MATHEMATICS 378
Note that a given line in space can be extended in two opposite directions and so it
has two sets of direction cosines. In order to have a unique set of direction cosines for
a given line in space, we must take the given line as a directed line. These unique
direction cosines are denoted by l, m and n.
Remark If the given line in space does not pass through the origin, then, in order to find
its direction cosines, we draw a line through the origin and parallel to the given line.
Now take one of the directed lines from the origin and find its direction cosines as two
parallel line have same set of direction cosines.
Any three numbers which are proportional to the direction cosines of a line are
called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are
direction ratios of a line, then a = ?l, b=?m and c = ?n, for any nonzero ? ? R.
A
Note   Some authors also call direction ratios as direction numbers.
Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines
(d.c’s) of the line. Then
l
a
 = 
m
b
 =
n
k
c
=
 (say), k being a constant.
Therefore l = ak, m = bk, n = ck ... (1)
But l
2
 + m
2
 + n
2
 = 1
Therefore k
2
 (a
2
 + b
2
 + c
2
) = 1
or k =
2 2 2
1
a b c
±
+ +
Fig 11.1
Rationalised 2023-24
THREE DIMENSIONAL GEOMETRY 379
Hence, from (1), the d.c.’s of the line are
2 2 2 2 2 2 2 2 2
, ,
a b c
l m n
a b c a b c a b c
= ± = ± = ±
+ + + + + +
where, depending on the desired sign of k, either a positive or a negative sign is to be
taken for l, m and n.
For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k ? 0 is also a
set of direction ratios. So, any two sets of direction ratios of a line are also proportional.
Also, for any line there are infinitely many sets of direction ratios.
11.2.1  Direction cosines of a line passing through two points
Since one and only one line passes through two given points, we can determine the
direction cosines of a line passing through the given points P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
)
as follows (Fig 11.2 (a)).
Fig 11.2
Let l, m, n be the direction cosines of the line PQ and let it makes angles a, ß and  ?
with the x, y and z-axis, respectively.
Draw perpendiculars from P and Q to XY-plane to meet at R and S. Draw a
perpendicular from P to QS to meet at N. Now, in right angle triangle PNQ, ?PQN= ?
(Fig 11.2 (b).
Therefore, cos ? =
2 1
NQ
PQ PQ
z z -
=
Similarly cosa =
2 1 2 1
and cos
PQ PQ
x x y y - -
ß=
Hence, the direction cosines of the line segment joining the points P(x
1
, y
1
, z
1
) and
Q(x
2
, y
2
, z
2
) are
Rationalised 2023-24
 MATHEMATICS 380
2 1
PQ
x x -
, 
2 1
PQ
y y -
, 
2 1
PQ
z z -
where PQ =
( )
2
2 2
2 1 2 1 2 1
( ) ( ) x x y y z z - + - + -
A
Note   The direction ratios of the line segment joining P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
)
may be taken as
x
2 
– x
1
, y
2 
– y
1
, z
2 
– z
1
 or x
1 
– x
2
, y
1 
– y
2
, z
1 
– z
2
Example 1 If a line makes angle 90°, 60° and 30° with the positive direction of x, y and
z-axis respectively, find its direction cosines.
Solution Let the d.c.'s of the lines be l , m, n. Then l = cos 90
0
 = 0, m = cos 60
0
 = 
1
2
,
n = cos 30
0
 = 
2
3
.
Example 2 If a line has direction ratios 2, – 1, – 2, determine its direction cosines.
Solution Direction cosines are
2 2 2
) 2 ( ) 1 ( 2
2
- + - +
,  
2 2 2
) 2 ( ) 1 ( 2
1
- + - +
-
,  
( )
2 2 2
) 2 ( 1 2
2
- + - +
-
or 
2 1 2
, ,
3 3 3
- -
Example 3 Find the direction cosines of the line passing through  the two points
(– 2, 4, – 5) and (1, 2, 3).
Solution We know the direction cosines of the line passing through  two points
P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) are given by
2 1 2 1 2 1
, ,
PQ PQ PQ
x x y y z z - - -
where PQ =
( )
2
1 2
2
1 2
2
1 2
) ( ) ( z z y y x x - + - + -
Here P is (– 2, 4, – 5) and Q is (1, 2, 3).
So PQ =
2 2 2
(1 ( 2)) (2 4) (3 ( 5)) - - + - + - - = 
77
Thus, the direction cosines of the line joining two points is
3 2 8
, ,
77 77 77
-
Rationalised 2023-24
THREE DIMENSIONAL GEOMETRY 381
Example 4 Find the direction cosines of x, y and z-axis.
Solution The x-axis makes angles 0°, 90° and 90° respectively with x, y and z-axis.
Therefore, the direction cosines of x-axis are cos 0°, cos 90°, cos 90° i.e., 1,0,0.
Similarly, direction cosines of y-axis and z-axis are 0, 1, 0 and 0, 0, 1 respectively.
Example 5 Show that the points A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11) are
collinear.
Solution Direction ratios of line joining A and B are
1 – 2, – 2 – 3, 3 + 4 i.e., – 1, – 5, 7.
The direction ratios of line joining B and C are
3 –1, 8 + 2, – 11 – 3, i.e., 2, 10, – 14.
It is clear that direction ratios of AB and BC are proportional, hence, AB is parallel
to BC. But point B is common to both AB and BC. Therefore, A, B, C are
collinear points.
EXERCISE 11.1
1. If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its
direction cosines.
2. Find the direction cosines of a line which makes equal angles with the coordinate
axes.
3. If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?
4. Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) 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).
11.3 Equation of a Line in Space
We have studied equation of  lines in two dimensions in Class XI, we shall now study
the vector and cartesian equations of a line in space.
A line is uniquely determined if
(i) it passes through a given point and has given direction, or
(ii) it passes through two given points.
11.3.1 Equation of a line through a given point and parallel to  given vector 
Let  be the position vector of the given point A with respect to the origin O of the
rectangular coordinate system. Let l be the line which passes through the point A and
is parallel to a given vector . Let   be the position vector of an arbitrary point P on the
line (Fig 11.3).
Rationalised 2023-24