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1 | 6118-6121 | e ˆ
ˆ
x i
+y j
=
ˆ
ˆ
ˆ
(3
2
)
( 4
3 )
(1
5
)
i
j
k
+
λ
+
−
λ
+
+
λ
Equating the like coefficients of
ˆ
ˆ
,ˆ
and
i
j
k , we have
x = 3 + 2 λ
y = 4 – 3 λ
0 = 1 + 5 λ
Solving the above equations, we get
x = 13
23
and
5
5
y =
Hence, the coordinates of the required point are
5,0
135,23 Miscellaneous Exercise on Chapter 11
1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the
line determined by the points (3, 5, – 1), (4, 3, – 1) |
1 | 6119-6122 | ˆ
ˆ
x i
+y j
=
ˆ
ˆ
ˆ
(3
2
)
( 4
3 )
(1
5
)
i
j
k
+
λ
+
−
λ
+
+
λ
Equating the like coefficients of
ˆ
ˆ
,ˆ
and
i
j
k , we have
x = 3 + 2 λ
y = 4 – 3 λ
0 = 1 + 5 λ
Solving the above equations, we get
x = 13
23
and
5
5
y =
Hence, the coordinates of the required point are
5,0
135,23 Miscellaneous Exercise on Chapter 11
1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the
line determined by the points (3, 5, – 1), (4, 3, – 1) 2 |
1 | 6120-6123 | Miscellaneous Exercise on Chapter 11
1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the
line determined by the points (3, 5, – 1), (4, 3, – 1) 2 If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular
lines, show that the direction cosines of the line perpendicular to both of these
are
1
2
2
1
1
2
2
1
1
2
2
1
,
,
m n
m n
n l
n l
l m
l m
−
−
−
© NCERT
not to be republished
MATHEMATI CS
498
3 |
1 | 6121-6124 | Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the
line determined by the points (3, 5, – 1), (4, 3, – 1) 2 If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular
lines, show that the direction cosines of the line perpendicular to both of these
are
1
2
2
1
1
2
2
1
1
2
2
1
,
,
m n
m n
n l
n l
l m
l m
−
−
−
© NCERT
not to be republished
MATHEMATI CS
498
3 Find the angle between the lines whose direction ratios are a, b, c and
b – c, c – a, a – b |
1 | 6122-6125 | 2 If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular
lines, show that the direction cosines of the line perpendicular to both of these
are
1
2
2
1
1
2
2
1
1
2
2
1
,
,
m n
m n
n l
n l
l m
l m
−
−
−
© NCERT
not to be republished
MATHEMATI CS
498
3 Find the angle between the lines whose direction ratios are a, b, c and
b – c, c – a, a – b 4 |
1 | 6123-6126 | If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two mutually perpendicular
lines, show that the direction cosines of the line perpendicular to both of these
are
1
2
2
1
1
2
2
1
1
2
2
1
,
,
m n
m n
n l
n l
l m
l m
−
−
−
© NCERT
not to be republished
MATHEMATI CS
498
3 Find the angle between the lines whose direction ratios are a, b, c and
b – c, c – a, a – b 4 Find the equation of a line parallel to x-axis and passing through the origin |
1 | 6124-6127 | Find the angle between the lines whose direction ratios are a, b, c and
b – c, c – a, a – b 4 Find the equation of a line parallel to x-axis and passing through the origin 5 |
1 | 6125-6128 | 4 Find the equation of a line parallel to x-axis and passing through the origin 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and
(2, 9, 2) respectively, then find the angle between the lines AB and CD |
1 | 6126-6129 | Find the equation of a line parallel to x-axis and passing through the origin 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and
(2, 9, 2) respectively, then find the angle between the lines AB and CD 6 |
1 | 6127-6130 | 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and
(2, 9, 2) respectively, then find the angle between the lines AB and CD 6 If the lines
1
2
3
1
1
6
and
3
2
2
3
1
5
x
y
z
x
y
z
k
k
−
−
−
−
−
−
=
=
=
=
−
−
are perpendicular,,
find the value of k |
1 | 6128-6131 | If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and
(2, 9, 2) respectively, then find the angle between the lines AB and CD 6 If the lines
1
2
3
1
1
6
and
3
2
2
3
1
5
x
y
z
x
y
z
k
k
−
−
−
−
−
−
=
=
=
=
−
−
are perpendicular,,
find the value of k 7 |
1 | 6129-6132 | 6 If the lines
1
2
3
1
1
6
and
3
2
2
3
1
5
x
y
z
x
y
z
k
k
−
−
−
−
−
−
=
=
=
=
−
−
are perpendicular,,
find the value of k 7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to
the plane
0
9
)ˆ
5
2ˆ
ˆ
( |
1 | 6130-6133 | If the lines
1
2
3
1
1
6
and
3
2
2
3
1
5
x
y
z
x
y
z
k
k
−
−
−
−
−
−
=
=
=
=
−
−
are perpendicular,,
find the value of k 7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to
the plane
0
9
)ˆ
5
2ˆ
ˆ
( =
+
−
+
k
j
i
rr |
1 | 6131-6134 | 7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to
the plane
0
9
)ˆ
5
2ˆ
ˆ
( =
+
−
+
k
j
i
rr 8 |
1 | 6132-6135 | Find the vector equation of the line passing through (1, 2, 3) and perpendicular to
the plane
0
9
)ˆ
5
2ˆ
ˆ
( =
+
−
+
k
j
i
rr 8 Find the equation of the plane passing through (a, b, c) and parallel to the plane
ˆ
ˆ
ˆ
(
)
2 |
1 | 6133-6136 | =
+
−
+
k
j
i
rr 8 Find the equation of the plane passing through (a, b, c) and parallel to the plane
ˆ
ˆ
ˆ
(
)
2 r
i
j
k
⋅ + + =
r
9 |
1 | 6134-6137 | 8 Find the equation of the plane passing through (a, b, c) and parallel to the plane
ˆ
ˆ
ˆ
(
)
2 r
i
j
k
⋅ + + =
r
9 Find the shortest distance between lines
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
6
2
2
(
2
2
)
r
i
j
k
i
j
k
=
+
+
+ λ
−
+
r
and
ˆ
ˆ
ˆ
ˆ
ˆ
4
(3
2
2
)
r
i
k
i
j
k
= −
−
+ µ
−
−
r |
1 | 6135-6138 | Find the equation of the plane passing through (a, b, c) and parallel to the plane
ˆ
ˆ
ˆ
(
)
2 r
i
j
k
⋅ + + =
r
9 Find the shortest distance between lines
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
6
2
2
(
2
2
)
r
i
j
k
i
j
k
=
+
+
+ λ
−
+
r
and
ˆ
ˆ
ˆ
ˆ
ˆ
4
(3
2
2
)
r
i
k
i
j
k
= −
−
+ µ
−
−
r 10 |
1 | 6136-6139 | r
i
j
k
⋅ + + =
r
9 Find the shortest distance between lines
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
6
2
2
(
2
2
)
r
i
j
k
i
j
k
=
+
+
+ λ
−
+
r
and
ˆ
ˆ
ˆ
ˆ
ˆ
4
(3
2
2
)
r
i
k
i
j
k
= −
−
+ µ
−
−
r 10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)
crosses the YZ-plane |
1 | 6137-6140 | Find the shortest distance between lines
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
6
2
2
(
2
2
)
r
i
j
k
i
j
k
=
+
+
+ λ
−
+
r
and
ˆ
ˆ
ˆ
ˆ
ˆ
4
(3
2
2
)
r
i
k
i
j
k
= −
−
+ µ
−
−
r 10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)
crosses the YZ-plane 11 |
1 | 6138-6141 | 10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)
crosses the YZ-plane 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)
crosses the ZX-plane |
1 | 6139-6142 | Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1)
crosses the YZ-plane 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)
crosses the ZX-plane 12 |
1 | 6140-6143 | 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)
crosses the ZX-plane 12 Find the coordinates of the point where the line through (3, – 4, – 5) and
(2, – 3, 1) crosses the plane 2x + y + z = 7 |
1 | 6141-6144 | Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)
crosses the ZX-plane 12 Find the coordinates of the point where the line through (3, – 4, – 5) and
(2, – 3, 1) crosses the plane 2x + y + z = 7 13 |
1 | 6142-6145 | 12 Find the coordinates of the point where the line through (3, – 4, – 5) and
(2, – 3, 1) crosses the plane 2x + y + z = 7 13 Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular
to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 |
1 | 6143-6146 | Find the coordinates of the point where the line through (3, – 4, – 5) and
(2, – 3, 1) crosses the plane 2x + y + z = 7 13 Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular
to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 14 |
1 | 6144-6147 | 13 Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular
to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 14 If the points (1, 1, p) and (– 3, 0, 1) be equidist ant from the plane
ˆ
ˆ
ˆ
(3
4
12 )
13
0,
⋅ + −
+ =
rr
i
j
k
then find the value of p |
1 | 6145-6148 | Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular
to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 14 If the points (1, 1, p) and (– 3, 0, 1) be equidist ant from the plane
ˆ
ˆ
ˆ
(3
4
12 )
13
0,
⋅ + −
+ =
rr
i
j
k
then find the value of p 15 |
1 | 6146-6149 | 14 If the points (1, 1, p) and (– 3, 0, 1) be equidist ant from the plane
ˆ
ˆ
ˆ
(3
4
12 )
13
0,
⋅ + −
+ =
rr
i
j
k
then find the value of p 15 Find the equation of the plane passing through the line of intersection of the
planes
ˆ
ˆ
ˆ
(
)
1
r
i
j
k
⋅
+
+
=
r
and
ˆ
ˆ
ˆ
(2
3
)
4
0
r
i
j
k
⋅
+
−
+
=
r
and parallel to x-axis |
1 | 6147-6150 | If the points (1, 1, p) and (– 3, 0, 1) be equidist ant from the plane
ˆ
ˆ
ˆ
(3
4
12 )
13
0,
⋅ + −
+ =
rr
i
j
k
then find the value of p 15 Find the equation of the plane passing through the line of intersection of the
planes
ˆ
ˆ
ˆ
(
)
1
r
i
j
k
⋅
+
+
=
r
and
ˆ
ˆ
ˆ
(2
3
)
4
0
r
i
j
k
⋅
+
−
+
=
r
and parallel to x-axis 16 |
1 | 6148-6151 | 15 Find the equation of the plane passing through the line of intersection of the
planes
ˆ
ˆ
ˆ
(
)
1
r
i
j
k
⋅
+
+
=
r
and
ˆ
ˆ
ˆ
(2
3
)
4
0
r
i
j
k
⋅
+
−
+
=
r
and parallel to x-axis 16 If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of
the plane passing through P and perpendicular to OP |
1 | 6149-6152 | Find the equation of the plane passing through the line of intersection of the
planes
ˆ
ˆ
ˆ
(
)
1
r
i
j
k
⋅
+
+
=
r
and
ˆ
ˆ
ˆ
(2
3
)
4
0
r
i
j
k
⋅
+
−
+
=
r
and parallel to x-axis 16 If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of
the plane passing through P and perpendicular to OP 17 |
1 | 6150-6153 | 16 If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of
the plane passing through P and perpendicular to OP 17 Find the equation of the plane which contains the line of intersection of the planes
ˆ
ˆ
ˆ
(
2
3 )
4
0
r
i
j
k
⋅
+
+
−
=
r
,
ˆ
ˆ
ˆ
(2
)
5
0
r
i
j
k
⋅
+
−
+
=
r
and which is perpendicular to the
plane
ˆ
ˆ
ˆ
(5
3
6 )
8
0
r
i
j
k
⋅
+
−
+
=
r |
1 | 6151-6154 | If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of
the plane passing through P and perpendicular to OP 17 Find the equation of the plane which contains the line of intersection of the planes
ˆ
ˆ
ˆ
(
2
3 )
4
0
r
i
j
k
⋅
+
+
−
=
r
,
ˆ
ˆ
ˆ
(2
)
5
0
r
i
j
k
⋅
+
−
+
=
r
and which is perpendicular to the
plane
ˆ
ˆ
ˆ
(5
3
6 )
8
0
r
i
j
k
⋅
+
−
+
=
r © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
499
18 |
1 | 6152-6155 | 17 Find the equation of the plane which contains the line of intersection of the planes
ˆ
ˆ
ˆ
(
2
3 )
4
0
r
i
j
k
⋅
+
+
−
=
r
,
ˆ
ˆ
ˆ
(2
)
5
0
r
i
j
k
⋅
+
−
+
=
r
and which is perpendicular to the
plane
ˆ
ˆ
ˆ
(5
3
6 )
8
0
r
i
j
k
⋅
+
−
+
=
r © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
499
18 Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the
line
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
2
(3
4
2
)
r
i
j
k
i
j
k
=
−
+
+ λ
+
+
r
and the plane
ˆ
ˆ
ˆ
(
)
5
r
i
j
k
⋅
−
+
=
r |
1 | 6153-6156 | Find the equation of the plane which contains the line of intersection of the planes
ˆ
ˆ
ˆ
(
2
3 )
4
0
r
i
j
k
⋅
+
+
−
=
r
,
ˆ
ˆ
ˆ
(2
)
5
0
r
i
j
k
⋅
+
−
+
=
r
and which is perpendicular to the
plane
ˆ
ˆ
ˆ
(5
3
6 )
8
0
r
i
j
k
⋅
+
−
+
=
r © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
499
18 Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the
line
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
2
(3
4
2
)
r
i
j
k
i
j
k
=
−
+
+ λ
+
+
r
and the plane
ˆ
ˆ
ˆ
(
)
5
r
i
j
k
⋅
−
+
=
r 19 |
1 | 6154-6157 | © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
499
18 Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the
line
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
2
(3
4
2
)
r
i
j
k
i
j
k
=
−
+
+ λ
+
+
r
and the plane
ˆ
ˆ
ˆ
(
)
5
r
i
j
k
⋅
−
+
=
r 19 Find the vector equation of the line passing through (1, 2, 3) and parallel to the
planes
ˆ
ˆ
ˆ
(
2 )
5
r
i
j
k
⋅
−
+
=
r
and
ˆ
ˆ
ˆ
(3
)
6
r
i
j
k
⋅
+
+
=
r |
1 | 6155-6158 | Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the
line
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
2
(3
4
2
)
r
i
j
k
i
j
k
=
−
+
+ λ
+
+
r
and the plane
ˆ
ˆ
ˆ
(
)
5
r
i
j
k
⋅
−
+
=
r 19 Find the vector equation of the line passing through (1, 2, 3) and parallel to the
planes
ˆ
ˆ
ˆ
(
2 )
5
r
i
j
k
⋅
−
+
=
r
and
ˆ
ˆ
ˆ
(3
)
6
r
i
j
k
⋅
+
+
=
r 20 |
1 | 6156-6159 | 19 Find the vector equation of the line passing through (1, 2, 3) and parallel to the
planes
ˆ
ˆ
ˆ
(
2 )
5
r
i
j
k
⋅
−
+
=
r
and
ˆ
ˆ
ˆ
(3
)
6
r
i
j
k
⋅
+
+
=
r 20 Find the vector equation of the line passing through the point (1, 2, – 4) and
perpendicular to the two lines:
7
10
16
19
3
8
−
=
−
+
=
−
z
y
x
and
15
3
x −
=
29
5
8
5
y
z
−
−
=
− |
1 | 6157-6160 | Find the vector equation of the line passing through (1, 2, 3) and parallel to the
planes
ˆ
ˆ
ˆ
(
2 )
5
r
i
j
k
⋅
−
+
=
r
and
ˆ
ˆ
ˆ
(3
)
6
r
i
j
k
⋅
+
+
=
r 20 Find the vector equation of the line passing through the point (1, 2, – 4) and
perpendicular to the two lines:
7
10
16
19
3
8
−
=
−
+
=
−
z
y
x
and
15
3
x −
=
29
5
8
5
y
z
−
−
=
− 21 |
1 | 6158-6161 | 20 Find the vector equation of the line passing through the point (1, 2, – 4) and
perpendicular to the two lines:
7
10
16
19
3
8
−
=
−
+
=
−
z
y
x
and
15
3
x −
=
29
5
8
5
y
z
−
−
=
− 21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from
the origin, then
2
2
2
2
1
1
1
1
p
c
b
a
=
+
+ |
1 | 6159-6162 | Find the vector equation of the line passing through the point (1, 2, – 4) and
perpendicular to the two lines:
7
10
16
19
3
8
−
=
−
+
=
−
z
y
x
and
15
3
x −
=
29
5
8
5
y
z
−
−
=
− 21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from
the origin, then
2
2
2
2
1
1
1
1
p
c
b
a
=
+
+ Choose the correct answer in Exercises 22 and 23 |
1 | 6160-6163 | 21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from
the origin, then
2
2
2
2
1
1
1
1
p
c
b
a
=
+
+ Choose the correct answer in Exercises 22 and 23 22 |
1 | 6161-6164 | Prove that if a plane has the intercepts a, b, c and is at a distance of p units from
the origin, then
2
2
2
2
1
1
1
1
p
c
b
a
=
+
+ Choose the correct answer in Exercises 22 and 23 22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units
(B) 4 units
(C) 8 units
(D)
2
29
units
23 |
1 | 6162-6165 | Choose the correct answer in Exercises 22 and 23 22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units
(B) 4 units
(C) 8 units
(D)
2
29
units
23 The planes: 2x – y + 4z = 5 and 5x – 2 |
1 | 6163-6166 | 22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units
(B) 4 units
(C) 8 units
(D)
2
29
units
23 The planes: 2x – y + 4z = 5 and 5x – 2 5y + 10z = 6 are
(A) Perpendicular
(B) Parallel
(C) intersect y-axis
(D) passes through
5
0,0,
4
Summary
� Direction cosines of a line are the cosines of the angles made by the line
with the positive directions of the coordinate axes |
1 | 6164-6167 | Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units
(B) 4 units
(C) 8 units
(D)
2
29
units
23 The planes: 2x – y + 4z = 5 and 5x – 2 5y + 10z = 6 are
(A) Perpendicular
(B) Parallel
(C) intersect y-axis
(D) passes through
5
0,0,
4
Summary
� Direction cosines of a line are the cosines of the angles made by the line
with the positive directions of the coordinate axes � If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 |
1 | 6165-6168 | The planes: 2x – y + 4z = 5 and 5x – 2 5y + 10z = 6 are
(A) Perpendicular
(B) Parallel
(C) intersect y-axis
(D) passes through
5
0,0,
4
Summary
� Direction cosines of a line are the cosines of the angles made by the line
with the positive directions of the coordinate axes � If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 � Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are
2
1
2
1
2
1
,
,
PQ
PQ
PQ
x
x
y
y
z
z
−
−
−
where PQ =
(
)
12
2
2
1
2
2
1
2
)
(
)
(
z
z
y
y
x
x
−
+
−
+
−
� Direction ratios of a line are the numbers which are proportional to the
direction cosines of a line |
1 | 6166-6169 | 5y + 10z = 6 are
(A) Perpendicular
(B) Parallel
(C) intersect y-axis
(D) passes through
5
0,0,
4
Summary
� Direction cosines of a line are the cosines of the angles made by the line
with the positive directions of the coordinate axes � If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 � Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are
2
1
2
1
2
1
,
,
PQ
PQ
PQ
x
x
y
y
z
z
−
−
−
where PQ =
(
)
12
2
2
1
2
2
1
2
)
(
)
(
z
z
y
y
x
x
−
+
−
+
−
� Direction ratios of a line are the numbers which are proportional to the
direction cosines of a line � If l, m, n are the direction cosines and a, b, c are the direction ratios of a line
© NCERT
not to be republished
MATHEMATI CS
500
then
l =
2
2
2
c
b
a
a
+
+
; m =
2
2
2
c
b
a
b
+
+
; n =
2
2
2
c
b
a
c
+
+
� Skew lines are lines in space which are neither parallel nor intersecting |
1 | 6167-6170 | � If l, m, n are the direction cosines of a line, then l2 + m 2 + n2 = 1 � Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are
2
1
2
1
2
1
,
,
PQ
PQ
PQ
x
x
y
y
z
z
−
−
−
where PQ =
(
)
12
2
2
1
2
2
1
2
)
(
)
(
z
z
y
y
x
x
−
+
−
+
−
� Direction ratios of a line are the numbers which are proportional to the
direction cosines of a line � If l, m, n are the direction cosines and a, b, c are the direction ratios of a line
© NCERT
not to be republished
MATHEMATI CS
500
then
l =
2
2
2
c
b
a
a
+
+
; m =
2
2
2
c
b
a
b
+
+
; n =
2
2
2
c
b
a
c
+
+
� Skew lines are lines in space which are neither parallel nor intersecting They lie in different planes |
1 | 6168-6171 | � Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are
2
1
2
1
2
1
,
,
PQ
PQ
PQ
x
x
y
y
z
z
−
−
−
where PQ =
(
)
12
2
2
1
2
2
1
2
)
(
)
(
z
z
y
y
x
x
−
+
−
+
−
� Direction ratios of a line are the numbers which are proportional to the
direction cosines of a line � If l, m, n are the direction cosines and a, b, c are the direction ratios of a line
© NCERT
not to be republished
MATHEMATI CS
500
then
l =
2
2
2
c
b
a
a
+
+
; m =
2
2
2
c
b
a
b
+
+
; n =
2
2
2
c
b
a
c
+
+
� Skew lines are lines in space which are neither parallel nor intersecting They lie in different planes � Angle between skew lines is the angle between two intersecting lines
drawn from any point (preferably through the origin) parallel to each of the
skew lines |
1 | 6169-6172 | � If l, m, n are the direction cosines and a, b, c are the direction ratios of a line
© NCERT
not to be republished
MATHEMATI CS
500
then
l =
2
2
2
c
b
a
a
+
+
; m =
2
2
2
c
b
a
b
+
+
; n =
2
2
2
c
b
a
c
+
+
� Skew lines are lines in space which are neither parallel nor intersecting They lie in different planes � Angle between skew lines is the angle between two intersecting lines
drawn from any point (preferably through the origin) parallel to each of the
skew lines � If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and θ is the
acute angle between the two lines; then
cosθ = |l1l2 + m 1m 2 + n1n2|
� If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the
acute angle between the two lines; then
cosθ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
a a
b b
c c
a
b
c
a
b
c
+
+
+
+
+
+
� Vector equation of a line that passes through the given point whose position
vector is ar and parallel to a given vector b
r
is r
a
b
=
+ λ
r
r
r |
1 | 6170-6173 | They lie in different planes � Angle between skew lines is the angle between two intersecting lines
drawn from any point (preferably through the origin) parallel to each of the
skew lines � If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and θ is the
acute angle between the two lines; then
cosθ = |l1l2 + m 1m 2 + n1n2|
� If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the
acute angle between the two lines; then
cosθ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
a a
b b
c c
a
b
c
a
b
c
+
+
+
+
+
+
� Vector equation of a line that passes through the given point whose position
vector is ar and parallel to a given vector b
r
is r
a
b
=
+ λ
r
r
r � Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is
1
1
1
x
x
y
y
z
z
l
m
n
−
−
−
=
=
� The vector equation of a line which passes through two points whose position
vectors are ar and b
r
is
(
)
r
a
b
a
=
+ λ
−
r
r
r
r |
1 | 6171-6174 | � Angle between skew lines is the angle between two intersecting lines
drawn from any point (preferably through the origin) parallel to each of the
skew lines � If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and θ is the
acute angle between the two lines; then
cosθ = |l1l2 + m 1m 2 + n1n2|
� If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the
acute angle between the two lines; then
cosθ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
a a
b b
c c
a
b
c
a
b
c
+
+
+
+
+
+
� Vector equation of a line that passes through the given point whose position
vector is ar and parallel to a given vector b
r
is r
a
b
=
+ λ
r
r
r � Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is
1
1
1
x
x
y
y
z
z
l
m
n
−
−
−
=
=
� The vector equation of a line which passes through two points whose position
vectors are ar and b
r
is
(
)
r
a
b
a
=
+ λ
−
r
r
r
r � Cartesian equation of a line that passes through two points (x1, y1, z1) and
(x2, y2, z2) is
1
1
1
2
1
2
1
2
1
x
x
y
y
z
z
x
x
y
y
z
z
−
−
−
=
=
−
−
− |
1 | 6172-6175 | � If l1, m 1, n1 and l2, m 2, n2 are the direction cosines of two lines; and θ is the
acute angle between the two lines; then
cosθ = |l1l2 + m 1m 2 + n1n2|
� If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the
acute angle between the two lines; then
cosθ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
a a
b b
c c
a
b
c
a
b
c
+
+
+
+
+
+
� Vector equation of a line that passes through the given point whose position
vector is ar and parallel to a given vector b
r
is r
a
b
=
+ λ
r
r
r � Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is
1
1
1
x
x
y
y
z
z
l
m
n
−
−
−
=
=
� The vector equation of a line which passes through two points whose position
vectors are ar and b
r
is
(
)
r
a
b
a
=
+ λ
−
r
r
r
r � Cartesian equation of a line that passes through two points (x1, y1, z1) and
(x2, y2, z2) is
1
1
1
2
1
2
1
2
1
x
x
y
y
z
z
x
x
y
y
z
z
−
−
−
=
=
−
−
− � If θ is the acute angle between
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ λ
r
r
r
, then
1
2
1
2
cos
|
| |
|
bb b
b
⋅
θ =
r
r
r
r
� If
1
1
1
1
1
1
n
z
z
m
y
y
l
x
x
−
=
−
=
−
and
2
2
2
2
2
2
n
z
z
m
y
y
l
x
x
−
=
−
=
−
are the equations of two lines, then the acute angle between the two lines is
given by cos θ = |l1l2 + m 1m 2 + n1n2| |
1 | 6173-6176 | � Equation of a line through a point (x1, y1, z1) and having direction cosines l, m, n is
1
1
1
x
x
y
y
z
z
l
m
n
−
−
−
=
=
� The vector equation of a line which passes through two points whose position
vectors are ar and b
r
is
(
)
r
a
b
a
=
+ λ
−
r
r
r
r � Cartesian equation of a line that passes through two points (x1, y1, z1) and
(x2, y2, z2) is
1
1
1
2
1
2
1
2
1
x
x
y
y
z
z
x
x
y
y
z
z
−
−
−
=
=
−
−
− � If θ is the acute angle between
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ λ
r
r
r
, then
1
2
1
2
cos
|
| |
|
bb b
b
⋅
θ =
r
r
r
r
� If
1
1
1
1
1
1
n
z
z
m
y
y
l
x
x
−
=
−
=
−
and
2
2
2
2
2
2
n
z
z
m
y
y
l
x
x
−
=
−
=
−
are the equations of two lines, then the acute angle between the two lines is
given by cos θ = |l1l2 + m 1m 2 + n1n2| © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
501
� Shortest distance between two skew lines is the line segment perpendicular
to both the lines |
1 | 6174-6177 | � Cartesian equation of a line that passes through two points (x1, y1, z1) and
(x2, y2, z2) is
1
1
1
2
1
2
1
2
1
x
x
y
y
z
z
x
x
y
y
z
z
−
−
−
=
=
−
−
− � If θ is the acute angle between
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ λ
r
r
r
, then
1
2
1
2
cos
|
| |
|
bb b
b
⋅
θ =
r
r
r
r
� If
1
1
1
1
1
1
n
z
z
m
y
y
l
x
x
−
=
−
=
−
and
2
2
2
2
2
2
n
z
z
m
y
y
l
x
x
−
=
−
=
−
are the equations of two lines, then the acute angle between the two lines is
given by cos θ = |l1l2 + m 1m 2 + n1n2| © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
501
� Shortest distance between two skew lines is the line segment perpendicular
to both the lines � Shortest distance between
1
1
r
a
b
= +λ
r
r
r
and
2
2
r
a
b
= +µ
r
r
r
is
1
2
2
1
1
2
(
) (
–
)
|
|
b
b
a
a
b
b
×
⋅
×
r
r
r
r
r
r
� Shortest distance between the lines:
1
1
1
1
1
1
x
x
y
y
z
z
a
b
c
−
−
−
=
=
and
2
2
2
2
x
x
y
y
a
b
−
−
=
=
2
2
z
z
−c
is
2
1
2
1
2
1
1
1
1
2
2
2
2
2
2
1 2
2 1
1 2
2 1
1 2
2 1
(
)
(
)
(
)
x
x
y
y
z
z
a
b
c
a
b
c
bc
b c
c a
c a
a b
a b
−
−
−
−
+
−
+
−
� Distance between parallel lines
1
r
a
b
= +λ
r
r
r
and
2
r
a
b
= +µ
r
r
r
is
2
1
(
)
|
|
b
a
a
b
×
−
r
r
r
r
� In the vector form, equation of a plane which is at a distance d from the
origin, and nˆ is the unit vector normal to the plane through the origin is
r nˆ
d
⋅
=
r |
1 | 6175-6178 | � If θ is the acute angle between
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ λ
r
r
r
, then
1
2
1
2
cos
|
| |
|
bb b
b
⋅
θ =
r
r
r
r
� If
1
1
1
1
1
1
n
z
z
m
y
y
l
x
x
−
=
−
=
−
and
2
2
2
2
2
2
n
z
z
m
y
y
l
x
x
−
=
−
=
−
are the equations of two lines, then the acute angle between the two lines is
given by cos θ = |l1l2 + m 1m 2 + n1n2| © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
501
� Shortest distance between two skew lines is the line segment perpendicular
to both the lines � Shortest distance between
1
1
r
a
b
= +λ
r
r
r
and
2
2
r
a
b
= +µ
r
r
r
is
1
2
2
1
1
2
(
) (
–
)
|
|
b
b
a
a
b
b
×
⋅
×
r
r
r
r
r
r
� Shortest distance between the lines:
1
1
1
1
1
1
x
x
y
y
z
z
a
b
c
−
−
−
=
=
and
2
2
2
2
x
x
y
y
a
b
−
−
=
=
2
2
z
z
−c
is
2
1
2
1
2
1
1
1
1
2
2
2
2
2
2
1 2
2 1
1 2
2 1
1 2
2 1
(
)
(
)
(
)
x
x
y
y
z
z
a
b
c
a
b
c
bc
b c
c a
c a
a b
a b
−
−
−
−
+
−
+
−
� Distance between parallel lines
1
r
a
b
= +λ
r
r
r
and
2
r
a
b
= +µ
r
r
r
is
2
1
(
)
|
|
b
a
a
b
×
−
r
r
r
r
� In the vector form, equation of a plane which is at a distance d from the
origin, and nˆ is the unit vector normal to the plane through the origin is
r nˆ
d
⋅
=
r � Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d |
1 | 6176-6179 | © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
501
� Shortest distance between two skew lines is the line segment perpendicular
to both the lines � Shortest distance between
1
1
r
a
b
= +λ
r
r
r
and
2
2
r
a
b
= +µ
r
r
r
is
1
2
2
1
1
2
(
) (
–
)
|
|
b
b
a
a
b
b
×
⋅
×
r
r
r
r
r
r
� Shortest distance between the lines:
1
1
1
1
1
1
x
x
y
y
z
z
a
b
c
−
−
−
=
=
and
2
2
2
2
x
x
y
y
a
b
−
−
=
=
2
2
z
z
−c
is
2
1
2
1
2
1
1
1
1
2
2
2
2
2
2
1 2
2 1
1 2
2 1
1 2
2 1
(
)
(
)
(
)
x
x
y
y
z
z
a
b
c
a
b
c
bc
b c
c a
c a
a b
a b
−
−
−
−
+
−
+
−
� Distance between parallel lines
1
r
a
b
= +λ
r
r
r
and
2
r
a
b
= +µ
r
r
r
is
2
1
(
)
|
|
b
a
a
b
×
−
r
r
r
r
� In the vector form, equation of a plane which is at a distance d from the
origin, and nˆ is the unit vector normal to the plane through the origin is
r nˆ
d
⋅
=
r � Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d � The equation of a plane through a point whose position vector is ar and
perpendicular to the vector N
ur
is (
) |
1 | 6177-6180 | � Shortest distance between
1
1
r
a
b
= +λ
r
r
r
and
2
2
r
a
b
= +µ
r
r
r
is
1
2
2
1
1
2
(
) (
–
)
|
|
b
b
a
a
b
b
×
⋅
×
r
r
r
r
r
r
� Shortest distance between the lines:
1
1
1
1
1
1
x
x
y
y
z
z
a
b
c
−
−
−
=
=
and
2
2
2
2
x
x
y
y
a
b
−
−
=
=
2
2
z
z
−c
is
2
1
2
1
2
1
1
1
1
2
2
2
2
2
2
1 2
2 1
1 2
2 1
1 2
2 1
(
)
(
)
(
)
x
x
y
y
z
z
a
b
c
a
b
c
bc
b c
c a
c a
a b
a b
−
−
−
−
+
−
+
−
� Distance between parallel lines
1
r
a
b
= +λ
r
r
r
and
2
r
a
b
= +µ
r
r
r
is
2
1
(
)
|
|
b
a
a
b
×
−
r
r
r
r
� In the vector form, equation of a plane which is at a distance d from the
origin, and nˆ is the unit vector normal to the plane through the origin is
r nˆ
d
⋅
=
r � Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d � The equation of a plane through a point whose position vector is ar and
perpendicular to the vector N
ur
is (
) N
0
r
−a
=
ur
r
r |
1 | 6178-6181 | � Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d � The equation of a plane through a point whose position vector is ar and
perpendicular to the vector N
ur
is (
) N
0
r
−a
=
ur
r
r � Equation of a plane perpendicular to a given line with direction ratios A, B, C
and passing through a given point (x1, y1, z1) is
A (x – x1) + B (y – y1) + C (z – z1 ) = 0
� Equation of a plane passing through three non collinear points (x1, y1, z1),
© NCERT
not to be republished
MATHEMATI CS
502
(x2, y2, z2) and (x3, y3, z3) is
1
3
1
3
1
3
1
2
1
2
1
2
1
1
1
z
z
y
y
x
x
z
z
y
y
x
x
z
z
y
y
x
x
−
−
−
−
−
−
−
−
−
= 0
� Vector equation of a plane that contains three non collinear points having
position vectors
b
a
r,r
and cr is (
) |
1 | 6179-6182 | � The equation of a plane through a point whose position vector is ar and
perpendicular to the vector N
ur
is (
) N
0
r
−a
=
ur
r
r � Equation of a plane perpendicular to a given line with direction ratios A, B, C
and passing through a given point (x1, y1, z1) is
A (x – x1) + B (y – y1) + C (z – z1 ) = 0
� Equation of a plane passing through three non collinear points (x1, y1, z1),
© NCERT
not to be republished
MATHEMATI CS
502
(x2, y2, z2) and (x3, y3, z3) is
1
3
1
3
1
3
1
2
1
2
1
2
1
1
1
z
z
y
y
x
x
z
z
y
y
x
x
z
z
y
y
x
x
−
−
−
−
−
−
−
−
−
= 0
� Vector equation of a plane that contains three non collinear points having
position vectors
b
a
r,r
and cr is (
) [(
)
(
) ]
0
r
a
b
a
c
a
−
−
×
−
=
r
r
r
r
r
r
� Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and
(0, 0, c) is
=1
+
+
c
z
b
y
a
x
� Vector equation of a plane t hat passes through t he intersection of
planes
1
1
2
2
and
r n
d
r n
d
⋅
=
⋅
=
r r
r r
is
1
2
1
2
(
)
r
n
n
d
d
⋅
+ λ
=
+ λ
r
r
r
, where λ is any
nonzero constant |
1 | 6180-6183 | N
0
r
−a
=
ur
r
r � Equation of a plane perpendicular to a given line with direction ratios A, B, C
and passing through a given point (x1, y1, z1) is
A (x – x1) + B (y – y1) + C (z – z1 ) = 0
� Equation of a plane passing through three non collinear points (x1, y1, z1),
© NCERT
not to be republished
MATHEMATI CS
502
(x2, y2, z2) and (x3, y3, z3) is
1
3
1
3
1
3
1
2
1
2
1
2
1
1
1
z
z
y
y
x
x
z
z
y
y
x
x
z
z
y
y
x
x
−
−
−
−
−
−
−
−
−
= 0
� Vector equation of a plane that contains three non collinear points having
position vectors
b
a
r,r
and cr is (
) [(
)
(
) ]
0
r
a
b
a
c
a
−
−
×
−
=
r
r
r
r
r
r
� Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and
(0, 0, c) is
=1
+
+
c
z
b
y
a
x
� Vector equation of a plane t hat passes through t he intersection of
planes
1
1
2
2
and
r n
d
r n
d
⋅
=
⋅
=
r r
r r
is
1
2
1
2
(
)
r
n
n
d
d
⋅
+ λ
=
+ λ
r
r
r
, where λ is any
nonzero constant � Cartesian equation of a plane that passes through the intersection of two
given planes A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0
is (A1 x + B1 y + C1 z + D1) + λ(A2 x + B2 y + C2 z + D2) = 0 |
1 | 6181-6184 | � Equation of a plane perpendicular to a given line with direction ratios A, B, C
and passing through a given point (x1, y1, z1) is
A (x – x1) + B (y – y1) + C (z – z1 ) = 0
� Equation of a plane passing through three non collinear points (x1, y1, z1),
© NCERT
not to be republished
MATHEMATI CS
502
(x2, y2, z2) and (x3, y3, z3) is
1
3
1
3
1
3
1
2
1
2
1
2
1
1
1
z
z
y
y
x
x
z
z
y
y
x
x
z
z
y
y
x
x
−
−
−
−
−
−
−
−
−
= 0
� Vector equation of a plane that contains three non collinear points having
position vectors
b
a
r,r
and cr is (
) [(
)
(
) ]
0
r
a
b
a
c
a
−
−
×
−
=
r
r
r
r
r
r
� Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and
(0, 0, c) is
=1
+
+
c
z
b
y
a
x
� Vector equation of a plane t hat passes through t he intersection of
planes
1
1
2
2
and
r n
d
r n
d
⋅
=
⋅
=
r r
r r
is
1
2
1
2
(
)
r
n
n
d
d
⋅
+ λ
=
+ λ
r
r
r
, where λ is any
nonzero constant � Cartesian equation of a plane that passes through the intersection of two
given planes A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0
is (A1 x + B1 y + C1 z + D1) + λ(A2 x + B2 y + C2 z + D2) = 0 � Two lines
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ µ
r
r
r
are coplanar if
2
1
1
2
(
) (
)
a
a
b
b
−
⋅
×
r
r
r
r
= 0
� In the cartesian form above lines passing through the points A (x1, y1, z1) and
B
2
2
2
(
,
,
)
x
y
z
2
2
2
2
–
–
y
y
z
z
b
C
=
=
are coplanar if
2
2
2
1
1
1
1
2
1
2
1
2
c
b
a
c
b
a
z
z
y
y
x
x
−
−
−
= 0 |
1 | 6182-6185 | [(
)
(
) ]
0
r
a
b
a
c
a
−
−
×
−
=
r
r
r
r
r
r
� Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and
(0, 0, c) is
=1
+
+
c
z
b
y
a
x
� Vector equation of a plane t hat passes through t he intersection of
planes
1
1
2
2
and
r n
d
r n
d
⋅
=
⋅
=
r r
r r
is
1
2
1
2
(
)
r
n
n
d
d
⋅
+ λ
=
+ λ
r
r
r
, where λ is any
nonzero constant � Cartesian equation of a plane that passes through the intersection of two
given planes A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0
is (A1 x + B1 y + C1 z + D1) + λ(A2 x + B2 y + C2 z + D2) = 0 � Two lines
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ µ
r
r
r
are coplanar if
2
1
1
2
(
) (
)
a
a
b
b
−
⋅
×
r
r
r
r
= 0
� In the cartesian form above lines passing through the points A (x1, y1, z1) and
B
2
2
2
(
,
,
)
x
y
z
2
2
2
2
–
–
y
y
z
z
b
C
=
=
are coplanar if
2
2
2
1
1
1
1
2
1
2
1
2
c
b
a
c
b
a
z
z
y
y
x
x
−
−
−
= 0 � In the vector form, if θ is the angle between the two planes,
1
1
r n
d
⋅
r r=
and
2
2
r n
d
⋅
r r=
, then θ = cos–1
1
2
1
2
|
|
|
||
|
n
n
n
r⋅n
r
r
r |
1 | 6183-6186 | � Cartesian equation of a plane that passes through the intersection of two
given planes A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0
is (A1 x + B1 y + C1 z + D1) + λ(A2 x + B2 y + C2 z + D2) = 0 � Two lines
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ µ
r
r
r
are coplanar if
2
1
1
2
(
) (
)
a
a
b
b
−
⋅
×
r
r
r
r
= 0
� In the cartesian form above lines passing through the points A (x1, y1, z1) and
B
2
2
2
(
,
,
)
x
y
z
2
2
2
2
–
–
y
y
z
z
b
C
=
=
are coplanar if
2
2
2
1
1
1
1
2
1
2
1
2
c
b
a
c
b
a
z
z
y
y
x
x
−
−
−
= 0 � In the vector form, if θ is the angle between the two planes,
1
1
r n
d
⋅
r r=
and
2
2
r n
d
⋅
r r=
, then θ = cos–1
1
2
1
2
|
|
|
||
|
n
n
n
r⋅n
r
r
r � The angle φ between the line r
a
b
=
+ λ
r
r
r
and the plane
r nˆ
d
⋅
=
r
is
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
503
ˆ
sin
| | | |ˆ
b n
b n
⋅
φ =
r
r
� The angle θ between the planes A1x + B1y + C1z + D1 = 0 and
A2 x + B2 y + C2 z + D2 = 0 is given by
cos θ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
A A
B B
C C
A
B
C
A
B
C
+
+
+
+
+
+
� The distance of a point whose position vector is ar from the plane
r nˆ
d
⋅
=
r
is
ˆ
|
|
d
−a n
⋅
r
� The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is
1
1
1
2
2
2
A
B
C
D
A
B
C
x
y
z
+
+
+
+
+ |
1 | 6184-6187 | � Two lines
1
1
r
a
b
=
+ λ
r
r
r
and
2
2
r
a
b
=
+ µ
r
r
r
are coplanar if
2
1
1
2
(
) (
)
a
a
b
b
−
⋅
×
r
r
r
r
= 0
� In the cartesian form above lines passing through the points A (x1, y1, z1) and
B
2
2
2
(
,
,
)
x
y
z
2
2
2
2
–
–
y
y
z
z
b
C
=
=
are coplanar if
2
2
2
1
1
1
1
2
1
2
1
2
c
b
a
c
b
a
z
z
y
y
x
x
−
−
−
= 0 � In the vector form, if θ is the angle between the two planes,
1
1
r n
d
⋅
r r=
and
2
2
r n
d
⋅
r r=
, then θ = cos–1
1
2
1
2
|
|
|
||
|
n
n
n
r⋅n
r
r
r � The angle φ between the line r
a
b
=
+ λ
r
r
r
and the plane
r nˆ
d
⋅
=
r
is
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
503
ˆ
sin
| | | |ˆ
b n
b n
⋅
φ =
r
r
� The angle θ between the planes A1x + B1y + C1z + D1 = 0 and
A2 x + B2 y + C2 z + D2 = 0 is given by
cos θ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
A A
B B
C C
A
B
C
A
B
C
+
+
+
+
+
+
� The distance of a point whose position vector is ar from the plane
r nˆ
d
⋅
=
r
is
ˆ
|
|
d
−a n
⋅
r
� The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is
1
1
1
2
2
2
A
B
C
D
A
B
C
x
y
z
+
+
+
+
+ —�—
© NCERT
not to be republished
504
MATHEMATICS
�The mathematical experience of the student is incomplete if he never had
the opportunity to solve a problem invented by himself |
1 | 6185-6188 | � In the vector form, if θ is the angle between the two planes,
1
1
r n
d
⋅
r r=
and
2
2
r n
d
⋅
r r=
, then θ = cos–1
1
2
1
2
|
|
|
||
|
n
n
n
r⋅n
r
r
r � The angle φ between the line r
a
b
=
+ λ
r
r
r
and the plane
r nˆ
d
⋅
=
r
is
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
503
ˆ
sin
| | | |ˆ
b n
b n
⋅
φ =
r
r
� The angle θ between the planes A1x + B1y + C1z + D1 = 0 and
A2 x + B2 y + C2 z + D2 = 0 is given by
cos θ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
A A
B B
C C
A
B
C
A
B
C
+
+
+
+
+
+
� The distance of a point whose position vector is ar from the plane
r nˆ
d
⋅
=
r
is
ˆ
|
|
d
−a n
⋅
r
� The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is
1
1
1
2
2
2
A
B
C
D
A
B
C
x
y
z
+
+
+
+
+ —�—
© NCERT
not to be republished
504
MATHEMATICS
�The mathematical experience of the student is incomplete if he never had
the opportunity to solve a problem invented by himself – G |
1 | 6186-6189 | � The angle φ between the line r
a
b
=
+ λ
r
r
r
and the plane
r nˆ
d
⋅
=
r
is
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
503
ˆ
sin
| | | |ˆ
b n
b n
⋅
φ =
r
r
� The angle θ between the planes A1x + B1y + C1z + D1 = 0 and
A2 x + B2 y + C2 z + D2 = 0 is given by
cos θ =
1
2
1
2
1
2
2
2
2
2
2
2
1
1
1
2
2
2
A A
B B
C C
A
B
C
A
B
C
+
+
+
+
+
+
� The distance of a point whose position vector is ar from the plane
r nˆ
d
⋅
=
r
is
ˆ
|
|
d
−a n
⋅
r
� The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is
1
1
1
2
2
2
A
B
C
D
A
B
C
x
y
z
+
+
+
+
+ —�—
© NCERT
not to be republished
504
MATHEMATICS
�The mathematical experience of the student is incomplete if he never had
the opportunity to solve a problem invented by himself – G POLYA �
12 |
1 | 6187-6190 | —�—
© NCERT
not to be republished
504
MATHEMATICS
�The mathematical experience of the student is incomplete if he never had
the opportunity to solve a problem invented by himself – G POLYA �
12 1 Introduction
In earlier classes, we have discussed systems of linear
equations and their applications in day to day problems |
1 | 6188-6191 | – G POLYA �
12 1 Introduction
In earlier classes, we have discussed systems of linear
equations and their applications in day to day problems In
Class XI, we have studied linear inequalities and systems
of linear inequalities in two variables and their solutions by
graphical method |
1 | 6189-6192 | POLYA �
12 1 Introduction
In earlier classes, we have discussed systems of linear
equations and their applications in day to day problems In
Class XI, we have studied linear inequalities and systems
of linear inequalities in two variables and their solutions by
graphical method Many applications in mathematics
involve systems of inequalities/equations |
1 | 6190-6193 | 1 Introduction
In earlier classes, we have discussed systems of linear
equations and their applications in day to day problems In
Class XI, we have studied linear inequalities and systems
of linear inequalities in two variables and their solutions by
graphical method Many applications in mathematics
involve systems of inequalities/equations In this chapter,
we shall apply the systems of linear inequalities/equations
to solve some real life problems of the type as given below:
A furniture dealer deals in only two items–tables and
chairs |
1 | 6191-6194 | In
Class XI, we have studied linear inequalities and systems
of linear inequalities in two variables and their solutions by
graphical method Many applications in mathematics
involve systems of inequalities/equations In this chapter,
we shall apply the systems of linear inequalities/equations
to solve some real life problems of the type as given below:
A furniture dealer deals in only two items–tables and
chairs He has Rs 50,000 to invest and has storage space
of at most 60 pieces |
1 | 6192-6195 | Many applications in mathematics
involve systems of inequalities/equations In this chapter,
we shall apply the systems of linear inequalities/equations
to solve some real life problems of the type as given below:
A furniture dealer deals in only two items–tables and
chairs He has Rs 50,000 to invest and has storage space
of at most 60 pieces A table costs Rs 2500 and a chair
Rs 500 |
1 | 6193-6196 | In this chapter,
we shall apply the systems of linear inequalities/equations
to solve some real life problems of the type as given below:
A furniture dealer deals in only two items–tables and
chairs He has Rs 50,000 to invest and has storage space
of at most 60 pieces A table costs Rs 2500 and a chair
Rs 500 He estimates that from the sale of one table, he
can make a profit of Rs 250 and that from the sale of one
chair a profit of Rs 75 |
1 | 6194-6197 | He has Rs 50,000 to invest and has storage space
of at most 60 pieces A table costs Rs 2500 and a chair
Rs 500 He estimates that from the sale of one table, he
can make a profit of Rs 250 and that from the sale of one
chair a profit of Rs 75 He wants to know how many tables and chairs he should buy
from the available money so as to maximise his total profit, assuming that he can sell all
the items which he buys |
1 | 6195-6198 | A table costs Rs 2500 and a chair
Rs 500 He estimates that from the sale of one table, he
can make a profit of Rs 250 and that from the sale of one
chair a profit of Rs 75 He wants to know how many tables and chairs he should buy
from the available money so as to maximise his total profit, assuming that he can sell all
the items which he buys Such type of problems which seek to maximise (or, minimise) profit (or, cost) form
a general class of problems called optimisation problems |
1 | 6196-6199 | He estimates that from the sale of one table, he
can make a profit of Rs 250 and that from the sale of one
chair a profit of Rs 75 He wants to know how many tables and chairs he should buy
from the available money so as to maximise his total profit, assuming that he can sell all
the items which he buys Such type of problems which seek to maximise (or, minimise) profit (or, cost) form
a general class of problems called optimisation problems Thus, an optimisation
problem may involve finding maximum profit, minimum cost, or minimum use of
resources etc |
1 | 6197-6200 | He wants to know how many tables and chairs he should buy
from the available money so as to maximise his total profit, assuming that he can sell all
the items which he buys Such type of problems which seek to maximise (or, minimise) profit (or, cost) form
a general class of problems called optimisation problems Thus, an optimisation
problem may involve finding maximum profit, minimum cost, or minimum use of
resources etc A special but a very important class of optimisation problems is linear programming
problem |
1 | 6198-6201 | Such type of problems which seek to maximise (or, minimise) profit (or, cost) form
a general class of problems called optimisation problems Thus, an optimisation
problem may involve finding maximum profit, minimum cost, or minimum use of
resources etc A special but a very important class of optimisation problems is linear programming
problem The above stated optimisation problem is an example of linear programming
problem |
1 | 6199-6202 | Thus, an optimisation
problem may involve finding maximum profit, minimum cost, or minimum use of
resources etc A special but a very important class of optimisation problems is linear programming
problem The above stated optimisation problem is an example of linear programming
problem Linear programming problems are of much interest because of their wide
applicability in industry, commerce, management science etc |
1 | 6200-6203 | A special but a very important class of optimisation problems is linear programming
problem The above stated optimisation problem is an example of linear programming
problem Linear programming problems are of much interest because of their wide
applicability in industry, commerce, management science etc In this chapter, we shall study some linear programming problems and their solutions
by graphical method only, though there are many other methods also to solve such
problems |
1 | 6201-6204 | The above stated optimisation problem is an example of linear programming
problem Linear programming problems are of much interest because of their wide
applicability in industry, commerce, management science etc In this chapter, we shall study some linear programming problems and their solutions
by graphical method only, though there are many other methods also to solve such
problems Chapter 12
LINEAR PROGRAMMING
L |
1 | 6202-6205 | Linear programming problems are of much interest because of their wide
applicability in industry, commerce, management science etc In this chapter, we shall study some linear programming problems and their solutions
by graphical method only, though there are many other methods also to solve such
problems Chapter 12
LINEAR PROGRAMMING
L Kantorovich
© NCERT
not to be republished
LINEAR PROGRAMMING 505
12 |
1 | 6203-6206 | In this chapter, we shall study some linear programming problems and their solutions
by graphical method only, though there are many other methods also to solve such
problems Chapter 12
LINEAR PROGRAMMING
L Kantorovich
© NCERT
not to be republished
LINEAR PROGRAMMING 505
12 2 Linear Programming Problem and its Mathematical Formulation
We begin our discussion with the above example of furniture dealer which will further
lead to a mathematical formulation of the problem in two variables |
1 | 6204-6207 | Chapter 12
LINEAR PROGRAMMING
L Kantorovich
© NCERT
not to be republished
LINEAR PROGRAMMING 505
12 2 Linear Programming Problem and its Mathematical Formulation
We begin our discussion with the above example of furniture dealer which will further
lead to a mathematical formulation of the problem in two variables In this example, we
observe
(i)
The dealer can invest his money in buying tables or chairs or combination thereof |
1 | 6205-6208 | Kantorovich
© NCERT
not to be republished
LINEAR PROGRAMMING 505
12 2 Linear Programming Problem and its Mathematical Formulation
We begin our discussion with the above example of furniture dealer which will further
lead to a mathematical formulation of the problem in two variables In this example, we
observe
(i)
The dealer can invest his money in buying tables or chairs or combination thereof Further he would earn different profits by following different investment
strategies |
1 | 6206-6209 | 2 Linear Programming Problem and its Mathematical Formulation
We begin our discussion with the above example of furniture dealer which will further
lead to a mathematical formulation of the problem in two variables In this example, we
observe
(i)
The dealer can invest his money in buying tables or chairs or combination thereof Further he would earn different profits by following different investment
strategies (ii)
There are certain overriding conditions or constraints viz |
1 | 6207-6210 | In this example, we
observe
(i)
The dealer can invest his money in buying tables or chairs or combination thereof Further he would earn different profits by following different investment
strategies (ii)
There are certain overriding conditions or constraints viz , his investment is
limited to a maximum of Rs 50,000 and so is his storage space which is for a
maximum of 60 pieces |
1 | 6208-6211 | Further he would earn different profits by following different investment
strategies (ii)
There are certain overriding conditions or constraints viz , his investment is
limited to a maximum of Rs 50,000 and so is his storage space which is for a
maximum of 60 pieces Suppose he decides to buy tables only and no chairs, so he can buy 50000 ÷ 2500,
i |
1 | 6209-6212 | (ii)
There are certain overriding conditions or constraints viz , his investment is
limited to a maximum of Rs 50,000 and so is his storage space which is for a
maximum of 60 pieces Suppose he decides to buy tables only and no chairs, so he can buy 50000 ÷ 2500,
i e |
1 | 6210-6213 | , his investment is
limited to a maximum of Rs 50,000 and so is his storage space which is for a
maximum of 60 pieces Suppose he decides to buy tables only and no chairs, so he can buy 50000 ÷ 2500,
i e , 20 tables |
1 | 6211-6214 | Suppose he decides to buy tables only and no chairs, so he can buy 50000 ÷ 2500,
i e , 20 tables His profit in this case will be Rs (250 × 20), i |
1 | 6212-6215 | e , 20 tables His profit in this case will be Rs (250 × 20), i e |
1 | 6213-6216 | , 20 tables His profit in this case will be Rs (250 × 20), i e , Rs 5000 |
1 | 6214-6217 | His profit in this case will be Rs (250 × 20), i e , Rs 5000 Suppose he chooses to buy chairs only and no tables |
1 | 6215-6218 | e , Rs 5000 Suppose he chooses to buy chairs only and no tables With his capital of Rs 50,000,
he can buy 50000 ÷ 500, i |
1 | 6216-6219 | , Rs 5000 Suppose he chooses to buy chairs only and no tables With his capital of Rs 50,000,
he can buy 50000 ÷ 500, i e |
1 | 6217-6220 | Suppose he chooses to buy chairs only and no tables With his capital of Rs 50,000,
he can buy 50000 ÷ 500, i e 100 chairs |
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