Important Three dimensional Geometry Formulas for JEE and NEET

 Page 1


Page # 35
3-DIMENSION
1. Vector representation of a point :
Position vector of point P (x, y, z) is x i
ˆ
 + y j
ˆ
 + z k
ˆ
.
2. Distance formula :
2
2 1
2
2 1
2
2 1
)z z ( ) y y ( )x x ( ? ?? ??
, AB = |
OB
 – 
OA
|
3. Distance of P from coordinate axes :
PA =
2 2
z y ? , PB =
2 2
x z ? , PC = 
2 2
y x ?
4. Section Formula :  x = 
n m
nx mx
1 2
?
?
, y = 
n m
ny my
1 2
?
?
, z = 
n m
nz mz
1 2
?
?
Mid point : 
2
z z
z ,
2
y y
y ,
2
x x
x
2 1 21 21
?
?
?
?
?
?
5. Direction Cosines And Direction Ratios
(i) Direction cosines: Let ??? ? ?? ? be the angles which a directed  line
makes with the positive directions of the axes of x, y and z  respectively,
then cos ?, cos ? ? cos ? are called the direction  cosines of the line. The
direction cosines are usually denoted  by ( ?, m, n). Thus ? = cos ?, m = cos
?, n = cos ?.
(ii) If ?, m, n be the direction cosines of a line, then ?
2
 + m
2
 + n
2
 = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines
?, m, n then a, b, c  are called the direction ratios.
(iv) If ?, m, n be the direction cosines and a, b, c be the direction ratios
of a vector, then
22 2 2 22 22 2
c b a
c
n ,
c b a
b
m ,
c b a
a
? ?
? ?
? ?
? ?
? ?
? ? ?
Page 2


Page # 35
3-DIMENSION
1. Vector representation of a point :
Position vector of point P (x, y, z) is x i
ˆ
 + y j
ˆ
 + z k
ˆ
.
2. Distance formula :
2
2 1
2
2 1
2
2 1
)z z ( ) y y ( )x x ( ? ?? ??
, AB = |
OB
 – 
OA
|
3. Distance of P from coordinate axes :
PA =
2 2
z y ? , PB =
2 2
x z ? , PC = 
2 2
y x ?
4. Section Formula :  x = 
n m
nx mx
1 2
?
?
, y = 
n m
ny my
1 2
?
?
, z = 
n m
nz mz
1 2
?
?
Mid point : 
2
z z
z ,
2
y y
y ,
2
x x
x
2 1 21 21
?
?
?
?
?
?
5. Direction Cosines And Direction Ratios
(i) Direction cosines: Let ??? ? ?? ? be the angles which a directed  line
makes with the positive directions of the axes of x, y and z  respectively,
then cos ?, cos ? ? cos ? are called the direction  cosines of the line. The
direction cosines are usually denoted  by ( ?, m, n). Thus ? = cos ?, m = cos
?, n = cos ?.
(ii) If ?, m, n be the direction cosines of a line, then ?
2
 + m
2
 + n
2
 = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines
?, m, n then a, b, c  are called the direction ratios.
(iv) If ?, m, n be the direction cosines and a, b, c be the direction ratios
of a vector, then
22 2 2 22 22 2
c b a
c
n ,
c b a
b
m ,
c b a
a
? ?
? ?
? ?
? ?
? ?
? ? ?
Page # 36
1
, y
1
, z
1
) and (x
2
, y
2
, z
2
) then the
direction ratios of  line PQ are, a = x
2
 ? x
1
, b = y
2
 ? y
1
 & c = z
2
 ? z
1
 and
the direction cosines of line PQ are ? ? =
| PQ |
x x
1 2
?
,
m =
| PQ |
y y
1 2
?
 and n = 
| PQ |
z z
1 2
?
6. Angle Between Two Line Segments:
cos ? =
2
2
2
2
2
2
2
1
2
1
2
1
21 21 2 1
cb a cb a
c c bb aa
?? ??
? ?
.
The line will be perpendicular if a
1
a
2
 + b
1
b
2
 + c
1
c
2
 = 0,
parallel if 
2
1
a
a
 =
2
1
b
b
 =
2
1
c
c
7. Projection of a line segment on a line
If P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) then the projection of  PQ on a line having
direction cosines ?, m, n is
)z z ( n ) y y (m ) x x (
1 2 1 2 1 2
? ? ? ? ? ?
8. Equation Of A Plane : General form: ax + by + cz + d = 0, where a, b, c
are not all zero,  a, b, c, d ? R.
(i) Normal form : ?x + my + nz = p
(ii) Plane through the point (x
1
, y
1
, z
1
) :
 a (x ? x
1
) + b( y ? y
1
) + c (z ? z
1
) = 0
(iii) Intercept Form: 1
c
z
b
y
a
x
? ? ?
(iv) Vector form: ( r
?
 ? a
?
). n
?
 = 0 or r
?
. n
?
 = a
?
. n
?
(v) Any plane parallel to the given plane ax + by + cz + d = 0 is
ax + by + cz + ? = 0. Distance between  ax + by + cz + d
1
 = 0 and
ax + by + cz + d
2
 = 0 is = 
222
2 1
c b a
| d d |
? ?
?
Page 3


Page # 35
3-DIMENSION
1. Vector representation of a point :
Position vector of point P (x, y, z) is x i
ˆ
 + y j
ˆ
 + z k
ˆ
.
2. Distance formula :
2
2 1
2
2 1
2
2 1
)z z ( ) y y ( )x x ( ? ?? ??
, AB = |
OB
 – 
OA
|
3. Distance of P from coordinate axes :
PA =
2 2
z y ? , PB =
2 2
x z ? , PC = 
2 2
y x ?
4. Section Formula :  x = 
n m
nx mx
1 2
?
?
, y = 
n m
ny my
1 2
?
?
, z = 
n m
nz mz
1 2
?
?
Mid point : 
2
z z
z ,
2
y y
y ,
2
x x
x
2 1 21 21
?
?
?
?
?
?
5. Direction Cosines And Direction Ratios
(i) Direction cosines: Let ??? ? ?? ? be the angles which a directed  line
makes with the positive directions of the axes of x, y and z  respectively,
then cos ?, cos ? ? cos ? are called the direction  cosines of the line. The
direction cosines are usually denoted  by ( ?, m, n). Thus ? = cos ?, m = cos
?, n = cos ?.
(ii) If ?, m, n be the direction cosines of a line, then ?
2
 + m
2
 + n
2
 = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines
?, m, n then a, b, c  are called the direction ratios.
(iv) If ?, m, n be the direction cosines and a, b, c be the direction ratios
of a vector, then
22 2 2 22 22 2
c b a
c
n ,
c b a
b
m ,
c b a
a
? ?
? ?
? ?
? ?
? ?
? ? ?
Page # 36
1
, y
1
, z
1
) and (x
2
, y
2
, z
2
) then the
direction ratios of  line PQ are, a = x
2
 ? x
1
, b = y
2
 ? y
1
 & c = z
2
 ? z
1
 and
the direction cosines of line PQ are ? ? =
| PQ |
x x
1 2
?
,
m =
| PQ |
y y
1 2
?
 and n = 
| PQ |
z z
1 2
?
6. Angle Between Two Line Segments:
cos ? =
2
2
2
2
2
2
2
1
2
1
2
1
21 21 2 1
cb a cb a
c c bb aa
?? ??
? ?
.
The line will be perpendicular if a
1
a
2
 + b
1
b
2
 + c
1
c
2
 = 0,
parallel if 
2
1
a
a
 =
2
1
b
b
 =
2
1
c
c
7. Projection of a line segment on a line
If P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) then the projection of  PQ on a line having
direction cosines ?, m, n is
)z z ( n ) y y (m ) x x (
1 2 1 2 1 2
? ? ? ? ? ?
8. Equation Of A Plane : General form: ax + by + cz + d = 0, where a, b, c
are not all zero,  a, b, c, d ? R.
(i) Normal form : ?x + my + nz = p
(ii) Plane through the point (x
1
, y
1
, z
1
) :
 a (x ? x
1
) + b( y ? y
1
) + c (z ? z
1
) = 0
(iii) Intercept Form: 1
c
z
b
y
a
x
? ? ?
(iv) Vector form: ( r
?
 ? a
?
). n
?
 = 0 or r
?
. n
?
 = a
?
. n
?
(v) Any plane parallel to the given plane ax + by + cz + d = 0 is
ax + by + cz + ? = 0. Distance between  ax + by + cz + d
1
 = 0 and
ax + by + cz + d
2
 = 0 is = 
222
2 1
c b a
| d d |
? ?
?
Page # 37
(vi) Equation of a plane passing through a given point & parallel to
the given vectors:
r
?
 = a
?
+ ? ? b
?
 + ? c
?
  (parametric form)   where ? & ? are scalars.
or r
?
.
) c b (
?
?
?
 = a
?
. ) c b (
?
?
? (non parametric form)
9. A Plane & A Point
(i) Distance of the point (x ?, y ?, z ?) from the plane ax + by + cz+ d = 0 is
given by 
2 2 2
c b a
d ' cz ' by ' ax
? ?
? ? ?
.
(ii) Length of the perpendicular from a point ( a
?
) to plane r
?
. n
?
 = d
is given by p =
| n |
| d n . a |
?
? ?
?
.
(iii) Foot (x ?, y ?, z ?) of perpendicular drawn from the point (x
1
, y
1
, z
1
) to
the plane ax + by + cz + d = 0 is given by  
c
z ' z
b
y ' y
a
x ' x
1 1 1
?
?
?
?
?
= – 
222
1 1 1
c b a
) d cz by ax (
? ?
? ? ?
(iv) To find image of a point w.r.t. a plane:
Let P (x
1
, y
1
, z
1
) is a given point and ax + by + cz + d = 0 is given
plane Let (x ?, y ?, z ?) is the image point. then
c
z ' z
b
y ' y
a
x ' x
1 1 1
?
?
?
?
?
 = – 2 
222
1 1 1
c b a
) d cz by ax (
? ?
? ? ?
10. Angle Between Two Planes:
cos ? = 
2' 2' 2 ' 2 2 2
c b a c b a
' cc ' bb ' aa
? ? ? ?
? ?
Planes are perpendicular if aa ? + bb ? + cc ? = 0 and planes are parallel if
' a
a
=
' b
b
 =
' c
c
Page 4


Page # 35
3-DIMENSION
1. Vector representation of a point :
Position vector of point P (x, y, z) is x i
ˆ
 + y j
ˆ
 + z k
ˆ
.
2. Distance formula :
2
2 1
2
2 1
2
2 1
)z z ( ) y y ( )x x ( ? ?? ??
, AB = |
OB
 – 
OA
|
3. Distance of P from coordinate axes :
PA =
2 2
z y ? , PB =
2 2
x z ? , PC = 
2 2
y x ?
4. Section Formula :  x = 
n m
nx mx
1 2
?
?
, y = 
n m
ny my
1 2
?
?
, z = 
n m
nz mz
1 2
?
?
Mid point : 
2
z z
z ,
2
y y
y ,
2
x x
x
2 1 21 21
?
?
?
?
?
?
5. Direction Cosines And Direction Ratios
(i) Direction cosines: Let ??? ? ?? ? be the angles which a directed  line
makes with the positive directions of the axes of x, y and z  respectively,
then cos ?, cos ? ? cos ? are called the direction  cosines of the line. The
direction cosines are usually denoted  by ( ?, m, n). Thus ? = cos ?, m = cos
?, n = cos ?.
(ii) If ?, m, n be the direction cosines of a line, then ?
2
 + m
2
 + n
2
 = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines
?, m, n then a, b, c  are called the direction ratios.
(iv) If ?, m, n be the direction cosines and a, b, c be the direction ratios
of a vector, then
22 2 2 22 22 2
c b a
c
n ,
c b a
b
m ,
c b a
a
? ?
? ?
? ?
? ?
? ?
? ? ?
Page # 36
1
, y
1
, z
1
) and (x
2
, y
2
, z
2
) then the
direction ratios of  line PQ are, a = x
2
 ? x
1
, b = y
2
 ? y
1
 & c = z
2
 ? z
1
 and
the direction cosines of line PQ are ? ? =
| PQ |
x x
1 2
?
,
m =
| PQ |
y y
1 2
?
 and n = 
| PQ |
z z
1 2
?
6. Angle Between Two Line Segments:
cos ? =
2
2
2
2
2
2
2
1
2
1
2
1
21 21 2 1
cb a cb a
c c bb aa
?? ??
? ?
.
The line will be perpendicular if a
1
a
2
 + b
1
b
2
 + c
1
c
2
 = 0,
parallel if 
2
1
a
a
 =
2
1
b
b
 =
2
1
c
c
7. Projection of a line segment on a line
If P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) then the projection of  PQ on a line having
direction cosines ?, m, n is
)z z ( n ) y y (m ) x x (
1 2 1 2 1 2
? ? ? ? ? ?
8. Equation Of A Plane : General form: ax + by + cz + d = 0, where a, b, c
are not all zero,  a, b, c, d ? R.
(i) Normal form : ?x + my + nz = p
(ii) Plane through the point (x
1
, y
1
, z
1
) :
 a (x ? x
1
) + b( y ? y
1
) + c (z ? z
1
) = 0
(iii) Intercept Form: 1
c
z
b
y
a
x
? ? ?
(iv) Vector form: ( r
?
 ? a
?
). n
?
 = 0 or r
?
. n
?
 = a
?
. n
?
(v) Any plane parallel to the given plane ax + by + cz + d = 0 is
ax + by + cz + ? = 0. Distance between  ax + by + cz + d
1
 = 0 and
ax + by + cz + d
2
 = 0 is = 
222
2 1
c b a
| d d |
? ?
?
Page # 37
(vi) Equation of a plane passing through a given point & parallel to
the given vectors:
r
?
 = a
?
+ ? ? b
?
 + ? c
?
  (parametric form)   where ? & ? are scalars.
or r
?
.
) c b (
?
?
?
 = a
?
. ) c b (
?
?
? (non parametric form)
9. A Plane & A Point
(i) Distance of the point (x ?, y ?, z ?) from the plane ax + by + cz+ d = 0 is
given by 
2 2 2
c b a
d ' cz ' by ' ax
? ?
? ? ?
.
(ii) Length of the perpendicular from a point ( a
?
) to plane r
?
. n
?
 = d
is given by p =
| n |
| d n . a |
?
? ?
?
.
(iii) Foot (x ?, y ?, z ?) of perpendicular drawn from the point (x
1
, y
1
, z
1
) to
the plane ax + by + cz + d = 0 is given by  
c
z ' z
b
y ' y
a
x ' x
1 1 1
?
?
?
?
?
= – 
222
1 1 1
c b a
) d cz by ax (
? ?
? ? ?
(iv) To find image of a point w.r.t. a plane:
Let P (x
1
, y
1
, z
1
) is a given point and ax + by + cz + d = 0 is given
plane Let (x ?, y ?, z ?) is the image point. then
c
z ' z
b
y ' y
a
x ' x
1 1 1
?
?
?
?
?
 = – 2 
222
1 1 1
c b a
) d cz by ax (
? ?
? ? ?
10. Angle Between Two Planes:
cos ? = 
2' 2' 2 ' 2 2 2
c b a c b a
' cc ' bb ' aa
? ? ? ?
? ?
Planes are perpendicular if aa ? + bb ? + cc ? = 0 and planes are parallel if
' a
a
=
' b
b
 =
' c
c
Page # 38
The angle ? between the planes r .
1
n = d
1
 and r
?
.
2
n
?
 = d
2
 is given by, cos
? = 
| n | . | n |
n . n
2 1
2 1
? ?
? ?
Planes are perpendicular if
1
n
?
.
2
n
?
 = 0 & planes are parallel if
1
n
?
 = ?
2
n
?
, ? is a scalar
11. Angle Bisectors
(i) The equations of the planes bisecting the angle between two
given planes
a
1
x + b
1
y + c
1
z + d
1
 = 0 and a
2
x + b
2
y + c
2
z + d
2
 = 0 are
2
1
2
1
2
1
11 11
c b a
d z c y b x a
? ?
? ? ?
 = ± 
2
2
2
2
2
2
222 2
c b a
d z c y b x a
? ?
???
(ii) Bisector of acute/obtuse angle: First make both the constant terms
positive. Then
a
1
a
2
 + b
1
b
2
 + c
1
c
2
 > 0 ? origin lies on obtuse angle
a
1
a
2
 + b
1
b
2
 + c
1
c
2
 < 0 ? origin lies in acute angle
12. Family of Planes
(i) Any plane through the intersection of a
1
x + b
1
y + c
1
z + d
1 
= 0 &
a
2
x + b
2
y + c
2
z + d
2
 = 0 is
a
1
x + b
1
y + c
1
z + d
1
 + ? (a
2
x + b
2
y + c
2
z + d
2
) = 0
(ii) The equation of plane passing through the intersection of the
planes r
?
.
1
n
?
 = d
1
 & r
?
.
2
n
?
 = d
2
  is r
?
. (n
1
 + ?
2
n
?
) = d
1
 ? ? ?d
2
 where
? is arbitrary scalar
13. Volume Of A Tetrahedron: Volume of a tetrahedron with vertices
A (x
1
, y
1
, z
1
), B( x
2
, y
2
, z
2
), C (x
3
, y
3
, z
3
) and
D (x
4
, y
4
, z
4
) is given by V =
6
1
1 zyx
1 z y x
1 z y x
1 z y x
4 4 4
3 3 3
2 2 2
111
Page 5


Page # 35
3-DIMENSION
1. Vector representation of a point :
Position vector of point P (x, y, z) is x i
ˆ
 + y j
ˆ
 + z k
ˆ
.
2. Distance formula :
2
2 1
2
2 1
2
2 1
)z z ( ) y y ( )x x ( ? ?? ??
, AB = |
OB
 – 
OA
|
3. Distance of P from coordinate axes :
PA =
2 2
z y ? , PB =
2 2
x z ? , PC = 
2 2
y x ?
4. Section Formula :  x = 
n m
nx mx
1 2
?
?
, y = 
n m
ny my
1 2
?
?
, z = 
n m
nz mz
1 2
?
?
Mid point : 
2
z z
z ,
2
y y
y ,
2
x x
x
2 1 21 21
?
?
?
?
?
?
5. Direction Cosines And Direction Ratios
(i) Direction cosines: Let ??? ? ?? ? be the angles which a directed  line
makes with the positive directions of the axes of x, y and z  respectively,
then cos ?, cos ? ? cos ? are called the direction  cosines of the line. The
direction cosines are usually denoted  by ( ?, m, n). Thus ? = cos ?, m = cos
?, n = cos ?.
(ii) If ?, m, n be the direction cosines of a line, then ?
2
 + m
2
 + n
2
 = 1
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines
?, m, n then a, b, c  are called the direction ratios.
(iv) If ?, m, n be the direction cosines and a, b, c be the direction ratios
of a vector, then
22 2 2 22 22 2
c b a
c
n ,
c b a
b
m ,
c b a
a
? ?
? ?
? ?
? ?
? ?
? ? ?
Page # 36
1
, y
1
, z
1
) and (x
2
, y
2
, z
2
) then the
direction ratios of  line PQ are, a = x
2
 ? x
1
, b = y
2
 ? y
1
 & c = z
2
 ? z
1
 and
the direction cosines of line PQ are ? ? =
| PQ |
x x
1 2
?
,
m =
| PQ |
y y
1 2
?
 and n = 
| PQ |
z z
1 2
?
6. Angle Between Two Line Segments:
cos ? =
2
2
2
2
2
2
2
1
2
1
2
1
21 21 2 1
cb a cb a
c c bb aa
?? ??
? ?
.
The line will be perpendicular if a
1
a
2
 + b
1
b
2
 + c
1
c
2
 = 0,
parallel if 
2
1
a
a
 =
2
1
b
b
 =
2
1
c
c
7. Projection of a line segment on a line
If P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) then the projection of  PQ on a line having
direction cosines ?, m, n is
)z z ( n ) y y (m ) x x (
1 2 1 2 1 2
? ? ? ? ? ?
8. Equation Of A Plane : General form: ax + by + cz + d = 0, where a, b, c
are not all zero,  a, b, c, d ? R.
(i) Normal form : ?x + my + nz = p
(ii) Plane through the point (x
1
, y
1
, z
1
) :
 a (x ? x
1
) + b( y ? y
1
) + c (z ? z
1
) = 0
(iii) Intercept Form: 1
c
z
b
y
a
x
? ? ?
(iv) Vector form: ( r
?
 ? a
?
). n
?
 = 0 or r
?
. n
?
 = a
?
. n
?
(v) Any plane parallel to the given plane ax + by + cz + d = 0 is
ax + by + cz + ? = 0. Distance between  ax + by + cz + d
1
 = 0 and
ax + by + cz + d
2
 = 0 is = 
222
2 1
c b a
| d d |
? ?
?
Page # 37
(vi) Equation of a plane passing through a given point & parallel to
the given vectors:
r
?
 = a
?
+ ? ? b
?
 + ? c
?
  (parametric form)   where ? & ? are scalars.
or r
?
.
) c b (
?
?
?
 = a
?
. ) c b (
?
?
? (non parametric form)
9. A Plane & A Point
(i) Distance of the point (x ?, y ?, z ?) from the plane ax + by + cz+ d = 0 is
given by 
2 2 2
c b a
d ' cz ' by ' ax
? ?
? ? ?
.
(ii) Length of the perpendicular from a point ( a
?
) to plane r
?
. n
?
 = d
is given by p =
| n |
| d n . a |
?
? ?
?
.
(iii) Foot (x ?, y ?, z ?) of perpendicular drawn from the point (x
1
, y
1
, z
1
) to
the plane ax + by + cz + d = 0 is given by  
c
z ' z
b
y ' y
a
x ' x
1 1 1
?
?
?
?
?
= – 
222
1 1 1
c b a
) d cz by ax (
? ?
? ? ?
(iv) To find image of a point w.r.t. a plane:
Let P (x
1
, y
1
, z
1
) is a given point and ax + by + cz + d = 0 is given
plane Let (x ?, y ?, z ?) is the image point. then
c
z ' z
b
y ' y
a
x ' x
1 1 1
?
?
?
?
?
 = – 2 
222
1 1 1
c b a
) d cz by ax (
? ?
? ? ?
10. Angle Between Two Planes:
cos ? = 
2' 2' 2 ' 2 2 2
c b a c b a
' cc ' bb ' aa
? ? ? ?
? ?
Planes are perpendicular if aa ? + bb ? + cc ? = 0 and planes are parallel if
' a
a
=
' b
b
 =
' c
c
Page # 38
The angle ? between the planes r .
1
n = d
1
 and r
?
.
2
n
?
 = d
2
 is given by, cos
? = 
| n | . | n |
n . n
2 1
2 1
? ?
? ?
Planes are perpendicular if
1
n
?
.
2
n
?
 = 0 & planes are parallel if
1
n
?
 = ?
2
n
?
, ? is a scalar
11. Angle Bisectors
(i) The equations of the planes bisecting the angle between two
given planes
a
1
x + b
1
y + c
1
z + d
1
 = 0 and a
2
x + b
2
y + c
2
z + d
2
 = 0 are
2
1
2
1
2
1
11 11
c b a
d z c y b x a
? ?
? ? ?
 = ± 
2
2
2
2
2
2
222 2
c b a
d z c y b x a
? ?
???
(ii) Bisector of acute/obtuse angle: First make both the constant terms
positive. Then
a
1
a
2
 + b
1
b
2
 + c
1
c
2
 > 0 ? origin lies on obtuse angle
a
1
a
2
 + b
1
b
2
 + c
1
c
2
 < 0 ? origin lies in acute angle
12. Family of Planes
(i) Any plane through the intersection of a
1
x + b
1
y + c
1
z + d
1 
= 0 &
a
2
x + b
2
y + c
2
z + d
2
 = 0 is
a
1
x + b
1
y + c
1
z + d
1
 + ? (a
2
x + b
2
y + c
2
z + d
2
) = 0
(ii) The equation of plane passing through the intersection of the
planes r
?
.
1
n
?
 = d
1
 & r
?
.
2
n
?
 = d
2
  is r
?
. (n
1
 + ?
2
n
?
) = d
1
 ? ? ?d
2
 where
? is arbitrary scalar
13. Volume Of A Tetrahedron: Volume of a tetrahedron with vertices
A (x
1
, y
1
, z
1
), B( x
2
, y
2
, z
2
), C (x
3
, y
3
, z
3
) and
D (x
4
, y
4
, z
4
) is given by V =
6
1
1 zyx
1 z y x
1 z y x
1 z y x
4 4 4
3 3 3
2 2 2
111
Page # 39
1. Equation Of A Line
(i) A straight line is intersection of two planes.
it is reprsented by two planes a
1
x + b
1
y + c
1
z + d
1
 = 0 and
a
2
x + b
2
y + +c
2
z + d
2 
=0.
(ii) Symmetric form : 
a
x x
1
?
 =
b
y y
1
?
 =
c
z z
1
?
 = r. .
(iii) Vector equation: 
r
?
 = a
?
 + ? b
?
(iv) Reduction of cartesion form of equation of a line to vector form & vice
versa
a
x x
1
?
 =
b
y y
1
?
 =
c
z z
1
?
 ? ?
r
?
=(x
1 i
ˆ +y
1
j
ˆ
+ z
1 k
ˆ ) + ? (a
i
ˆ + b j
ˆ
 + c
k
ˆ ).
2. Angle Between A Plane And A Line:
(i) If ? is the angle between line
?
1
x x ?
 =
m
y y
1
?
 =
n
z z
1
?
 and the
plane ax + by + cz + d = 0, then
sin ? =
2 2 2 222
n m c b a
n c m b a
) ( ? ? ? ?
? ?
?
?
.
(ii) Vector form: If ? is the angle between a line r
?
 = ( a
?
 + ? b
?
) and
r
?
.
n
?
 = d then sin ? ? =
?
?
?
?
?
?
| n | | b |
n . b
?
?
?
?
.
(iii) Condition for perpendicularity  
a
?
 =
b
m
 = 
c
n
 ,  b
?
 x n
?
 = 0
(iv) Condition for parallel      a ? + bm + cn = 0
b
?
. n
?
 = 0
3. Condition For A Line To Lie In A Plane
(i) Cartesian form: Line
?
1
x x ?
 =
m
y y
1
?
 =
n
z z
1
?
 would lie in a plane
ax + by + cz + d = 0, if ax
1
 + by
1
 + cz
1
 + d = 0 &
a ? + bm + cn = 0.
(ii) Vector form: Line r
?
 = a
?
 + ? b
?
 would lie in the plane r
?
. n
?
= d if 
b
?
.
n
?
 = 0 & a
?
. n
?
 = d