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THREE DIMENSIONAL GEOMETRY
(I) Equation of a line through a given point and parallel to given vector:
VECTOR EQUATION:
Equation of a line passing through the point with position vector a and parallel to
the vector b is given by:
r  a  b
Where r is position vector of any point on the
line.
CARTESIAN EQUATION
Cartesian equation of a line passing through
the point  x1 , y1 , z1  and parallel to the line
having D-ratios  a, b, c  is given by
x  x1 y  y1 z  z1


a
b
c
NOTE: If we are given the D-cosine of the line  l , m, n  then equation of the line
x  x1 y  y1 z  z1


l
m
n
(II) Equation of a line passing through two points:
VECTOR EQUATION
Equation of a line passing through two points with position vector a and b is given
by
r  a   b  a ,  R


CARTESIAN EQUATION
Cartesian equation of a line passing through the points with  x1 , y1 , z1  and  x2 , y2 , z2 
is given by:
x  x1
y  y1
z  z1


x2  x1 y2  y1 z2  z1
(III) ANGLE BETWEEN TWO LINES
Angle between the lines with D-ratios  a1 , b1 , c1  and  a2 , b2 , c2  is given by:
1. In terms of cos :
cos  
a1a2  b1b2  c1c2
a  b12  c12 a2 2  b2 2  c2 2
2
1
 a1b2  a2b1    b1c2  b2c1    c1a2  c2 a1 
2
2. In terms of sin  
2
a12  a2 2  a32 b12  b2 2  b32
NOTE: Two lines with D-ratios  a1 , b1 , c1  and  a2 , b2 , c2 
(1) Will be perpendicular to each other iff a1a2  b1b2  c1c2  0
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2
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(2) Will be parallel to each other iff
a1 b1 c1
 
a2 b2 c2
Angle between the two lines with direction cosines  l1 , m1 , n1  and  l2 , m2 , n2  is given
by
(I) cos   l1l2  m1m2  n1n2
 l1m2  l2m1    m1n2  m2n1    n1l2  n2l1 
NOTE: Two lines with D-cosines  l1 , m1 , n1  and  l2 , m2 , n2 
(II) sin  
2
2
2
(1) Will be perpendicular to each other iff l1l2  m1m2  n1n2  0
(2) Will be parallel to each other iff l1  l2 , m1  m2 & n1  n2 .
(IV) SHORTEST DISTANCE BETWEEN TWO LINES
SKEW LINES: Two lines in the space which are neither parallel and nor
intersecting are called as skew lines.
RESULT:
1. If two lines are coplanar then they are either parallel or intersecting
2. Two lines are neither parallel nor intersecting iff they are non-coplanar.
(I) VECTOR FORM:
If r  a1  b1 and r  a2  b2 are the vector equations of two lines then the shortest
distance between them is the length of the line of the line segment perpendicular to
both of them, it is denoted by ‘d’ and given by:
 b  b  .  a2  a1 
d 1 2
b1  b2
NOTE: Two skew lines r  a1  b1 and r  a2  b2 will be
intersecting each other iff the shortest distance between them
is zero iff  b1  b2  .  a2  a1   0 iff the lines are coplanar.
(II) CARTESIAN FORM:
x  x1 y  y1 z  z1
x  x2 y  y2 z  z2




If
and
are the equation of two skew
a1
b1
c1
a2
b2
c2
lines then the shortest distance between them ‘d’ is given by:
x2  x1 y  y z2  z1
a1
b1
c1
a2
b2
c2
d
2
2
2
 a1b2  a2b1    b1c2  b2c1    c1a2  c2 a1 
NOTE: Two skew lines
x  x1 y  y1 z  z1
x  x2 y  y2 z  z2




and
will be
a1
b1
c1
a2
b2
c2
intersecting each other iff
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x2  x1
y y
z2  z1
a1
a2
b1
b2
c1
c2
 0.
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(V)DISTANCE BETWEEN TWO PARALLEL LINES
Distance between two parallel lines r  a1  b and r  a2  b and is given by:
d
b   a2  a1 
b
PLANE
NORMAL FORM OF THE EQUATION OF THE PLANE
(I) UNIT VECTOR NORMAL FORM
Equation of a plane having unit vector normal to the plane n̂ and at a
distance of‘d’ from the origin is
r .nˆ  d
Where,
r =Position vector of any point on the plane
n̂ = Unit vector normal to the plane
d = Distance of plane from the origin to the plane
CARTESIAN FORM
Equation of the plane with  l , m, n  as d-cosine of the normal to the plane is
lx  my  nz  d
Where, d = Distance of plane from the origin to the plane
(II) GENERAL FORM OF THE EQAUTION OF THE PLANE
VECTOR FORM
Equation of plane having normal vector to it as n is given by
r .n  p
Note:1. Here ‘p’ is not the distance from the origin to the plane
2. In order to convert the equation r .n  p to unit vector normal form we
n
p
divide the equation by n i.e. the unit vector normal form becomes r . 
n
n
p
and then , d = Distance of plane from the origin to the plane =
n
CARTESIAN FORM
Equation of plane with d-ratios of normal vector to it as  a, b, c  is
ax  by  cz  p  0
Note:1. Here ‘p’ is not the distance from the origin to the plane
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2. In order to convert the equation ax  by  cz  p  0 to normal form we
divide the equation by a2  b2  c2 i.e. the normal form becomes
a
b
c
p
x
y
z
0
a 2  b2  c2
a 2  b2  c2
a 2  b2  c2
a 2  b2  c2
and then , d = Distance of plane from the origin to the plane =
p
a  b2  c2
2
.
(III) EQUATION OF A PLANE PASSING THROUGH A POINT AND WITH
GIVEN NORMAL VECTOR
VECTOR FROM
Equation of a plane passing through the point a and having normal to the plane as
vector n is given by
 r  a  .n  0
CARTESIAN EQUATION
Equation of a plane passing through the point  x1 , y1 , z1 
and having  a, b, c  as the d-ratios of the normal to the
plane is a  x  x1   b  y  y1   c  z  z1   0
(IV) EQUATION OF A PLANE PASSING THROUGH THREE NON
COLLINEAR POINTS
VECTOR FORM
 
Equation of plane passing through three points A  a  , B b & C  c  , where
a , b & c are the position vectors of three points A, B & C , is given by:
 r  a  .b  a    c  a   0
CARTESIAN FORM
Equation of the plane passing through the three points
A  x1 , y1 , z1  , B  x2 , y2 , z2  and C  x3 , y3 , z3  is given by
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x  x1
y  y1
z  z1
x2  x1
x3  x1
y2  y1
y3  y1
z2  z1  0
z3  z1
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(V) INTERCEPT FORM OF THE EQUATION OF PALNE
Equation of a plane having x,y and z intercept as a, b & c respectively is given by
x y z
  1
a b c
(VI) EQUAITON OF A PLANE PASSING THROUGH THE INTERSECTION OF
TWO PLANES
VECTOR FORM
Equation of a plane passing through the intersection of two planes
r .n1  d1 & r .n2  d2 is given by
r .  n1  n2   d1  d2 , where  R
CARTESIAN FORM
Equation of plane passing through the intersection of the two planes
a1 x  b1 y  c1 z  d1  0 and a2 x  b2 y  c2 z  d2  0 is given by
a1 x  b1 y  c1z  d1    a2 x  b2 y  c2 z  d2   0 , where   R
(VII) COPLANARITY OF TWO LINES
VECTOR FORM
Two lines r  a1  b1 and r  a2  b2 will be coplanar iff they lie in a common plane


iff b1  b2 .  a2  a1   0 .
CARTESIAN FORM
Two lines
x  x1 y  y1 z  z1
x  x2 y  y2 z  z2




and
will be coplanar iff
a1
b1
c1
a2
b2
c2
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x2  x1
y2  y1
z2  z1
a1
a2
b1
b2
c1
c2
0
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(VIII) ANGLE BETWEEN TWO PLANES
VECTOR FORM
Angle between two intersecting planes is the angle between the normal to the planes
i.e. Angle ‘  ’ between the planes r .n1  d1 & r .n2  d2 is given by
n .n
cos   1 2
n1 n2
CARTESIAN FORM
Angle ‘  ’ between planes a1 x  b1 y  c1 z  d1  0 and a2 x  b2 y  c2 z  d2  0 is given by
a1a2  b1b2  c1c2
cos  
2
a1  b12  c12 a2 2  b2 2  c2 2
(IX) ANGLE BETWEEN A LINE AND A PLANE
VECTOR FORM
Angle ‘  ’between a line r  a  b and a plane r .n  d is given by
b .n
sin  
b .n
CARTESIAN FORM
Angle ‘  ’between a line
x  x1 y  y1 z  z1


and a plane ax  by  cz  d  0 is given
a1
b1
c1
by
sin  
aa1  bb1  cc1
a  b 2  c 2 a12  b12  c12
(X) DISTANCE OF A POINT FROM A LINE
2
VECTOR FORM
Distance of a point a (not on the plane) from a plane r .n  p is given by
D
r .n  p
n
CARTESIAN FORM
Distance of a point  x1 , y1 , z1  (not on the plane) from a plane ax  by  cz  d  0 is
given by
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D
ax1  by1  cz1  d
a 2  b2  c 2