Chapter 11
THREE DIMENSIONAL GEOMETRY
11.1 Overview
11.1.1 Direction cosines of a line are the cosines of the angles made by the line with
positive directions of the co-ordinate axes.
11.1.2 If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1
11.1.3 Direction cosines of a line joining two points P (x1, y1 , z1) and Q (x2, y2, z2) are
x2 − x1 y2 − y1 z2 − z1
, , ,
PQ PQ PQ
where PQ = (x2 – x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2
11.1.4 Direction ratios of a line are the numbers which are proportional to the direction
cosines of the line.
11.1.5 If l, m, n are the direction cosines and a, b, c are the direction ratios of a line,
±a ±b ±c
then l = ;m= ; n=
a 2 + b2 + c 2 a 2 + b2 + c 2 a 2 + b2 + c 2
11.1.6 Skew lines are lines in the space which are neither parallel nor interesecting.
They lie in the different planes.
11.1.7 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.
11.1.8 If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines and θ is the
acute angle between the two lines, then
cosθ = l1l2 + m1m2 + n1n2
11.1.9 If a1, b1, c1 and a2, b2, c2 are the directions ratios of two lines and θ is the
acute angle between the two lines, then
, THREE DIMENSIONAL GEOMETRY 221
a1a2 + b1b2 + c1c2
cos θ =
a12 + a22 + a32 . b12 + b22 + b32
11.1.10 Vector equation of a line that passes through the given point whose position
vector is a and parallel to a given vector b is r = a + λb .
11.1.11 Equation of a line through a point (x1, y1, z1) and having directions cosines
l, m, n (or, direction ratios a, b and c) is
x − x1 y − y1 z − z1 x − x1 y − y1 z − z1
= = or = = .
l m n a b c
11.1.12 The vector equation of a line that passes through two points whose positions
vectors are a and b is r = a + λ (b − a ) .
11.1.13 Cartesian equation of a line that passes through two points (x1, y1, z1) and
(x2, y2, z2) is
x − x1 y − y1 z − z1
= =
x2 − x1 y2 − y1 z2 − z1 .
11.1.14 If θ is the acute angle between the lines r = a1 + λ b1 and r = a2 + λ b2 , then
b1 . b2 b1 . b2
θ is given by cos θ = or θ = cos –1 .
b1 b2 b1 b2
x − x1 y − y1 z − z1 x − x2 y − y2 z − z2
11.1.15 If = = and = =
l1 m1 n1 l1 m2 n2 are equations of two
lines, then the acute angle θ between the two lines is given by
cosθ = l1 l2 + m1 m2 + n1 n2 .
11.1.16 The shortest distance between two skew lines is the length of the line segment
perpendicular to both the lines.
11.1.17 The shortest distance between the lines r = a1 + λ b1 and r = a2 + λ b2 is
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( b × b ) . ( a
1 2 2
– a1 )
.
b1 × b2
x − x1 y − y1 z − z1
11.1.18 Shortest distance between the lines: = = and
a1 b1 c1
x − x2 y − y2 z − z2
= =
a2 b2 c2 is
x2 − x1 y2 − y1 z2 − z1
a1 b1 c1
a2 b2 c2
(b1c2 − b2 c1 ) 2 + (c1a2 − c2 a1 ) 2 + (a1b2 − a2b1 ) 2
11.1.19 Distance between parallel lines r = a1 + µ b and r = a2 + λ b is
b × ( a2 – a1 )
.
b
11.1.20 The vector equation of a plane which is at a distance p from the origin, where
n̂ is the unit vector normal to the plane, is r . nˆ = p .
11.1.21 Equation of a plane which is at a distance p from the origin with direction
cosines of the normal to the plane as l, m, n is lx + my + nz = p.
11.1.22 The equation of a plane through a point whose position vector is a and
perpendicular to the vector n is (r – a ). n = 0 or r . n = d , where d = a . n.
11.1.23 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.
11.1.24 Equation of a plane passing through three non-collinear points (x1, y1, z1),
(x2, y2, z2) and (x3, y3, z3) is
THREE DIMENSIONAL GEOMETRY
11.1 Overview
11.1.1 Direction cosines of a line are the cosines of the angles made by the line with
positive directions of the co-ordinate axes.
11.1.2 If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1
11.1.3 Direction cosines of a line joining two points P (x1, y1 , z1) and Q (x2, y2, z2) are
x2 − x1 y2 − y1 z2 − z1
, , ,
PQ PQ PQ
where PQ = (x2 – x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2
11.1.4 Direction ratios of a line are the numbers which are proportional to the direction
cosines of the line.
11.1.5 If l, m, n are the direction cosines and a, b, c are the direction ratios of a line,
±a ±b ±c
then l = ;m= ; n=
a 2 + b2 + c 2 a 2 + b2 + c 2 a 2 + b2 + c 2
11.1.6 Skew lines are lines in the space which are neither parallel nor interesecting.
They lie in the different planes.
11.1.7 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.
11.1.8 If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines and θ is the
acute angle between the two lines, then
cosθ = l1l2 + m1m2 + n1n2
11.1.9 If a1, b1, c1 and a2, b2, c2 are the directions ratios of two lines and θ is the
acute angle between the two lines, then
, THREE DIMENSIONAL GEOMETRY 221
a1a2 + b1b2 + c1c2
cos θ =
a12 + a22 + a32 . b12 + b22 + b32
11.1.10 Vector equation of a line that passes through the given point whose position
vector is a and parallel to a given vector b is r = a + λb .
11.1.11 Equation of a line through a point (x1, y1, z1) and having directions cosines
l, m, n (or, direction ratios a, b and c) is
x − x1 y − y1 z − z1 x − x1 y − y1 z − z1
= = or = = .
l m n a b c
11.1.12 The vector equation of a line that passes through two points whose positions
vectors are a and b is r = a + λ (b − a ) .
11.1.13 Cartesian equation of a line that passes through two points (x1, y1, z1) and
(x2, y2, z2) is
x − x1 y − y1 z − z1
= =
x2 − x1 y2 − y1 z2 − z1 .
11.1.14 If θ is the acute angle between the lines r = a1 + λ b1 and r = a2 + λ b2 , then
b1 . b2 b1 . b2
θ is given by cos θ = or θ = cos –1 .
b1 b2 b1 b2
x − x1 y − y1 z − z1 x − x2 y − y2 z − z2
11.1.15 If = = and = =
l1 m1 n1 l1 m2 n2 are equations of two
lines, then the acute angle θ between the two lines is given by
cosθ = l1 l2 + m1 m2 + n1 n2 .
11.1.16 The shortest distance between two skew lines is the length of the line segment
perpendicular to both the lines.
11.1.17 The shortest distance between the lines r = a1 + λ b1 and r = a2 + λ b2 is
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( b × b ) . ( a
1 2 2
– a1 )
.
b1 × b2
x − x1 y − y1 z − z1
11.1.18 Shortest distance between the lines: = = and
a1 b1 c1
x − x2 y − y2 z − z2
= =
a2 b2 c2 is
x2 − x1 y2 − y1 z2 − z1
a1 b1 c1
a2 b2 c2
(b1c2 − b2 c1 ) 2 + (c1a2 − c2 a1 ) 2 + (a1b2 − a2b1 ) 2
11.1.19 Distance between parallel lines r = a1 + µ b and r = a2 + λ b is
b × ( a2 – a1 )
.
b
11.1.20 The vector equation of a plane which is at a distance p from the origin, where
n̂ is the unit vector normal to the plane, is r . nˆ = p .
11.1.21 Equation of a plane which is at a distance p from the origin with direction
cosines of the normal to the plane as l, m, n is lx + my + nz = p.
11.1.22 The equation of a plane through a point whose position vector is a and
perpendicular to the vector n is (r – a ). n = 0 or r . n = d , where d = a . n.
11.1.23 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.
11.1.24 Equation of a plane passing through three non-collinear points (x1, y1, z1),
(x2, y2, z2) and (x3, y3, z3) is
