THREE-DIMENSIONAL GEOMETRY QUICK NOTES
Prepared by Neha Agrawal MATHEMATICALLY INCLINED
•
DIRECTION ANGLES OF A VECTOR

Let  : angle OP makes with the positive directions of x axis.

 : angle OP makes with the positive directions of y axis.

 : angle OP makes with the positive directions of z axis.
are called the DIRECTION ANGLES
•
DIRECTION COSINES OF A VECTOR

Cosines of these Direction angles are called the DIRECTION COSINES of OP .
They are denoted by l , m and n respectively.
0  ,  ,  
l = cos  ; m = cos  ; n = cos 
l 2 + m2 + n2 = 1

Also PO makes angles  −  ,  −  ,  −  with OX,OY,OZ axes.

So, the direction cosines of PO are: - l , - m , - n
DIRECTION RATIOS OF A VECTOR

Let l , m and n be the direction cosines of a vector r and a, b and c be three numbers such
that
l m n
= =
a b c
(i.e if a,b,c are three numbers proportional to the d.c’s of a line then a,b,c are called the

direction ratios of vector r )
•
DCs are always UNIQUE and DRs are NOT UNIQUE.
•
If a, b ,c are the direction ratios of a vector, then its direction cosines are given by
a
b
c

, 
, 
a2 + b2 + c2
a2 + b2 + c2
a2 + b2 + c2
(signs should be taken all +ve or all -ve )
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THREE-DIMENSIONAL GEOMETRY QUICK NOTES
Prepared by Neha Agrawal MATHEMATICALLY INCLINED
LINES
CONCEPT
VECTOR EQUATION
CARTESIAN EQUATION
EQUATION OF LINES
POINT PARALLEL
FORM
TWO-POINT
FORM
ANGLE
BETWEEN
TWO LINES
CONDITION
FOR TWO
LINES TO BE
PARALLEL
CONDITION
FOR TWO
LINES TO BE
PERPENDICUL
AR
Line passing through a point whose

p.v is a and is parallel to a given

vector b
Line passing through a point (x1,y1,z1) and
x − x1 y − y1 z − z1
=
=
having DR’s a,b,c
a
b
c

 
r = a + b
(a,b,c can be replaced by l,m,n)
Line passing through two points


whose p.v are a and b
Line passing through two points (x1,y1,z1)
and (x2,y2,z2)
 
 
r = a +  (b − a )
x − x1
y − y1
z − z1
=
=
x2 − x1 y 2 − y1 z 2 − z1
Angle between two lines


 
 
r = a1 + b1 and r = a2 + b2
Angle between
x − x1 y − y1 z − z1
x − x2 y − y 2 z − z 2
=
=
and
=
=
a1
b1
c1
a2
b2
c2
 
b1 .b2
cos  =  
b1 b2

b1 = b2

b1 .b2 = 0
cos  =
a1 a 2 + b1b2 + c1c 2
a1 + b1 + c1
2
2
2
a 2 + b2 + c 2
2
2
2
a1 b1 c1
=
=
a 2 b2 c2
a1 a 2 + b1b2 + c1c 2 = 0
Skew lines: Two lines in space which are neither parallel nor intersecting are called Skew lines. They
lie in different planes.
KDS HO GAYA !!! Prepared by Neha Agrawal Mathematically Inclined
THREE-DIMENSIONAL GEOMETRY QUICK NOTES
Prepared by Neha Agrawal MATHEMATICALLY INCLINED
SHORTEST DISTANCE BETWEEN
TWO SKEW LINES
SHORTEST DISTANCE BETWEEN TWO
PARALLEL LINES

 
 
If r = a1 + b1 and r = a 2 + b2 are
  

b1  b2 .(a 2 − a1 )
two lines then
 
b1  b2
 

b  (a 2 − a1 )


 
 
r = a1 + b and r = a2 + b is

b
(
)
PLANES
CONCEPT
VECTOR EQUATION
CARTESIAN EQUATION
EQUATION OF PLANES
NORMAL FORM
A plane passing having n̂ as
a unit vector normal to it
and at a distance d from the

origin r .nˆ = d
lx + my + nz = d
POINT-NORMAL
FORM
Plane passing through a

point whose p.v is a and ┴

to the vector n
   
(r − a ).n = 0
Plane passing through a point (x1,y1,z1) and
direction ratios of the normal to the plane is a,b,c
a( x − x1 ) + b( y − y1 ) + c( z − z1 ) = 0

r .n = d
PLANE THROUGH
THREE NONCOLLINEAR
POINTS
( )( )
 
(r − a ).[ b − a X c − a ] = 0
INTERCEPT FORM
x − x1
y − y1
z − z1
x 2 − x1
x3 − x1
y 2 − y1
y3 − y1
z 2 − z1 = 0
z 3 − z1
Plane cutting off intercepts a,b,c from x,y,z axes
x y z
+ + =1
a b c
PLANE THROUGH
INTERSECTION OF
TWO PLANES
(r.n − d )+  (r.n − d ) = 0
1
1
2
2
( A1 x + B1 y + C1 z − D1 ) +  ( A2 x + B2 y + C2 z − D2 ) = 0
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THREE-DIMENSIONAL GEOMETRY QUICK NOTES
Prepared by Neha Agrawal MATHEMATICALLY INCLINED
ANGLE BETWEEN
TWO PLANES
Angle between two planes


r .n1 = d1 , r .n2 = d 2 is
 
n1 .n 2
cos  =   (Angle
n1 n 2
a1 a 2 + b1b2 + c1c 2
cos  =
a1 + b1 + c1
2
2
2
a 2 + b2 + c 2
2
2
2
between their normal’s)
CONDITION FOR
TWO PLANES TO
BE PARALLEL

 
n1  n2 = 0
OR
a1 b1 c1
=
=
a 2 b2 c2


n1 = n2
 
n1 .n2 = 0
a1 a 2 + b1b2 + c1c 2 = 0

a.n − d  
(r .n = d , where p.v

n

of P is a )
The length of the ┴ from P(x1,y1,z1) to the plane
ax1 + by1 + cz1 + d
ax+by+cz+d=0 is
a2 + b2 + c2
DISTANCE
BETWEEN TWO
PARALLEL
PLANES
d1 − d 2

if r .n = d1 and

n

r .n = d 2
The distance between two parallel planes
ax+by+cz+d1=0 and ax+by+cz+d2=0 is
d1 − d 2
CONDITION FOR
TWO LINES TO BE
CO-PLANAR

 
Two lines r = a1 + b1 and

 
r = a2 + b2 are coplanar if

   
(a2 − a1 ).(b1  b2 ) = 0
CONDITION FOR
TWO PLANES TO
BE
PERPENDICULAR
DISTANCE
BETWEEN A
POINT AND A
PLANE
EQUATION OF A
PLANE
CONTAINING TWO
LINES
a2 + b2 + c2
x 2 − x1
y 2 − y1
z 2 − z1
a1
a2
b1
b2
c1
c2
=0

   
(r − a1 ).(b1  b2 ) = 0
x − x1
y − y1
z − z1
OR
a1
a2
b1
b2
c1
c2
 

 
(r − a2 ).(b1  b2 ) = 0
=0
OR
KDS HO GAYA !!! Prepared by Neha Agrawal Mathematically Inclined
THREE-DIMENSIONAL GEOMETRY QUICK NOTES
Prepared by Neha Agrawal MATHEMATICALLY INCLINED
ANGLE BETWEEN
A LINE AND A
PLANE
Angle between the line

 

r = a + b and plane r .n = d

b .n
is sin  =  
bn
Angle between the line
and the plane A( x − x1 ) + B( y − y1 ) + C ( z − z1 ) = 0
is
sin  =
CONDITION FOR A
LINE AND A
PLANE TO BE
PARALLEL
CONDITION FOR A
LINE AND A
PLANE TO BE
PERPENDICULAR
 
n1 .n2 = 0

 
n1  n2 = 0
OR
x − x1 y − y1 z − z1
=
=
a
b
c
Aa + Bb + Cc
A2 + B 2 + C 2 a 2 + b 2 + c 2
a1 a 2 + b1b2 + c1c 2 = 0
a1 b1 c1
=
=
a 2 b2 c2


n1 = n2
KDS HO GAYA !!! Prepared by Neha Agrawal Mathematically Inclined