Three Dimensional
Geometry
Coordinate System
The three mutually perpendicular lines in a space which divides the
space into eight parts and if these perpendicular lines are the
coordinate axes, then it is said to be a coordinate system.
y=0 Z
X'
Y'
x=0
z=0
Y
X
Z' O (0, 0, 0)
Note The coordinates of any point on the X , Y and Z-axes will be the form
(x, 0, 0), (0, y, 0) and (0, 0, z ) respectively.
Sign Convention
Octant Coordinate x y z
OXYZ + + +
OX ′ YZ − + +
OXY ′ Z + − +
OXYZ′ + + −
OX ′ Y ′ Z − – +
OX ′ YZ ′ − + −
OXY ′ Z ′ + – −
OX ′ Y ′ Z ′ − − –
, Distance between Two Points
Let P ( x1 , y1 , z1 ) and Q( x2 , y2 , z 2 ) be two given points. Then, distance
between these points is given by
PQ = ( x2 − x1 )2 + ( y2 − y1 )2 + ( z 2 − z1 )2
The distance of a point P ( x , y , z ) from origin O is
OP = x 2 + y 2 + z 2
Section Formulae
(i) The coordinates of any point, which divides the join of points
P ( x1 , y1 , z1 ) and Q( x2 , y2 , z 2 ) in the ratio m : n internally are
mx2 + nx1 my2 + ny1 mz 2 + nz1
, ,
m+n m+n m+n
(ii) The coordinates of any point, which divides the join of points
P ( x1 , y1 , z1 ) and Q( x2 , y2 , z 2 ) in the ratio m : n externally are
mx2 − nx1 my2 − ny1 mz 2 − mz1
, ,
m−n m−n m−n
(iii) The coordinates of mid-point of P and Q are
x1 + x2 y1 + y2 z1 + z 2
, ,
2 2 2
(iv) Coordinates of the centroid of a triangle formed with vertices
P ( x1 , y1 , z1 ), Q ( x2 , y2 , z 2 ) and R ( x3 , y3 , z3 ) are
x1 + x2 + x3 y1 + y2 + y3 z1 + z 2 + z3
, ,
3 3 3
(v) Centroid of a Tetrahedron
If ( x1 , y1 , z1 ), ( x2 , y2 , z 2 ), ( x3 , y3 , z3 ) and ( x4 , y4 , z 4 ) are the vertices
of a tetrahedron, then its centroid G is given by
x1 + x2 + x3 + x4 y1 + y2 + y3 + y4 z1 + z 2 + z3 + z 4
, , .
4 4 4
Area of Triangle
If the vertices of a triangle be A( x1 , y1 , z1 ), B ( x2 , y2 , z 2 ) and C( x3 , y3 , z3 ),
then
Area of ∆ABC = ∆ xy
2
+ ∆ yz
2
+ ∆ zx
2
Geometry
Coordinate System
The three mutually perpendicular lines in a space which divides the
space into eight parts and if these perpendicular lines are the
coordinate axes, then it is said to be a coordinate system.
y=0 Z
X'
Y'
x=0
z=0
Y
X
Z' O (0, 0, 0)
Note The coordinates of any point on the X , Y and Z-axes will be the form
(x, 0, 0), (0, y, 0) and (0, 0, z ) respectively.
Sign Convention
Octant Coordinate x y z
OXYZ + + +
OX ′ YZ − + +
OXY ′ Z + − +
OXYZ′ + + −
OX ′ Y ′ Z − – +
OX ′ YZ ′ − + −
OXY ′ Z ′ + – −
OX ′ Y ′ Z ′ − − –
, Distance between Two Points
Let P ( x1 , y1 , z1 ) and Q( x2 , y2 , z 2 ) be two given points. Then, distance
between these points is given by
PQ = ( x2 − x1 )2 + ( y2 − y1 )2 + ( z 2 − z1 )2
The distance of a point P ( x , y , z ) from origin O is
OP = x 2 + y 2 + z 2
Section Formulae
(i) The coordinates of any point, which divides the join of points
P ( x1 , y1 , z1 ) and Q( x2 , y2 , z 2 ) in the ratio m : n internally are
mx2 + nx1 my2 + ny1 mz 2 + nz1
, ,
m+n m+n m+n
(ii) The coordinates of any point, which divides the join of points
P ( x1 , y1 , z1 ) and Q( x2 , y2 , z 2 ) in the ratio m : n externally are
mx2 − nx1 my2 − ny1 mz 2 − mz1
, ,
m−n m−n m−n
(iii) The coordinates of mid-point of P and Q are
x1 + x2 y1 + y2 z1 + z 2
, ,
2 2 2
(iv) Coordinates of the centroid of a triangle formed with vertices
P ( x1 , y1 , z1 ), Q ( x2 , y2 , z 2 ) and R ( x3 , y3 , z3 ) are
x1 + x2 + x3 y1 + y2 + y3 z1 + z 2 + z3
, ,
3 3 3
(v) Centroid of a Tetrahedron
If ( x1 , y1 , z1 ), ( x2 , y2 , z 2 ), ( x3 , y3 , z3 ) and ( x4 , y4 , z 4 ) are the vertices
of a tetrahedron, then its centroid G is given by
x1 + x2 + x3 + x4 y1 + y2 + y3 + y4 z1 + z 2 + z3 + z 4
, , .
4 4 4
Area of Triangle
If the vertices of a triangle be A( x1 , y1 , z1 ), B ( x2 , y2 , z 2 ) and C( x3 , y3 , z3 ),
then
Area of ∆ABC = ∆ xy
2
+ ∆ yz
2
+ ∆ zx
2
