Page 1
Three Dimensional Geometry
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this
surface is called a plane. This fixed line is called the normal to the plane.
Equation of a plane :
(i) Vector form : The equation (???  ???
0
) · ??? ? = 0 represents a plane containing the point with position
vector is a vector normal to the plane.
The above equation can also be written as ??? · ??? ? = ?? , where ?? = ???
0
· ??? ?
(ii) Cartesian form : The equation of a plane passing through the point (?? 1
, ?? 1
, ?? 1
) is given by
?? (??  ?? 1
) + ?? (??  ?? 1
) + ?? (??  ?? 1
) = 0 where ?? , ?? , ?? are the direction ratios of the normal to the plane.
(iii) Normal form : Vector equation of a plane normal to unit vector and at a distance ?? from the
origin is ??? · ??? ? = ?? . Normal form of the equation of a plane is ???? + ???? + ???? = ?? , where, ?? , ?? , ?? are the
direction cosines of the normal to the plane and ?? is the distance of the plane from the origin.
(iv) General form : ???? + ???? + ???? + ?? = 0 is the equation of a plane, where ?? , ?? , ?? are the direction
ratios of the normal to the plane.
(v) Plane through three points : The equation of the plane through three noncollinear points
(?? 1
, ?? 1
, ?? 1
), (?? 2
, ?? 2
, ?? 2
), (?? 3
, ?? 3
, ?? 3
) ???? ??  ?? 3
??  ?? 3
??  ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
 = 0
(vi) Intercept Form : The equation of a plane cutting intercept a, b, c on the axes is
?? ?? +
?? ?? +
?? ?? = 1
Note :
Equation of ???? plane, ???? plane and ???? plane is ?? = 0, ?? = 0 and ?? = 0
Transformation of the equation of a plane to the normal form: To reduce any equation ???? + ???? +
????  ?? = 0 to the normal form, first write the constant term on the right hand side and make it
positive, then divide each term by v?? 2
+ ?? 2
+ ?? 2
, where ?? , ?? , ?? are coefficients of ?? , ?? and ??
respectively e.g.
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
=
?? ±v?? 2
+ ?? 2
+ ?? 2
Where (+) sign is to be taken if ?? > 0 and () sign is to be taken if ?? < 0.
Page 2
Three Dimensional Geometry
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this
surface is called a plane. This fixed line is called the normal to the plane.
Equation of a plane :
(i) Vector form : The equation (???  ???
0
) · ??? ? = 0 represents a plane containing the point with position
vector is a vector normal to the plane.
The above equation can also be written as ??? · ??? ? = ?? , where ?? = ???
0
· ??? ?
(ii) Cartesian form : The equation of a plane passing through the point (?? 1
, ?? 1
, ?? 1
) is given by
?? (??  ?? 1
) + ?? (??  ?? 1
) + ?? (??  ?? 1
) = 0 where ?? , ?? , ?? are the direction ratios of the normal to the plane.
(iii) Normal form : Vector equation of a plane normal to unit vector and at a distance ?? from the
origin is ??? · ??? ? = ?? . Normal form of the equation of a plane is ???? + ???? + ???? = ?? , where, ?? , ?? , ?? are the
direction cosines of the normal to the plane and ?? is the distance of the plane from the origin.
(iv) General form : ???? + ???? + ???? + ?? = 0 is the equation of a plane, where ?? , ?? , ?? are the direction
ratios of the normal to the plane.
(v) Plane through three points : The equation of the plane through three noncollinear points
(?? 1
, ?? 1
, ?? 1
), (?? 2
, ?? 2
, ?? 2
), (?? 3
, ?? 3
, ?? 3
) ???? ??  ?? 3
??  ?? 3
??  ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
 = 0
(vi) Intercept Form : The equation of a plane cutting intercept a, b, c on the axes is
?? ?? +
?? ?? +
?? ?? = 1
Note :
Equation of ???? plane, ???? plane and ???? plane is ?? = 0, ?? = 0 and ?? = 0
Transformation of the equation of a plane to the normal form: To reduce any equation ???? + ???? +
????  ?? = 0 to the normal form, first write the constant term on the right hand side and make it
positive, then divide each term by v?? 2
+ ?? 2
+ ?? 2
, where ?? , ?? , ?? are coefficients of ?? , ?? and ??
respectively e.g.
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
=
?? ±v?? 2
+ ?? 2
+ ?? 2
Where (+) sign is to be taken if ?? > 0 and () sign is to be taken if ?? < 0.
A plane ???? + ???? + ???? + ?? = 0 divides the line segment joining (?? 1
, ?? 1
, ?? 1
) and (?? 2
, ?? 2
, ?? 2
). in the ratio
(
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? ?? 2
+?? )
Coplanarity of four points
The points ?? (?? 1
?? 1
?? 1
), ?? (?? 2
?? 2
?? 2
)?? (?? 3
?? 3
?? 3
) and ?? (?? 4
?? 4
?? 4
) are coplanar then
?? 2
 ?? 1
?? 2
 ?? 1
?? 2
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
 = 0
Problem 41 : Find the equation of the plane upon which the length of normal from origin is 10 and
direction ratios of this normal are 3,2,6.
Solution: If ?? be the length of perpendicular from origin to the plane and ?? , ?? , ?? be the direction
cosines of this normal, then its equation is
???? + ???? + ???? = 10
Direction ratios of normal to the plane are 3, 2, 6
? Direction cosines of normal to the required plane are ?? =
3
7
, ?? =
2
7
, ?? =
6
7
Equation of required plane is
3
7
?? +
2
7
?? +
6
7
?? = 10 or, 3?? + 2?? + 6?? = 70
Problem 42 :Find the plane through the points (2, 3,3), (5,2,0), (1, 7,1)
Solution : ??  2 ?? + 3 ??  3  5  2 2 + 3 0  3 1  2  7 + 3 1  3  = 0 or ??  2 ?? + 3 ??  3 
7 5  3  1  4  2  = 0 ? 2?? + ??  3?? + 8 = 0
Problem 43 : If ?? be any point on the plane ???? + ???? + ???? = ?? and ?? be a point on the line ???? such
that
???? . ???? = ?? 2
, show that the locus of the point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Solution : Let ?? = (?? , ?? , ?? ), ?? = (?? 1
, ?? 1
, ?? 1
)
Direction ratios of OP are ?? , ?? , ?? and direction ratios of ???? are ?? 1
, ?? 1
, ?? 1
.
Since ?? , ?? , ?? are collinear, we have
?? ?? 1
=
?? ?? 1
=
?? ?? 1
= ?? (say)
As ?? (?? , ?? , ?? ) lies on the plane ???? + ???? + ???? = ?? ,
???? + ???? + ???? = ?? or ?? (?? ?? 1
+ ????
1
+ ???? ?? 1
) = ??
Given ???? . ???? = ?? 2
? v?? 2
+ ?? 2
+ ?? 2
v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, v?? 2
(?? 1
2
+ ?? 1
2
+ ?? 1
2
)v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, ?? (?? 1
2
+ ?? 1
2
+ ?? 1
2
) = ?? 2
Page 3
Three Dimensional Geometry
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this
surface is called a plane. This fixed line is called the normal to the plane.
Equation of a plane :
(i) Vector form : The equation (???  ???
0
) · ??? ? = 0 represents a plane containing the point with position
vector is a vector normal to the plane.
The above equation can also be written as ??? · ??? ? = ?? , where ?? = ???
0
· ??? ?
(ii) Cartesian form : The equation of a plane passing through the point (?? 1
, ?? 1
, ?? 1
) is given by
?? (??  ?? 1
) + ?? (??  ?? 1
) + ?? (??  ?? 1
) = 0 where ?? , ?? , ?? are the direction ratios of the normal to the plane.
(iii) Normal form : Vector equation of a plane normal to unit vector and at a distance ?? from the
origin is ??? · ??? ? = ?? . Normal form of the equation of a plane is ???? + ???? + ???? = ?? , where, ?? , ?? , ?? are the
direction cosines of the normal to the plane and ?? is the distance of the plane from the origin.
(iv) General form : ???? + ???? + ???? + ?? = 0 is the equation of a plane, where ?? , ?? , ?? are the direction
ratios of the normal to the plane.
(v) Plane through three points : The equation of the plane through three noncollinear points
(?? 1
, ?? 1
, ?? 1
), (?? 2
, ?? 2
, ?? 2
), (?? 3
, ?? 3
, ?? 3
) ???? ??  ?? 3
??  ?? 3
??  ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
 = 0
(vi) Intercept Form : The equation of a plane cutting intercept a, b, c on the axes is
?? ?? +
?? ?? +
?? ?? = 1
Note :
Equation of ???? plane, ???? plane and ???? plane is ?? = 0, ?? = 0 and ?? = 0
Transformation of the equation of a plane to the normal form: To reduce any equation ???? + ???? +
????  ?? = 0 to the normal form, first write the constant term on the right hand side and make it
positive, then divide each term by v?? 2
+ ?? 2
+ ?? 2
, where ?? , ?? , ?? are coefficients of ?? , ?? and ??
respectively e.g.
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
=
?? ±v?? 2
+ ?? 2
+ ?? 2
Where (+) sign is to be taken if ?? > 0 and () sign is to be taken if ?? < 0.
A plane ???? + ???? + ???? + ?? = 0 divides the line segment joining (?? 1
, ?? 1
, ?? 1
) and (?? 2
, ?? 2
, ?? 2
). in the ratio
(
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? ?? 2
+?? )
Coplanarity of four points
The points ?? (?? 1
?? 1
?? 1
), ?? (?? 2
?? 2
?? 2
)?? (?? 3
?? 3
?? 3
) and ?? (?? 4
?? 4
?? 4
) are coplanar then
?? 2
 ?? 1
?? 2
 ?? 1
?? 2
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
 = 0
Problem 41 : Find the equation of the plane upon which the length of normal from origin is 10 and
direction ratios of this normal are 3,2,6.
Solution: If ?? be the length of perpendicular from origin to the plane and ?? , ?? , ?? be the direction
cosines of this normal, then its equation is
???? + ???? + ???? = 10
Direction ratios of normal to the plane are 3, 2, 6
? Direction cosines of normal to the required plane are ?? =
3
7
, ?? =
2
7
, ?? =
6
7
Equation of required plane is
3
7
?? +
2
7
?? +
6
7
?? = 10 or, 3?? + 2?? + 6?? = 70
Problem 42 :Find the plane through the points (2, 3,3), (5,2,0), (1, 7,1)
Solution : ??  2 ?? + 3 ??  3  5  2 2 + 3 0  3 1  2  7 + 3 1  3  = 0 or ??  2 ?? + 3 ??  3 
7 5  3  1  4  2  = 0 ? 2?? + ??  3?? + 8 = 0
Problem 43 : If ?? be any point on the plane ???? + ???? + ???? = ?? and ?? be a point on the line ???? such
that
???? . ???? = ?? 2
, show that the locus of the point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Solution : Let ?? = (?? , ?? , ?? ), ?? = (?? 1
, ?? 1
, ?? 1
)
Direction ratios of OP are ?? , ?? , ?? and direction ratios of ???? are ?? 1
, ?? 1
, ?? 1
.
Since ?? , ?? , ?? are collinear, we have
?? ?? 1
=
?? ?? 1
=
?? ?? 1
= ?? (say)
As ?? (?? , ?? , ?? ) lies on the plane ???? + ???? + ???? = ?? ,
???? + ???? + ???? = ?? or ?? (?? ?? 1
+ ????
1
+ ???? ?? 1
) = ??
Given ???? . ???? = ?? 2
? v?? 2
+ ?? 2
+ ?? 2
v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, v?? 2
(?? 1
2
+ ?? 1
2
+ ?? 1
2
)v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, ?? (?? 1
2
+ ?? 1
2
+ ?? 1
2
) = ?? 2
Hence the locus of point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Problem 44 : A moving plane passes through a fixed point (?? , ?? , ?? ) and cuts the coordinate axes A, B,
C . Find the locus of the centroid of the tetrahedron OABC.
Solution : Let the plane be
?? ?? +
?? ?? +
?? ?? = 1,0(0,0,0), ?? (?? , 0,0), ?? (0, ?? , 0)
C (0,0, ?? ). Centroid of ???????? is (
?? 4
,
?? 4
,
?? 4
)
The plane passes through (?? , ?? , ?? ) ?
?? ?? +
?? ?? +
?? ?? = 1
Centroid, ?? =
?? 4
, ?? =
?? 4
, ?? =
?? 4
or ?? = 4?? , ?? = 4?? , ?? = 4??
Now (1) gives the locus of ?? as
?? ?? +
?? ?? +
?? ?? = 4
Position of point with respect to plane :
A plane divides the three dimensional space in two equal parts. Two points ?? (?? 1
?? 1
?? 1
) and ?? (?? 2
?? 2
?? 2
)
are on the same side of the plane ???? + ???? + ???? + ?? = 0 if ?? ?? 1
+ ?? ?? 1
+ ?? ?? 1
+ ?? and ????
2
+ ????
2
+ ????
2
+ ??
are both positive or both negative and are opposite side of plane if both of these values are in
opposite sign
Problem 45 : Show that the points (1,2,3) and (2, 1,4) lie on opposite sides of the plane ?? + 4?? + ?? 
3 = 0
Solution : Since the numbers 1 + 4 × 2 + 3  3 = 9 and 2  4 + 4  3 = 1 are of opposite sign, then
points are on opposite sides of the plane.
A plane & a point
Let ?? = ???? + ???? + ???? + ?? = 0 is a given plane and ?? (?? 1
, ?? 1
, ?? 1
) is given point as shown in figure.
Let ?? (?? '
, ?? '
, ?? '
) be the foot of the point ?? (?? 1
, ?? 1
, ?? 1
) with respect to the plane ?? .
And ?? (?? ''
, ?? ''
, ?? ''
) be the reflection of point ?? (?? 1
, ?? 1
, ?? 1
) with respect to the plane ?? .
(i) Distance of the point (?? 1
, ?? 1
, ?? 1
) from the plane ???? + ???? + ???? + ?? = 0 is given by 
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? v?? 2
+?? 2
+?? 2
.
Page 4
Three Dimensional Geometry
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this
surface is called a plane. This fixed line is called the normal to the plane.
Equation of a plane :
(i) Vector form : The equation (???  ???
0
) · ??? ? = 0 represents a plane containing the point with position
vector is a vector normal to the plane.
The above equation can also be written as ??? · ??? ? = ?? , where ?? = ???
0
· ??? ?
(ii) Cartesian form : The equation of a plane passing through the point (?? 1
, ?? 1
, ?? 1
) is given by
?? (??  ?? 1
) + ?? (??  ?? 1
) + ?? (??  ?? 1
) = 0 where ?? , ?? , ?? are the direction ratios of the normal to the plane.
(iii) Normal form : Vector equation of a plane normal to unit vector and at a distance ?? from the
origin is ??? · ??? ? = ?? . Normal form of the equation of a plane is ???? + ???? + ???? = ?? , where, ?? , ?? , ?? are the
direction cosines of the normal to the plane and ?? is the distance of the plane from the origin.
(iv) General form : ???? + ???? + ???? + ?? = 0 is the equation of a plane, where ?? , ?? , ?? are the direction
ratios of the normal to the plane.
(v) Plane through three points : The equation of the plane through three noncollinear points
(?? 1
, ?? 1
, ?? 1
), (?? 2
, ?? 2
, ?? 2
), (?? 3
, ?? 3
, ?? 3
) ???? ??  ?? 3
??  ?? 3
??  ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
 = 0
(vi) Intercept Form : The equation of a plane cutting intercept a, b, c on the axes is
?? ?? +
?? ?? +
?? ?? = 1
Note :
Equation of ???? plane, ???? plane and ???? plane is ?? = 0, ?? = 0 and ?? = 0
Transformation of the equation of a plane to the normal form: To reduce any equation ???? + ???? +
????  ?? = 0 to the normal form, first write the constant term on the right hand side and make it
positive, then divide each term by v?? 2
+ ?? 2
+ ?? 2
, where ?? , ?? , ?? are coefficients of ?? , ?? and ??
respectively e.g.
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
=
?? ±v?? 2
+ ?? 2
+ ?? 2
Where (+) sign is to be taken if ?? > 0 and () sign is to be taken if ?? < 0.
A plane ???? + ???? + ???? + ?? = 0 divides the line segment joining (?? 1
, ?? 1
, ?? 1
) and (?? 2
, ?? 2
, ?? 2
). in the ratio
(
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? ?? 2
+?? )
Coplanarity of four points
The points ?? (?? 1
?? 1
?? 1
), ?? (?? 2
?? 2
?? 2
)?? (?? 3
?? 3
?? 3
) and ?? (?? 4
?? 4
?? 4
) are coplanar then
?? 2
 ?? 1
?? 2
 ?? 1
?? 2
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
 = 0
Problem 41 : Find the equation of the plane upon which the length of normal from origin is 10 and
direction ratios of this normal are 3,2,6.
Solution: If ?? be the length of perpendicular from origin to the plane and ?? , ?? , ?? be the direction
cosines of this normal, then its equation is
???? + ???? + ???? = 10
Direction ratios of normal to the plane are 3, 2, 6
? Direction cosines of normal to the required plane are ?? =
3
7
, ?? =
2
7
, ?? =
6
7
Equation of required plane is
3
7
?? +
2
7
?? +
6
7
?? = 10 or, 3?? + 2?? + 6?? = 70
Problem 42 :Find the plane through the points (2, 3,3), (5,2,0), (1, 7,1)
Solution : ??  2 ?? + 3 ??  3  5  2 2 + 3 0  3 1  2  7 + 3 1  3  = 0 or ??  2 ?? + 3 ??  3 
7 5  3  1  4  2  = 0 ? 2?? + ??  3?? + 8 = 0
Problem 43 : If ?? be any point on the plane ???? + ???? + ???? = ?? and ?? be a point on the line ???? such
that
???? . ???? = ?? 2
, show that the locus of the point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Solution : Let ?? = (?? , ?? , ?? ), ?? = (?? 1
, ?? 1
, ?? 1
)
Direction ratios of OP are ?? , ?? , ?? and direction ratios of ???? are ?? 1
, ?? 1
, ?? 1
.
Since ?? , ?? , ?? are collinear, we have
?? ?? 1
=
?? ?? 1
=
?? ?? 1
= ?? (say)
As ?? (?? , ?? , ?? ) lies on the plane ???? + ???? + ???? = ?? ,
???? + ???? + ???? = ?? or ?? (?? ?? 1
+ ????
1
+ ???? ?? 1
) = ??
Given ???? . ???? = ?? 2
? v?? 2
+ ?? 2
+ ?? 2
v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, v?? 2
(?? 1
2
+ ?? 1
2
+ ?? 1
2
)v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, ?? (?? 1
2
+ ?? 1
2
+ ?? 1
2
) = ?? 2
Hence the locus of point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Problem 44 : A moving plane passes through a fixed point (?? , ?? , ?? ) and cuts the coordinate axes A, B,
C . Find the locus of the centroid of the tetrahedron OABC.
Solution : Let the plane be
?? ?? +
?? ?? +
?? ?? = 1,0(0,0,0), ?? (?? , 0,0), ?? (0, ?? , 0)
C (0,0, ?? ). Centroid of ???????? is (
?? 4
,
?? 4
,
?? 4
)
The plane passes through (?? , ?? , ?? ) ?
?? ?? +
?? ?? +
?? ?? = 1
Centroid, ?? =
?? 4
, ?? =
?? 4
, ?? =
?? 4
or ?? = 4?? , ?? = 4?? , ?? = 4??
Now (1) gives the locus of ?? as
?? ?? +
?? ?? +
?? ?? = 4
Position of point with respect to plane :
A plane divides the three dimensional space in two equal parts. Two points ?? (?? 1
?? 1
?? 1
) and ?? (?? 2
?? 2
?? 2
)
are on the same side of the plane ???? + ???? + ???? + ?? = 0 if ?? ?? 1
+ ?? ?? 1
+ ?? ?? 1
+ ?? and ????
2
+ ????
2
+ ????
2
+ ??
are both positive or both negative and are opposite side of plane if both of these values are in
opposite sign
Problem 45 : Show that the points (1,2,3) and (2, 1,4) lie on opposite sides of the plane ?? + 4?? + ?? 
3 = 0
Solution : Since the numbers 1 + 4 × 2 + 3  3 = 9 and 2  4 + 4  3 = 1 are of opposite sign, then
points are on opposite sides of the plane.
A plane & a point
Let ?? = ???? + ???? + ???? + ?? = 0 is a given plane and ?? (?? 1
, ?? 1
, ?? 1
) is given point as shown in figure.
Let ?? (?? '
, ?? '
, ?? '
) be the foot of the point ?? (?? 1
, ?? 1
, ?? 1
) with respect to the plane ?? .
And ?? (?? ''
, ?? ''
, ?? ''
) be the reflection of point ?? (?? 1
, ?? 1
, ?? 1
) with respect to the plane ?? .
(i) Distance of the point (?? 1
, ?? 1
, ?? 1
) from the plane ???? + ???? + ???? + ?? = 0 is given by 
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? v?? 2
+?? 2
+?? 2
.
(ii) The length of the perpendicular from a point having position vector ?? ? to plane ??? · ??? ? = ?? is ?? =
?? ? ?·?? ? ??? 
?? ? ?
.
(iii) The coordinates of the foot (F) of perpendicular from the point (?? 1
, ?? 1
, ?? 1
) to the plane
???? + ???? + ???? + ?? = 0 are
?? ?? 1
?? =
?? ?? 1
?? =
?? ?? 1
?? = 
(?? ?? 1
+?? ?? 1
+?? ?? 1
+?? )
?? 2
+?? 2
+?? 2
(iv) The coordinates of the Image (R) of point (?? 1
, ?? 1
, ?? 1
) to the plane
???? + ???? + ???? + ?? = 0 are
?? ?? 1
?? =
?? ?? 1
?? =
?? ?? 1
?? = 
2(?? ?? 1
+?? ?? 1
+?? ?? 1
+?? )
?? 2
+?? 2
+?? 2
Problem 46 : Find the image of the point ?? (3,5,7) in the plane 2?? + ?? + ?? = 0.
Solution: Given plane is 2?? + ?? + ?? = 0
Direction ratios of normal to plane (1) are 2,1,1
Let ?? be the image of point ?? in plane (1). Let ???? meet plane (1) in ?? then ???? ? plane (1)
Let ?? = (2?? + 3, ?? + 5, ?? + 7)
Since ?? lies on plane (1)
? 2(2?? + 3) + ?? + 5 + ?? + 7 = 0 or, 6?? + 18 = 0 ? ?? = 3
? ?? = (3,2,4)
Let ?? = (?? , ?? , ?? )
Since ?? is the middle point of ????
? 3 =
?? + 3
2
? ?? = 9 ?????? 2 =
?? + 5
2
? ?? = 1 ?????? 4 =
?? + 7
2
? ?? = 1 ? ?? = (9, 1,1).
Problem 47 : A plane passes through a fixed point (?? , ?? , ?? ). Show that the locus of the foot of
perpendicular to it from the origin is the sphere ?? 2
+ ?? 2
+ ?? 2
 ????  ????  ???? = 0
Solution : Let the equation of the variable plane be ???? + ???? + ???? + ?? = 0
Plane passes through the fixed point (?? , ?? , ?? ) ? ???? + ???? + ???? + ?? = 0
Let ?? (?? , ?? , ?? ) be the foot of perpendicular from origin to plane (1).
Direction ratios of OP are
Page 5
Three Dimensional Geometry
A PLANE
If line joining any two points on a surface lies completely on it then the surface is a plane.
OR
If line joining any two points on a surface is perpendicular to some fixed straight line. Then this
surface is called a plane. This fixed line is called the normal to the plane.
Equation of a plane :
(i) Vector form : The equation (???  ???
0
) · ??? ? = 0 represents a plane containing the point with position
vector is a vector normal to the plane.
The above equation can also be written as ??? · ??? ? = ?? , where ?? = ???
0
· ??? ?
(ii) Cartesian form : The equation of a plane passing through the point (?? 1
, ?? 1
, ?? 1
) is given by
?? (??  ?? 1
) + ?? (??  ?? 1
) + ?? (??  ?? 1
) = 0 where ?? , ?? , ?? are the direction ratios of the normal to the plane.
(iii) Normal form : Vector equation of a plane normal to unit vector and at a distance ?? from the
origin is ??? · ??? ? = ?? . Normal form of the equation of a plane is ???? + ???? + ???? = ?? , where, ?? , ?? , ?? are the
direction cosines of the normal to the plane and ?? is the distance of the plane from the origin.
(iv) General form : ???? + ???? + ???? + ?? = 0 is the equation of a plane, where ?? , ?? , ?? are the direction
ratios of the normal to the plane.
(v) Plane through three points : The equation of the plane through three noncollinear points
(?? 1
, ?? 1
, ?? 1
), (?? 2
, ?? 2
, ?? 2
), (?? 3
, ?? 3
, ?? 3
) ???? ??  ?? 3
??  ?? 3
??  ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 1
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
?? 2
 ?? 3
 = 0
(vi) Intercept Form : The equation of a plane cutting intercept a, b, c on the axes is
?? ?? +
?? ?? +
?? ?? = 1
Note :
Equation of ???? plane, ???? plane and ???? plane is ?? = 0, ?? = 0 and ?? = 0
Transformation of the equation of a plane to the normal form: To reduce any equation ???? + ???? +
????  ?? = 0 to the normal form, first write the constant term on the right hand side and make it
positive, then divide each term by v?? 2
+ ?? 2
+ ?? 2
, where ?? , ?? , ?? are coefficients of ?? , ?? and ??
respectively e.g.
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
+
????
±v?? 2
+ ?? 2
+ ?? 2
=
?? ±v?? 2
+ ?? 2
+ ?? 2
Where (+) sign is to be taken if ?? > 0 and () sign is to be taken if ?? < 0.
A plane ???? + ???? + ???? + ?? = 0 divides the line segment joining (?? 1
, ?? 1
, ?? 1
) and (?? 2
, ?? 2
, ?? 2
). in the ratio
(
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? ?? 2
+?? )
Coplanarity of four points
The points ?? (?? 1
?? 1
?? 1
), ?? (?? 2
?? 2
?? 2
)?? (?? 3
?? 3
?? 3
) and ?? (?? 4
?? 4
?? 4
) are coplanar then
?? 2
 ?? 1
?? 2
 ?? 1
?? 2
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 3
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
?? 4
 ?? 1
 = 0
Problem 41 : Find the equation of the plane upon which the length of normal from origin is 10 and
direction ratios of this normal are 3,2,6.
Solution: If ?? be the length of perpendicular from origin to the plane and ?? , ?? , ?? be the direction
cosines of this normal, then its equation is
???? + ???? + ???? = 10
Direction ratios of normal to the plane are 3, 2, 6
? Direction cosines of normal to the required plane are ?? =
3
7
, ?? =
2
7
, ?? =
6
7
Equation of required plane is
3
7
?? +
2
7
?? +
6
7
?? = 10 or, 3?? + 2?? + 6?? = 70
Problem 42 :Find the plane through the points (2, 3,3), (5,2,0), (1, 7,1)
Solution : ??  2 ?? + 3 ??  3  5  2 2 + 3 0  3 1  2  7 + 3 1  3  = 0 or ??  2 ?? + 3 ??  3 
7 5  3  1  4  2  = 0 ? 2?? + ??  3?? + 8 = 0
Problem 43 : If ?? be any point on the plane ???? + ???? + ???? = ?? and ?? be a point on the line ???? such
that
???? . ???? = ?? 2
, show that the locus of the point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Solution : Let ?? = (?? , ?? , ?? ), ?? = (?? 1
, ?? 1
, ?? 1
)
Direction ratios of OP are ?? , ?? , ?? and direction ratios of ???? are ?? 1
, ?? 1
, ?? 1
.
Since ?? , ?? , ?? are collinear, we have
?? ?? 1
=
?? ?? 1
=
?? ?? 1
= ?? (say)
As ?? (?? , ?? , ?? ) lies on the plane ???? + ???? + ???? = ?? ,
???? + ???? + ???? = ?? or ?? (?? ?? 1
+ ????
1
+ ???? ?? 1
) = ??
Given ???? . ???? = ?? 2
? v?? 2
+ ?? 2
+ ?? 2
v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, v?? 2
(?? 1
2
+ ?? 1
2
+ ?? 1
2
)v?? 1
2
+ ?? 1
2
+ ?? 1
2
= ?? 2
or, ?? (?? 1
2
+ ?? 1
2
+ ?? 1
2
) = ?? 2
Hence the locus of point ?? is ?? (???? + ???? + ???? ) = ?? 2
+ ?? 2
+ ?? 2
.
Problem 44 : A moving plane passes through a fixed point (?? , ?? , ?? ) and cuts the coordinate axes A, B,
C . Find the locus of the centroid of the tetrahedron OABC.
Solution : Let the plane be
?? ?? +
?? ?? +
?? ?? = 1,0(0,0,0), ?? (?? , 0,0), ?? (0, ?? , 0)
C (0,0, ?? ). Centroid of ???????? is (
?? 4
,
?? 4
,
?? 4
)
The plane passes through (?? , ?? , ?? ) ?
?? ?? +
?? ?? +
?? ?? = 1
Centroid, ?? =
?? 4
, ?? =
?? 4
, ?? =
?? 4
or ?? = 4?? , ?? = 4?? , ?? = 4??
Now (1) gives the locus of ?? as
?? ?? +
?? ?? +
?? ?? = 4
Position of point with respect to plane :
A plane divides the three dimensional space in two equal parts. Two points ?? (?? 1
?? 1
?? 1
) and ?? (?? 2
?? 2
?? 2
)
are on the same side of the plane ???? + ???? + ???? + ?? = 0 if ?? ?? 1
+ ?? ?? 1
+ ?? ?? 1
+ ?? and ????
2
+ ????
2
+ ????
2
+ ??
are both positive or both negative and are opposite side of plane if both of these values are in
opposite sign
Problem 45 : Show that the points (1,2,3) and (2, 1,4) lie on opposite sides of the plane ?? + 4?? + ?? 
3 = 0
Solution : Since the numbers 1 + 4 × 2 + 3  3 = 9 and 2  4 + 4  3 = 1 are of opposite sign, then
points are on opposite sides of the plane.
A plane & a point
Let ?? = ???? + ???? + ???? + ?? = 0 is a given plane and ?? (?? 1
, ?? 1
, ?? 1
) is given point as shown in figure.
Let ?? (?? '
, ?? '
, ?? '
) be the foot of the point ?? (?? 1
, ?? 1
, ?? 1
) with respect to the plane ?? .
And ?? (?? ''
, ?? ''
, ?? ''
) be the reflection of point ?? (?? 1
, ?? 1
, ?? 1
) with respect to the plane ?? .
(i) Distance of the point (?? 1
, ?? 1
, ?? 1
) from the plane ???? + ???? + ???? + ?? = 0 is given by 
?? ?? 1
+?? ?? 1
+?? ?? 1
+?? v?? 2
+?? 2
+?? 2
.
(ii) The length of the perpendicular from a point having position vector ?? ? to plane ??? · ??? ? = ?? is ?? =
?? ? ?·?? ? ??? 
?? ? ?
.
(iii) The coordinates of the foot (F) of perpendicular from the point (?? 1
, ?? 1
, ?? 1
) to the plane
???? + ???? + ???? + ?? = 0 are
?? ?? 1
?? =
?? ?? 1
?? =
?? ?? 1
?? = 
(?? ?? 1
+?? ?? 1
+?? ?? 1
+?? )
?? 2
+?? 2
+?? 2
(iv) The coordinates of the Image (R) of point (?? 1
, ?? 1
, ?? 1
) to the plane
???? + ???? + ???? + ?? = 0 are
?? ?? 1
?? =
?? ?? 1
?? =
?? ?? 1
?? = 
2(?? ?? 1
+?? ?? 1
+?? ?? 1
+?? )
?? 2
+?? 2
+?? 2
Problem 46 : Find the image of the point ?? (3,5,7) in the plane 2?? + ?? + ?? = 0.
Solution: Given plane is 2?? + ?? + ?? = 0
Direction ratios of normal to plane (1) are 2,1,1
Let ?? be the image of point ?? in plane (1). Let ???? meet plane (1) in ?? then ???? ? plane (1)
Let ?? = (2?? + 3, ?? + 5, ?? + 7)
Since ?? lies on plane (1)
? 2(2?? + 3) + ?? + 5 + ?? + 7 = 0 or, 6?? + 18 = 0 ? ?? = 3
? ?? = (3,2,4)
Let ?? = (?? , ?? , ?? )
Since ?? is the middle point of ????
? 3 =
?? + 3
2
? ?? = 9 ?????? 2 =
?? + 5
2
? ?? = 1 ?????? 4 =
?? + 7
2
? ?? = 1 ? ?? = (9, 1,1).
Problem 47 : A plane passes through a fixed point (?? , ?? , ?? ). Show that the locus of the foot of
perpendicular to it from the origin is the sphere ?? 2
+ ?? 2
+ ?? 2
 ????  ????  ???? = 0
Solution : Let the equation of the variable plane be ???? + ???? + ???? + ?? = 0
Plane passes through the fixed point (?? , ?? , ?? ) ? ???? + ???? + ???? + ?? = 0
Let ?? (?? , ?? , ?? ) be the foot of perpendicular from origin to plane (1).
Direction ratios of OP are
i.e. ?? , ?? , ??
From equation (1), it is clear that the direction ratios of normal to the plane i.e. OP are ?? , ?? , ?? ; ?? , ?? , ??
and ?? , ?? , ?? are the direction ratios of the same line OP
?
?? ?? =
?? ?? =
?? ?? =
1
?? (?????? ) ? ?? = ???? , ?? = ???? , ?? = ???? #(3)
Putting the values of ?? , ?? , ?? in equation (2), we get ka ?? + ?????? + ?????? + ?? = 0
Since ?? , ?? , ?? lies in plane (1) ? ???? + ???? + ???? + ?? = 0
Putting the values of ?? , ?? , ?? from (3) in (5), we get ?? ?? 2
+ ?? ?? 2
+ ?? ?? 2
+ ?? = 0
or
?? ?? 2
+ ?? ?? 2
+ ?? ?? 2
 ??????  ??????  ?????? = 0 [putting the value of ?? from (4) in (6)]
or ?? 2
+ ?? 2
+ ?? 2
 ????  ????  ???? = 0
Therefore, locus of foot of perpendicular ?? (?? , ?? , ?? ) is ?? 2
+ ?? 2
+ ?? 2
 ????  ????  ???? = 0
Angle between two planes :
(i) Consider two planes ???? + ???? + ???? + ?? = 0 and ?? '
?? + ?? '
?? + ?? '
?? + ?? '
= 0. Angle between these
planes is the angle between their normals. Since direction ratios of their normals are (?? , ?? , ?? ) and
(?? '
, ?? '
, ?? '
) respectively, hence ?? , the angle between them, is given by
?????? ?? =
?? ?? '
+ ?? ?? '
+ ?? ?? '
v?? 2
+ ?? 2
+ ?? 2
v?? '2
+ ?? '2
+ ?? '2
Planes are perpendicular if ????
'
+ ????
'
+ ????
'
= 0 and planes are parallel if
?? ?? '
=
?? ?? '
=
?? ?? '
(ii) The angle ?? between the planes ??? · ??? ?
1
= ?? 1
and ??? · ??? ?
2
= ?? 2
is given by, ?????? ?? =
?? ? ?
1
·?? ? ?
2
?? ? ?
1
?? ? ?
2

Planes are perpendicular if ??? ?
1
· ??? ?
2
= 0 & planes are parallel if ??? ?
1
= ?? ??? ?
2
.
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