Three Dimentional Geometry (Solved MCQs and Notes)

Notes 76 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

141
Unit - 12
Three Dimensional Geometry
Important Point
• Distance formula in R
3
: If
 
 
1 1 1 2 2 3
y z, , b , y , z a x x
 
2 1 2 1 2 1
AB b a , y y , z z     

x x
2 2 2
2 1 2 1 2 1
AB AB ) (y y ) (z z )x x       

• Division of line segment :
Suppose position vector of A & B be
b,
a
respectively if
P(r)
divides
AB
from A
in  ratio. where
 
P A, P B 
Co-ordinate of P is
b
r , 0, -1
1
 
  
 
a
• Co-ordinates of mid point of
AB

b
2


a
• In






ABC;If A a , B b ,C c
then posintion vector of centroid is
b c
,
3
 

a
g
Page 1
142
• Co-ordinates of Incentre : In ABC, if co-ordinate of position vector A, B & C are
, b & ca
and BC = a, CA = b, AB = c
Then position vector of incentre is
bb cc
b c
aa
a
 
 
• For equilateral triangle centroid and Incentre are equal.
• Direction co-sine & direction angle:
If vector
 
3
r , b, c Ra  makes angle  with unit vectors i, j & k then 
 are called direction angles and cos, cos, cos are called direction co-sine
of
r
.
2 2 2 2 2 2 2 2 2
b c
cos , m cos , n cos
b c b c b c
a
l
a a a
        
     
• If l, m and n are direction co-sine of
 
2 2 2
r , b, c , then m n    a l
2 2 2
cos + cos cos     
• If unit vector in the direction of
 
r , b, c :  a
 
b c
ˆ
r , , , m, n
| r | | r | | r |
 
 
 
 
a
l
• Direction ratio : if & m for mx x    , mx
1
, mx
2
, mx
3
is called direction
ratio.
• Vector equation of line:
If direction of line is
l
passes through
A( ) then equation of line is : r k , k R  a a l
• Parametric equation of line:
1 2 1 2 1 3
k , y y k , z z k , k Rx x l l l       are the parametric equations of line
passing through
 
 
1 1 1 1 2 3
, y , z & with direction , ,  a x l l l l
• Cartesian equation of line
   
 
1 1 1 1 2 3
r , y, z , , y , z & , ,   x a x l l l l
Page 2
143
     
1 1 1 1 2 3
, y, z , , y , z & direction , , x x a x l l l l  
1 1 1
1 2 3
y y z zx x
l l l
  
 
• Equation of line passing through
A( ) and B(b)a
:
     
1 1 1 2 2 2
, y , z b , y , z & r , y, za x x x  
Vector equation of line
k(b ) k Rr a a   
Cartesion equation of line
1 1 1
2 1 2 1 2 1
y y z z
y y z z
x x
x x
  
 
  
• Paramaetric equation of line:
   
1 2 1 1 2 1 1 2 1
k , y y k(y y ), z z k z zx x x x        
,
k R
If l
1
= 0 & l
2
 0, l
3
 0 then
1 1 1 1 1
1
2 3 1 3
y y z z y y z z
, OR
0
x x
x x
l l l l
    
   
• Angle between two lines in space R
3
:
r k , r b km k Ra l    
If two lines are parallel & direction of lines
& m is l
same of opposite.
m OR km k R {0}    l and l
If two lines are perpendicular then
. ml  
If angle between two lines is

then
. m
cos 0 <
m
  
   

l
l
• To obtain angle between two lines it is not necessary that two lines are intersecting
(in R
3
only):
In R
3
condtion for two lines
, r b kmr a kl   
, kR to intersect is
( b) . ( m) where , m 0      a l l
In R
3
, condition for two lines
k & r b + kmr a l  
, k R to interset in
cartesion form






1 1 1 2 2 2 1 2 3
y z , b , y , z , , ,   a x x l l l l
Page 3
144
 
1 2 1 2 1 2
1 2 3 1 2 3
1 2 3
y y z z
m , m , m is
m m m
  
  
x x
m l l l
• Condition that
lines
k , r b km, k R    r a l 0, m are co-planer is   l
( b) . ( m)   a l
• Non-coplaner lines :
If for any two lines l & m there does not exist plane containing them then they are
non-coplanar.
• Condition for two lines to be co-planer or non-coplaner
r k & r b + km, k Ra l   
       
1 1 1 2 2 2 1 2 3 1 2 3
, y , z , b , y , z , , , , m m , m , ma x x l l l l  
(1) For Co-planer line :
   
b . m 0
a l  
vector form
Cartesian form
1 2 1 2 1 2
1 2 3
1 2 3
y y z z
m m m
x x
l l l
  
 
(2) For non-co-planerline :




a b . m 0  
Cartesian form
1 2 1 2 1 2
1 2 3
1 2 3
x x y y z z
0
m m m
  
  
• Perpendicular distance of a line from point :
Perpendicular distance of
r a k l , k R  
from point


P p
is
(1)
 
AP
P
PM

 
 
 

l
a l
l
l
(2) Cartesian Form
 
 
2 2 2 1 1 1 1 2 3
, y , z P( , y , z ), , ,  a x x l l l l
PM =
1 2 1 2 1 2
1 2 3
j k
y y z z  
l
x x
l l l
• Perpendicular distance between parallel lines:
k , r b k l, k R ,    r a l
is =
( ) b a l
l
Page 4
145
• Distance between two skew lines
(b ) . ( m)
k & r b + km, k R, then p
| m |
   
    

a l
r a l
l
In R
3
relation between two lines
L : k , k R , M : r b + km, k Rr a l    
using
ml 
. we will get relation.
Plane :
• Vector equation of plane :
If plane passes through A( ), B(b), C(c)a then vector equation is
r + m (b ) + n(c ), m, n Ra a a   
• Parametric Form r + mb + nc where + m + n 1al l 
• Cartesian parametric form
     
1 1 1 2 2 2 3 3 3
r , y, z), , y , z , b , y , z , c , y , zx a x x x   
x = lx
1
+ mx
2
+ nx
3
where l + m + n = 1, l, m, n R
y = ly
1
+ my
2
+ ny
3
z = lz
1
+ mz
2
+ nz
3
• Cartesian equation :
 
r . (b ) (c )a a a
 
     
 
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
y y z z
y y z z 0
y y z z
x x
x x
x x
  
   
  
m
m 0  m 0 
lines are parallel OR Co- Inside Lines Skew OR Intersecting Lines



b a 0
  
(b a) 0   (b a).( m) 0   (b a).( m) 0  
Parallel Lines Co- Inside Lines Skew Lines Intersecting Lines

Page 5
146
• If A(x
1
, y
1
, z
1
), B(x
2
, y
2
, z
2
), C(x
3
, y
3
, z
3
), D(x
4
, y
4
, z
4
) are co-planer then
2 1 2 1 2 1
3 1 3 1 3 1
4 1 4 1 4 1
y y z z
y y z z 0
y y z z
x x
x x
x x
  
   
  
• Equation of plane with intercepts a, b, c with X, Y and Z axis repectively is
y z
1
b c
x
a
  
(a, b, c  0)
• Equation of plane passing through A( )a with normal
n
is
. n . nr a
cartesian form
 
r , y, z , n , b, c) + by + cz = d (d . n)x a ax a    
• If angle between two planes is 
1 2
1 2
| n . n |
t h e n c o s 0
| n | | n |

    

• If planes are perpendicular then
1 2
n . n  
• The equation of plane passing through two parallel lines :
, k R & r b + km, k Rr a kl    
The equation of plane is
 
. (b ) 0r a a l
 
   
 
Cartesian form
1 1 1
2 1 2 1 2 1 1 1 1 2 2 2 1 2 3
1 2 3
y y z z
y y z z 0 ( ,y ,z ), b ( ,y ,z ), ( , , )
  
 
      
 
x x
x x a x x l l l l
l l l
• The equation of plane passing through two intersecting lines
+ k and r b + km, (r ) . ( m)= 0r a l a l   
Cartesian form
1 1 1
1 2 3
1 2 3
y y z z
0
m m m
x x
l l l
  

1 2 3 1 2 3 1 2 3
where ( , , ), ( , , ) & m (m ,m ,m )a x x x l l l l  
Page 6
147
• Perpendicular distance from point
P(p)
to plane
| p . n d |
. n d is
n

r
• =
1 1 1
2 2 2
| + by + cz d |
b c
ax
a

 
(Cartesian form)
• Perpendicular distance between two planes
1 2
r . n d and r . n d is  
1 2
| d
-
d |
| n |
• Angle between line
+ k , k R, plane r . n d  r a l
1
. n
sin 0
n
l
l

  
    
    
• For two plane
1 1 1 2 2 1
: r . n d and : r . n d

   
intersection is line then equation of line is
1 2
+ kn, k R, n n + nr a  
• For two plane a
1
x + b
1
y + c
1
z + d
1
= 0 and a
2
x + b
2
y + c
2
z + d
2
= 0 equation of
plane passing through the intersection of two planes
1 1 1 1 2 2 2 2
( + b y + c z + d ) + ( + b y + c z + d )a x a x  
, 0, 1  
Page 7
148
Question Bank
1. The point on x-axis equidistance from A(2, -5, 7) and B(1, 3, 6) is ....
(a) (-16, 0, 0) (b) (16, 0, 0) (c) (6, 0, 0) (d) none of these
2. The equation of the locus of point which are equdistance from (4, 5, 2) and (1, 6,
3) is ....
(a) 6x - 2y - 2z + 1 = 0(b) 6x + 2y - 2z + 1 = 0
(c) 6x + 2y + 2z + 1 = 0 (d) 6x - 2y - 2z - 1 = 0
3.
I f the posi ti on vector of A , B , C in R
3
are (-1, 2, 0), (1, 2, 3) and (4, 2, 1) then type
of ABC is ...............
(a) Right angled (b) Isosceles right angled
(c) Euilateral (d) Isosceles
4. If the vertices of quadrilatral are (1, 1, 1), (-2, 4, 1), (-1, 5, 5), (2, 2, 5) then it
is.....
(a) rectangle (b) square (c) parallelogram (d) rhombus
5. A(1, 1, 2), B(2, 3, 5), C(1, 3, 4) and D(0, 1, 1) forms ..... and its area is .........
(a) Square,
2 3
(b) Parallelogram,
2 3
(c) Rectangle,
2 3
(d) Parallelogram,
3
6. For A(7, -3, 1) and B(4, 9, 8), the point that divides
AB
from B in the ratio 2:5
is....
(a)


34 39 42
7 7 7
, , (b)


34 39 42
7 7 7
, ,

(c)


34 39 42
7 7 7
, ,
 
(d)


34 39 42
7 7 7
, ,
  
7. For A(1, 5, 6), B(3, 1, 2) and C(4, -1, 0), B divides
AC
from A in ...... ratio
(A) -2 : 3 (b) 2 : 3 (c) 2 : 1 (d) -2 : 1
8. A(0, -1, 4), B(1, 2, 3), C(5, 4, -1), then the foot of perpendicular from A on BC
is.......
(a) (-3, 3, 1) (b) (3, -3, 1) (c) (3, 3, 1) (d) (3, 3, -1)
9. If A(a, 1, 3), B(-1, b, 2), C(1, 0, c) are the vertices of ABC whose centroid is
(2, 3, 5), then values of a, b, c are respectively .......
(a) 10, 8, 6 (b) 6, 10, 8
(c) 8, 6, 10 (d) 6, 8, 10
Page 8
149
10. If A(6, 4, 6), B(12, 4, 0), C(4, 2, -1) are the vertices of triangle, then it’s incentre
is....
(a)


1022 4
3 3 3
, , (b)


1022 4
3 3 3
, ,

(c)


1022 4
3 3 3
, ,

(d)


1022 4
3 3 3
, ,

11. If the mid points of sides of ABC are P(9, 2, 5), Q(-7, 6, 1), R(8, -9, 3) then the
centroid of ABC is .......
(A)


10 1 2
3 3 3
, ,

(b)


10 1 2
3 3 3
, ,
  
(c)
2
1, 1,
3
 
 
 
 
(d) None of these
12. For ABC, A(-1, -2, -3), B(1, 2, 3), C(1, 2, 1) the length of median through A is
.... and centroid is ......
(a)
 
1 2 1
3 3 3
3 3, , ,
(b)
 
1 2 1
3 3 3
3 5, , ,
(c)
 
1 2 1
3 3 3
5, , ,
(d)


1 2 1
3 3 3
3, , ,
13. The co-ordinates of the points of trisection of
AB
is ..... where A(-5, 7, 2), B(1,
3, 7)
(a)




16 11 11
3 2 3
1, 4, 3, ,   (b)




16 11 11
3 2 3
1, 4, 3, , 
(c)




16 11 11
3 2 3
1, 4, 3, ,
 
  (d) None of these
14. If
m B


 
in ABC and P, Q are points of trisection of hypotenuse AC , then
BP
2
+ BQ
2
= ...........
(a)
5
9
AC
2
(b)
5
9
AC (c)
25
81
AC
2
(D)
25
81
AC
15. If G (0) is centroid of ABC, then
GA + GB + GC  
  
(a)
0
(b) 0 (c)
+ y zx
(d)
+ y z
3
x
16. If A - P - B and
AP m
PB n

, then for every point ‘O’ in space ......
(a) (m - n)
OP

(b) (m + n)
OP

(c) m
OP

(d) n
OP

Page 9
150
17. In ABC, if mid points of
AB
and
AC
are D and E respectively, then
BE + DC  
 
(a)
3
2
BE

(b)
2
3
BE

(c)
3
2
BC (D)
2
3
BC
18. In parallelogram ABCD, AB
2
+ BC
2
+ CD
2
+ DA
2
= k(AC
2
+ BD
2
), then k = .......
(a) 4 (b) 16 (d) 2 (d) 1
19. If sides of regular hexagon ABCDEF,
AB and BC
 
are
and ba
respectively,
then
AF  

(a)
b a
(b)
b
a 
(c)
b
a 
(d)
a
20. For regular hexagon ABCDEF,
AB + AC + AD + AE + AF  
    
(a)
0
(b)
3 AD

(c)
2 AD

(d)
4 AD

21. For regular hexagon ABCDEF,
AB + BC + CD + AF + EF + ED  
     
(a)
3 AD

(b)
2 AD

(c)
0
(d)
2 AD

22. If the centroid of ABC and PQR is G and G’ respectively then
AP + BQ + CR  
  
(a)
GG'

(b) 3
GG'

(c) 2
GG'

(d) 4
GG'

23. If three vertices of rhombus are (6, 0, 1) (8, -3, 7) (2, -5, 10), then forth vertices
= ....
(a) (0, -2, -4) (b) (0, -2, 4) (c) (0, 2, 4) (d) (0, 2, -4)
24. If vector
r
forms an angle  with x, y, z-axis then sin
2
 + sin
2
 + sin
2
 =
...........
(a) 1 (b) 2 (c) -1 (d) -2
25. If  are direction co-sines of x , then cos 2 + cos 2 + cos 2 = ..........
(a) 1 (b) 2 (c) -1 (d) -2
26. If vector
r
form angles


and
2

with x and z axis respectively, then angle with
y-axis is........
(a)
3
4
,
 

(b)
4
,
 


(c)
3
4
,
 


(d)
3
3
,
 


27. If


is an angle with positive direction of x-axis in R
3
the no. of such vectors
are...
(a) 1 (b) 2 (c) 3 (d) infinite
Page 10

Three Dimentional Geometry (Solved MCQs and Notes)

Three Dimentional Geometry (Solved MCQs and Notes)

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