0% found this document useful (0 votes)

30 views48 pages

Vect&3d 1

Uploaded by

raghavbanga8

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

0% found this document useful (0 votes)

30 views48 pages

Vect&3d 1

Uploaded by

raghavbanga8

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

You are on page 1/ 48

VE CT OR & 3 -DIM ENS IONA L GE OMET RY

SECTION-A [FUNDAMENTAL]
VECTORS

 
Q.1 Find the value of so that the vectors a  2î  ĵ  k̂ and b  î  2 ĵ  3k̂ are perpendicular to each
other.

    
Q.2 If a + b + c = 0 and | a | 3 , | b | 5 and | c | 7 , show that the angle between a and b is 60º.

 
Q.3 Find the angle between the vector a  î  ˆj  k̂ and b  î  ˆj  k̂ .

 
Q.4 Write the value of p for which a  3î  2 ĵ  9k̂ and b  î  pˆj  3k̂ are parallel vector..


Q.5 Find the value of p, if ( 2î  6 ĵ  27 k̂ ) × ( î  3ˆj  pk̂ ) = 0 .

             
Q.6 If a , b, c are three vectors such that a . b  a . c and a  b  a  c , a  0 , then show that b  c .


Q.7 Write a vector of magnitude 15 units in the direction of vector i  2ˆj  2k̂ .

Q.8 What is the cosine of theangle which the vector 2 î  ĵ  k̂ makes with y-axis?

     
Q.9 Let a  î  ĵ , b  3ˆj  k̂ and c  7î  k̂ . Find a vector d which isperpendicular to both a and b and

c.d =1.

Q.10 Find the projection of the vector î  ˆj on the vector î  ˆj .

Q.11 What are the direction cosines of a line, which makesequal angles with the co-ordinate axes ?

  
Q.12 Find a unit vector perpendicular to each of the vectors a  b and a  b , where a  3î  2ˆj  2k̂ and

b  î  2 ĵ  2k̂ .

 
Q.13 Write the value of î  ˆj · k̂  î · ĵ .

   
Q.14 Let a  î  4ˆj  2k̂ , b  3î  2ˆj  7k̂ and c  2î  ˆj  4k̂ . Find a vector p which is perpendicular to
   
both a and b and p · c = 18.

     
Q.15 Find | x | , if for a unit vector a , ( x  a ) · ( x  a ) = 15.

73
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

   
Q.16 L and M are two points with position vectors 2a  b and a  2b respectively. Write the position vector
of a point N which divides the line segment LM in the ratio 2 : 1 externally.

Q.17 Find the value of ‘p’ for which the vectors 3iˆ  2ˆj  9kˆ and ˆi  2pjˆ  3kˆ are parallel.

    
ˆ b  ˆi  2jˆ  kˆ and c  3iˆ  ˆj  2kˆ
Q.18 Find a.(b  c) , if a  2iˆ  ˆj  3k,

Q.19 Show that the four points A, B, C and D with position vectors 4iˆ  5jˆ  k,
ˆ ˆj  k,3
ˆ ˆi  9 ˆj 4 kˆ and

4( ˆi  ˆj  k)
ˆ respectively are coplanar..


Q.20 The scalar product of the vector a  ˆi  ˆj  kˆ with a unit vector along the sum of vectors
 
b  2iˆ  4ˆj  5kˆ and c   ˆi  2ˆj  3kˆ is equal to one. Find the value of  and hence find the unit vector
 
along b  c


Q.21 Find a vector of magnitude 171 which is perpendicular to both of the vectors a  ˆi  2ˆj  3kˆ

and b  3iˆ  ˆj  2kˆ .

Q.22 Find  and  if

î  3 ĵ  9k̂  3î  ĵ   k̂   0
   
Q.23 If a  4î  ˆj  k̂ and b  2 î  2 ĵ  k̂ , then find a unit vector parallel to the vector a  b .

Q.24 Show that the points A, B, C with position vectors 2î  ˆj  k̂ , î  3 ĵ  5k̂ and 3î  4ˆj  4k̂ respectively,,
are the vertices of a right-angled triangle. Hence find the area of the triangle.

     
Q.25 If a  2î  ˆj  2k̂ and b  7 î  2 ĵ  3k̂ , then express b in the form of b  b1  b 2 ,
   
where b1 is parallel to a and b2 is perpendicular to a.

   
Q.26 Find | a  b | , if a  î  7 ĵ  7 k̂ and b  3î  2ˆj  2k̂ .

Q.27 Writetheposition vector of apoint dividing theline segment joining pointsA and B with positionvectors
   
a and b externally in the ratio 1 : 4 where a  2î  3ˆj  4k̂ and b   î  ˆj  k̂ .

Q.28 Write thevalue of î.(ˆj  k̂ )  ĵ.( k̂  î )  k̂.( ĵ  î ) .

     
Q.29 Let a , b and c be three vectors such that | a | 3 , | b | 4 , | c | 5 and each one of them being
  
perpendicular to thesum of the other two, find | a  b  c | .

74
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

            
Q.30 Threevectors a , b and c satisfy thecondition a  b  c  0 . Evaluatethequantity = a . b  b . c  c . a ,
  
if | a |  1 , | b | 4 and | c | 2 .


Q.31 Find the area of the parallelogram whose adjacent sides are determined by the vectors a  î  ˆj  3k̂

and b  2î  7ˆj  k̂ .

 
Q.32 In a regular hexagon ABCDEF, if AB  a and BC  b then express CD , DE , EF , FA , AC , AD ,
 
AE and CE in terms of a and b .

Q.33 Show that the pointsA ( 2î  3 ĵ  5k̂ ) , B ( î  2ˆj  3k̂ ) and C (7î  k̂ ) are collinear..

   1 
Q.34 If a and b are unit vectors and  is the angle between them, then prove that cos = | a  b |.
2 2

  
Q.35 Find the volume of the parallelopiped whose adjacent sides are represented by a , b and c where
  
a  3î  2ˆj  5k̂ , b  2î  2 ĵ  k̂ , c  4î  3 ĵ  2k̂ .

Q.36 If the position vectors of three consecutive vertices of any parallelogram are respectively
î  ˆj  k̂ , î  3 ĵ  5k̂ , 7î  9 ĵ  11 k̂ then, find the position vector of its fourth vertex.

  xy 2
Q.37 ˆ ˆ
If a  x î  2 j  5k̂ and b  î  yj  zk̂ are linearly dependent, then find the value of .
z

   
Q.38 If a & b are non collinear vectors such that , p  ( x  4 y )a  ( 2 x  y  1) b &
    
q  ( y  2 x  2) a  ( 2 x  3y  1) b , find x & y such that 3p  2q .

    
Q.39 a , b, c are three non zero vectors no two of them are parallel. If a  b is collinear to c and b  c is
  
collinear to a, then a  b  c is equal to

Q.40 Determine vector of magnitude 9 which is perpendicular to both the vectors:

4î  ĵ  3k̂ &  2î  ˆj  2k̂

Q.41 A triangle has vertices (1, 1, 1) ; (2, 2, 2), (1, 1, y) and has the area equal to csc  4 sq. units. Find the
value of y.

  
Q.42 Vector V is perpendicular to the plane of vectors a  2î  3ˆj  k̂ and b  î  2 ĵ  3k̂ and satisfies the
 
condition V · ( î  2 ĵ  7 k̂ ) = 10. Find | V |2 .

75
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

THREE DIMENSION GEOMETRY

x  1 3y  5 3  z
Q.43 Find the angle between the line   and the plane 10x + 2y – 11z = 3.
2 9 6
Q.44 Find the direction cosines of the linepassing through two points (–2, 4, –5) and (1, 2, 3).

Q.45 Find theanglebetween the pair of linesgiven by

 
r  3î  2 ĵ  4k̂   ( î  2ˆj  2k̂ ) and r  5î  2ˆj   (3î  2 ĵ  6k̂ ) .

Q.46 Writethevector equation of thefollowing line.

x 5 y4 6z
= =
3 7 2
Q.47 Find the coordinatesof the point wherethe line through thepointsA (3, 4, 1) and B (5, 1, 6) crosses the
XY-plane.
3  x y  4 2z  6
Q.48 If the cartesian equations of a line are   , write the vector equation for the line.
5 7 4

Q.49 Find the angle between the lines 2x = 3y = –z and 6x = – y = –4z.


 
Q.50 Write the sum of intercepts cut off by the plane r · 2î  ˆj  k̂ – 5 = 0 on the three axes.

Q.51 Show that the four points A (4, 5, 1), B (0, –1, –1), C (3, 9, 4) and D (–4, 4, 4) are coplanar.

Q.52 If thex-coordinate of a point P on the join of Q(2, 2, 1) and R(5, 1, –2) is 4, then find its z-coordinate.

Q.53 Find the equation of the plane passing through the intersection of the planes, 3x – y + 2z – 4 = 0 and
x + y + z – 2 = 0 and the point (2, 2, 1).

Q.54 If the point (1, 1, P) and (–3, 0, 1) be equidistant from the plane

r .(3î  4 ĵ  12k̂ ) + 13 = 0, then find the value of P.

Q.55 Find thevector equation of theplane passing through the intersection of theplanes r .( î  ĵ  k̂ )  6 and

r .( 2î  3 ĵ  4k̂ )  5 , and the point (1, 1, 1).
 
Q.56 Show that theline r  ( 2î  2 ĵ  3k̂ )   ( î  ˆj  4k̂ ) is parallel to the plane r ·( î  5 ĵ  k̂ )  5 . Also, find
the distancebetween them.

Q.57 The equation of the plane which has the property that the point Q (5, 4, 5) is the reflection of
point P (1, 2, 3) through that plane, is ax + by + cz = d where a, b, c, d  N.
Find the least value of (a + b + c + d).

Q.58 Let P = (1, 0, – 1) ; Q = (1, 1, 1) and R = (2, 1, 3) are three points.

(i) Find the area of the triangle having P, Q and R as its vertices.
(ii) Give the equation of the plane through P, Q and R in the form ax + by + cz = 1.
(iii) Where does the plane in part (b) intersect the y-axis.
(iv) Give parametric equations for the line through R that is perpendicular to the plane in part (b).

76
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

SECTION-B [JEE (MAIN)]

VECTOR
Q.1 The position vector of a point C with respect to B is î  ˆj and that of B with respect to A is î  ˆj .
The position vector of C with respect to A is- [3011711878]
(A) 2 î (B) – 2 î (C) 2 ˆj (D) – 2 ˆj

Q.2 If D,E,F are mid points of sides BC, CA, and AB respectively of a triangle ABC, and
î  ˆj , ĵ  k̂ , k̂  î are p.v. of points A,B and C respectively, then p.v. of centroid of DEF is-

î  ĵ  k̂ 2(î  ˆj  k̂ )
(A) (B) î  ˆj  k̂ (C) 2 ( î  ˆj  k̂ ) (D)
3 3
[3011711929]

Q.3 If D,E and F are midpoints of sides BC, CA and AB of a triangle ABC, then AD + BE + CF is equal
to-
(A) 0 (B) 2 BC (C) 2 AB (D) 2 CA [3011711980]

Q.4 If a line makes angle  with the co-ordinate axis then cos2  + cos2  + cos 2equals to–
(A) –2 (B) –1 (C) 1 (D) 2 [3011712031]

Q.5 If E is the intersection point of diagonals of parallelogram ABCD and

OA  OB  OC  OD  x OE then x is equal to (where O represents origin)-
(A) 3 (B) 4 (C) 5 (D) 6 [3011712082]

 
Q.6 Let p is the p.v. of the orthocentre & g is the p.v. of the centroid of the triangle ABC where
 
circumcentre is the origin. If p = K g , then K =
(A) 3 (B) 2 (C) 1/3 (D) 2/3 [3011711765]


Q.7 A vector a has components 2p & 1 with respect to a rectangular cartesian system. The system is
rotated through a certain angle about the origin in the counterclockwise sense. If with respect to

the new system, a has components p + 1 & 1 then ,
(A) p = 0 (B) p = 1 or p =  1/3
(C) p =  1 or p = 1/3 (D) p = 1 or p =  1 [3011711816]

Q.8 The position vector of the points A and B are a and b respectively. If P divides AB is the ratio 3:
1 and Q is the mid point of AP, then the position vector of Q is- [3011711867]
       
ab a b 5a  3b 5a  3b
(A) (B) (C) (D)
2 2 8 8

77
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

 
Q.9 If A = (x + 1) a + (2y – 3) b and B = 5 a – 2 b are two vectors such that 2 A = 3 B , & a , b are
non collinear vectors then- [3011711918]
(A) x = 13/2, y = 0 (B) x = 0, y = 3 (C) x = –13/2, y = 0 (D) None of these

Q.10 If the points P,Q,R,S are respectively î  k̂ ,  î  2ˆj , 2î  3k̂ and 3î  2ˆj  k̂ , then projection of

PQ on RS is- [3011711969]
(A) 4/3 (B) – 4/3 (C) 3/4 (D) –3/4

      
Q.11 Two vectors p , q on a plane satisfy p  q  13 , p  q  1 and p  3 .
 
The angle between p and q , is equal to [3011712020]
   
(A) (B) (C) (D)
6 4 3 2

Q.12 The set of values of c for which the angle between the vectors cx i  6 j  3k & x i  2 j  2 cx k is
acute for every x  R is [3011712071]
(A) (0, 4/3) (B) [0, 4/3] (C) (11/9, 4/3) (D) [0, 4/3)

       
Q.13 Let u, v, w be such that u  1, v  2, w  3 . If the projection of v along u is equal to that
      
of w along u and vectors v , w are perpendicular to each other then u  v  w equals

(A) 2 (B) 7 (C) 14 (D) 14 [3011712154]

        
Q.14 If p & s are not perpendicular to each other and r x p  qx p & r . s = 0, then r = [3011712205]
   
    q . p    q . s   
(A) p . s (B) q      p (C) q      p (D) q   p for all scalars 
 p . s  p . s

  
Q.15 Let a  a1 i  a2 j  a3 k ; b  b1 i  b2 j  b3 k ; c  c1 i  c2 j  c3 k be three non-zero vectors such

     
that c is a unit vector perpendicular to both a & b . If the angle between a & b is then
6

2
a1 b1 c1
a2 b2 c2 =
a3 b3 c3

(A) 0 (B) 1 [3011712103]

1 3
(C) (a 2 + a22 + a32) (b12 + b22 + b32) (D) (a 2 + a22 + a32) (b12 + b22 + b32) (c12 + c22 + c32)
4 1 4 1

78
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.16 If the volume of the tetrahedron with edges î  ˆj  k̂ , î  aĵ  k̂ and î  2ˆj  k̂ is 6 cubic units, then
a is-
(A) 1 (B) – 1 (C) 2 (D) – 17 [3011712307]
  
Q.17 If a , b, c be any three unit vectors such that 3a  4b  5c = 0, then- [3011712256]
     
(A) a || b (B) b || c (C) a  b (D) None of these

Q.18 If a , b, c be any three unit vectors such that a and b are perpendicular to each other and
  
2a  3b  c , then value of  is- [3011712358]
(A) 1 (B) 5 (C) 13 (D) 13

Q.19 If a , b, c are three non-zero vectors such that a + b + c= 0 and m = a  b + b c+ c a , then
(A) m < 0 (B) m > 0 (C) m = 0 (D) m = 3 [3011712112]
Q.20 Let AB  3î  ˆj , AC  2î  3 ĵ and DE  4î  2 ĵ . A

The area of the shaded region in the adjacent figure, is D

(A) 5 (B) 6
(C) 7 (D) 8 [3011712409]
E
B C

Q.21 The altitude of a parallelopiped whose three coterminous edges are the vectors, A  î  ˆj  k̂ ;
   
B  2î  4 ĵ  k̂ & C  î  ĵ  3k̂ with A and B as the sides of the base of the parallelopiped, is

(A) 2 19 (B) 4 19 (C) 2 38 19 (D) none [3011712163]

Q.22 Position vectors of the four angular points of a tetrahedron ABCD are A(3, – 2, 1); B(3, 1, 5);
C(4, 0, 3) and D(1, 0, 0). Acute angle between the plane faces ADC and ABC is [3011712265]
(A) tan–1 5 2 (B) cos–1 2 5 (C) cosec–1 5 2 (D) cot–1 3 2 
  
Q.23 a , b and c be three vectors having magnitudes 1, 1 and 2 respectively. If
     
a × ( a × c ) + b = 0, then the acute angle between a & c is [3011712367]
(A) /6 (B) /4 (C) /3 (D) 5 12

Q.24 A vector of magnitude 5 5 coplanar with vectors î 2 ĵ & ĵ 2k̂ and the perpendicular vector 2î  ĵ 2k̂
is

(A) ± 5 5î  6ˆj  8k̂  (B) ± 
5 5î 6ˆj8k̂  [3011712316]


(C) ± 5 5 5î  6ˆj  8k̂  
(D) ± 5î 6ˆj8k̂ 
Q.25 ( a + 2 b– c) {( a – b) × ( a – b– c)} is equal to [3011712418]

(A) [ a b c] (B) 2[ a b c] (C) 3[ a b c] (D) 0

79
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

3-DIMENSIONAL G E O M E T RY

Q.1 The equation of the plane which is parallel to the plane x – 2y + 2z = 5 and whose distance from
the point (1, 2, 3) is 1, is [3011712322]
(A) x – 2y + 2z = 3 (B) x – 2y + 2z + 3 = 0
(C) x – 2y + 2z = 6 (D) x – 2y + 2z + 6 = 0

 
Q.2 The intercept made by the plane r . n  q on the x-axis is [3011712373]

q 
î . n  q
(A) 
î . n
(B)
q
 
(C) î . n q (D) | n |

Q.3 If the plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(k) with x-axis, then k is equal to
(A) 3 2 (B) 2/7 (C) 2 3 (D) 1 [3011712424]

Q.4 A variable plane forms a tetrahedron of constant volume 64 K3 with the coordinate planes and the
origin, then locus of the centroid of the tetrahedron is [3011712128]
3 3 3
(A) x + y + z = 6K 3 (B) xyz = 6k3

(C) x2 + y2 + z2 = 4K2 (D) x–2 + y–2 + z–2 = 4K–2

Q.5 Which of the following planes are parallel but not identical? [3011712179]
P1 : 4x – 2y + 6z = 3
P2 : 4x – 2y – 2z = 6
P3 : –6x + 3y – 9z = 5
P4 : 2x – y – z = 3

(A) P2 & P3 (B) P2 & P4 (C) P1 & P3 (D) P1 & P4

x  2 y  9 z  13 x a y7 z2
Q.6 The value of 'a' for which the lines =  and   intersect, is
1 2 3 1 2 3
(A) – 5 (B) – 2 (C) 5 (D) – 3
[3011711239]

x 1 y  2 z  3
Q.7 For the line   , which one of the following is incorrect? [3011712230]
1 2 3
(A) it lies in the plane x – 2y + z = 0
x y z
(B) it is same as line  
1 2 3
(C) it passes through (2, 3, 5)
(D) it is parallel to the plane x – 2y + z – 6 = 0

80
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.8 Given planes

P1 : cy + bz = x
P2 : az + cx = y
P3 : bx + ay = z
P1, P2 and P3 pass through one line, if [3011712332]
(A) a2 + b2 + c2 = ab + bc + ca (B) a2 + b2 + c2 + 2abc = 1
(C) a2 + b2 + c2 = 1 (D) a2 + b2 + c2 + 2ab + 2bc + 2ca + 2abc = 1

Q.9 The plane ax + by + cz = 1 meets the co-ordinate axes in A, B and C. The centroid of the triangle
is
a b c  3 3 3 1 1 1
(A) (3a, 3b, 3c) (B)  , ,  (C)  , ,  (D)  , , 
 3 3 3 a b c  3a 3b 3c 
[3011712281]

Q.10 The equation of the plane passing through the point (–1, 3, 2) and perpendicular to each of the
planes x + 2y + 3z = 5 and 3x + 3y + z = 0, is -
(A) 7x – 8y + 3z – 25 = 0 (B) 7x – 8y + 3z + 25 = 0
(C) –7x + 8y – 3z + 5 = 0 (D) 7x – 8y – 3z + 5 = 0 [3011712340]

Q.11 The equation of the plane through intersection of planes x + 2y + 3z = 4 and 2x + y – z = – 5 and
perpendicular to the plane 5x + 3y + 6z + 8 = 0 is
(A) 7x – 2y + 3z + 81 = 0 (B) 23x + 14y – 9z + 48 = 0
(C) 51x + 15y – 50z + 173 = 0 (D) None of these [3011712289]

Q.12 The equation of the plane passing through the line of intersection of the planes
x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to x-axis is - [3011712238]
(A) y – 3z – 6 = 0 (B) y – 3z + 6 = 0 (C) y – z – 1 = 0 (D) y – z + 1 = 0

Q.13 Image point of (1, 3, 4) in the plane 2x – y + z + 3 = 0 is - [3011712187]

(A) (–3, 5, 2) (B) (3, 5, – 2) (C) (3, – 5, 3) (D) none of these

Q.14 The graph of the equation x2 + y2 = 0 in three dimensional space is - [3011712136]

(A) x-axis (B) y-axis (C) z-axis (D) xy-plane

1 1 1
Q.15 If the direction cosines of a line are  , ,  , then - [3011712283]
c c c

(A) c > 0 (B) c =  3 (C) 0 < c < 1 (D) c > 2

81
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.16 The equations of the line passing through the point (1, 2, –4) and perpendicular to the two lines

x  8 y  19 z  10 x  15 y  29 z  5
  and   will be [3011712434]
3  16 7 3 8 5

x 1 y  2 z  4 x 1 y  2 z  4
(A)   (B)  
2 3 6 2 3 8

x 1 y  2 z  4
(C)   (D) None of these
3 2 8


Q.17 Equation of the plane through (3, 4, –1) which is parallel to the plane r . ( 2î  3ˆj  5k̂ ) + 7 = 0 is
[3011712391]
 
(A) r . ( 2î  3ˆj  5k̂ ) + 11 = 0 (B) r . (3î  4 ĵ  k̂ ) + 11 = 0
 
(C) r . (3î  4 ĵ  k̂ ) + 7 = 0 (D) r . ( 2î  3ˆj  5k̂ ) –7 = 0

 
Q.18 Equation of the plane containing the lines. r = î  2ˆj  k̂   ( î  2ˆj  k̂ ) and r = î  2ˆj  k̂   ( î  ˆj  3k̂ )
is
[3011712442]

(A) r . (7 î  4 ĵ  k̂ ) = 0 (B) 7(x – 1) – 4(y – 1) – (z + 3) = 0
 
(C) r . ( î  2 ĵ  k̂ ) = 0 (D) r . ( î  ˆj  3k̂ ) = 0

x  3 y  5 z 1
Q.19 If the line = = is parallel to the plane 6x + 8y + 2z – 4 = 0, then k
2 k 2k
(A) 1 (B) –1 [3011712142]
(C) 2 (D) 3

 
Q.20 If x & y are two non collinear vectors and a, b, c represent the sides of a  ABC satisfying
   
(a  b) x + (b  c) y + (c  a) x  y = 0 then  ABC is [3011712244]
(A) an acute angle triangle (B) an obtuse angle triangle
(C) a right angle triangle (D) a scalene triangle

82
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[JEE ADVANCED]

VECTOR
[ SINGLE CORRECT CHOICE T Y PE]
Q.1 Consider  ABC with A  (a ) ; B  ( b) & C  ( c) . If b . (a  c) = b . b  a . c ; b  a = 3;
 
c  b =4thentheanglebetweenthemedians A M & BD is [3011712214]

 1   1 
(A)  cos1   (B)  cos1  

 5 13   13 5 
 1   1 
(C) cos1  
 (D) cos1  

 5 13   13 5 

         
Q.2    
Let r  a  b sin x  b  c cos y  2  c  a  , where a , b, c are non-zero and non-coplanar vectors.
    20
If r is orthogonal to a  b  c , then the minimum value of 2 (x2 + y2), is [3011711032]

(A) 20 (B) 25 (C) 30 (D) 35


Q.3      
Let O be an interior point of ABC such that 2 O A  5 O B  10 O C  0 . If the ratio of the area of
 ABC to the area of  AOC is t, where 'O' is the origin. The value of [t], is
(Where [ ] denotes greatest integer function.) [3011711419]
(A) 1 (B) 2 (C) 3 (D) 4

         
Q.4 If a and b are two vectors such that | a | 1 , | b | 4 , a · b  2 . If c  ( 2a  b)  3b then the angle
 
between b and c , is [3011710930]
  2 5
(A) (B) (C) (D)
6 3 3 6

Q.5 The minimum area of the triangle whose vertices are A(–1, 1, 2); B(1, 2, 3) and C(t, 1, 1)
where t is a real number, is [3011710879]
1 1 1 3
(A) (B) (C) (D)
2 2 3 2

Q.6 Points X & Y are taken on the sides QR & RS , respectively of a parallelogram PQRS, so that
     
QX  4 XR & RY  4 YS . The line XY cuts the line PR at Z . If PZ  k PR then k is equal to.
[3011710861]
20 21 22 23
(A) (B) (C) (D)
27 25 23 20

83
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.7 The three vectors î  ˆj , ĵ  k̂ , k̂  î taken two at a time form three planes. The three unit vectors drawn
perpendicular to these three planes form a parallelopiped of volume : [3011712193]
(A) 1/3 (B) 4 (C) 3 3 4 (D) 4 3 3

Q.8 In a ABC, the side length BC, CA and AB are consecutive positive integers in increasing order. Let
      
a , b and c be the position vectors of vertices A, B and C respectively. If ( c  a ) · ( b  c ) = 0, then
     
the value of a  b  b  c  c  a is equal to
(A) 3 (B) 6 (C) 12 (D) 24

[PARAGRAPH TYPE]
Paragraph for questions nos. 9 to 11
   
Consider three vectors p  î  ˆj  k̂ , q  2î  4 ĵ  k̂ and r  î  ˆj  3k̂ and let s be a unit vector, then
[3011712295]
  
Q.9 p, q and r are
(A) linearly dependent
(B) can form the sides of a possible triangle
  
(C) such that the vectors (q  r ) is orthogonal to p
(D) such that each one of these can be expressed as a linear combination of the other two

     
Q.10 if ( p  q ) × r = up  vq  w r , then (u + v + w) equals to
(A) 8 (B) 2 (C) – 2 (D) 4

           
Q.11 The magnitude of the vector ( p · s )(q  r ) + (q · s )( r  p) + ( r · s )( p  q) is
(A) 4 (B) 8 (C) 18 (D) 2

Paragraph for question nos. 12 to 14

      
Let a = b = 2 and c = 1. Also (a  c) · ( b  c) = 0 [3011712397]

  2   
Q.12 a  b  2c · (a  b) has the value equal to
(A) 12 (B) 10 (C) 8 (D) 6

   2
Q.13 a bc equals
(A) 5 (B) 6 (C) 7 (D) 8

 
Q.14 Difference between of the maximum and minimum value of a  b is equal to
(A) 2 (B) 3 (C) 4 (D) 1

84
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ REASONING T Y PE]
  
Q.15 Consider three vectors a , b and c
       
     
Statement-1: a  b  ( î  a ) · b î  (ˆj  a ) · b ĵ  ( k̂  a ) · b k̂
   
Statement-2: c  ( î · c ) î  (ˆj· c) ĵ  ( k̂ · c) k̂
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true. [3011712171]


Q.16 Statement-1: Let the vector a  î  ĵ  k̂ be vertical. The line of greatest slope on a plane with

normal b  2î  ˆj  k̂ is along the vector î  4ˆj  2k̂ .
 
Statement-2: If a is vertical, then the line of greatest slope on a plane with normal b is along the
  
vector (a  b)  b .
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true. [3011712222]

[MULTIPLE CORRECT CHOICE TYPE]

  
Q.17 Given the following information about the non zero vectors A, B and C
     
(i) ( A  B)  A  0 (ii) B· B  4
   
(iii) A · B  6 (iv) B· C  6
Which one of the following holds good? [3011712273]
         
(A) A  B  0 (B) A · ( B  C)  0 (C) A · A  8 (D) A · C  9

 
Q.18 Let a and b be two non-zero and non-collinear vectors then which of the following is/are always
correct?
    
(A) a  b = a b î î + a b ˆj ĵ + a b k̂ k̂
     
       
(B) a · b = (a · î ) ( b · î ) + (a · ĵ) ( b · ĵ) + (a · k̂ ) ( b · k̂ )
 
(C) if u = â  (â · b̂) b̂ and v  â  b̂ then | u || v |

          
(D) if c  a  (a  b) and d  b  (a  b) then c  d  0 . [3011712375]

Q.19 L et O be an interior point of ABC such that OA  2OB  3OC  0 and areas of ABC, AOB,
1
BOC and AOC are denoted by 1, 2, 3 and 4 respectively then is
min . ( 2 ,  3 ,  4 )
divisible by
(A) 2 (B) 3 (C) 4 (D) 6

85
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ MAT RIX T YPE]

Q.20 If A(0, 1, 0), B (0, 0, 0), C(1, 0, 1) are the vertices of a ABC. Match the entries of column-I with column-II.
Column-I Column-II
2
(A) Orthocentre of ABC. (P)
2
3
(B) Circumcentre of ABC. (Q)
2

3
(C) Area (ABC). (R)
3
3
(D) Distance between orthocentre and centroid. (S)
6
(E) Distance between orthocentre and circumcentre. (T) (0, 0, 0)
1 1 1
(F) Distance between circumcentre and centroid. (U)  , , 
2 2 2
1 1 1
(G) Incentre of ABC. (V)  , , 
 3 3 3

 1 2 1 
(H) Centroid of ABC (W)  , , 

 1  2  3 1  2  3 1  2  3 
[3011712426]

Q.21 Column I Column II

(A) P is point in the plane of the triangle ABC. pv’s of A, B and C are (P) centroid
  
a , b and c respectively with respect to P as the origin.
       
  
If b  c · b  c = 0 and c  a · c  a  = 0, then w.r.t. the (Q) orthocentre
triangle ABC, P is its
  
(B) If a , b, c are the position vectors of the three non collinear (R) Incentre
points A, B and C respectively such that the vector

V  P A  P B  P C is a null vector then w.r.t.
the ABC, P is its
(C) If P is a point inside the ABC such that the vector (S) circumcentre

R  (BC)(P A )  (CA )(P B)  (AB)(P C) is a null vector then
w.r.t. the ABC, P is its
(D) If P is a point in the plane of the triangle ABC such that the
scalar product P A · C B and P B · A C vanishes, then w.r.t.
the ABC, P is its [3011712126]

86
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.22 Column-I Column-II

(A) Centre of the parallelopiped whose 3 coterminous edges (P)   
abc
  
OA, OB and OC have position vectors a, b and c
respectively where O is the origin, is
  
a bc
(B) OABC is a tetrahedron where O is the origin. Positions vectors (Q)
2
  
of its angular points A, B and C are a, b and c respectively..
Segments joining each vertex with the centroid of the opposite
face are concurrent at a point P whose p.v.'s are
  
a bc
(C) Let ABC be a triangle the position vectors of its angular points (R)
3
        
are a , b and c respectively. If a  b  b  c  c  a then
the p.v. of the orthocentre of the triangle is
  
   a bc
(D) Let a , b, c be 3 mutually perpendicular vectors of the same (S)
4

magnitude. If an unknown vectors x satisfies the equation
           
    
a  x  b  a  b  x  c   b  c  x  a   c   0 .

Then x is given by
(E) ABC is a triangle whose centroid is G, orthocentre is H and circumcentre
    
is the orign. If position vectos of A, B, C, G and H are a , b, c, g and h
   
respectively, then h in terms of a , b and c is equal to [3011712228]

[ INT EGER T YPE]

 
Q.23 Let V1 and V2 are two vectors such that
 
V1 = 2(sin  + cos ) î  ĵ and V2 = sin  î + cos  ˆj where  and  satisfy the relation 2(sin  +
cos )sin  = 3 – cos , find the value of (3 tan2 + 4 tan2). [3011710912]

Q.24 Let P and Q are two points in xy-plane on the curve y = x7 – 2x5 + 5x3 + 8x + 5 such that

OP . î = 2 and OQ . î = – 2 and the magnitude of the vector OP  OQ = 2M, (where O is origin)

then find the value of M. [3011710963]

Q.25 Given three points on the xy plane on O(0, 0), A(1, 0) and B(–1, 0). Point P is moving on the plane
  
satisfying the condition P A · P B + 3 O A · O B = 0 [3011710726]

If the maximum and minimum values of P A P B are M and m respectively then find the value of M2 + m2.

87
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.26 Given a tetrahedron D-ABC with AB = 12 , CD = 6. If the shortest distance between the skew lines

AB and CD is 8 and the angle between them is , then find the volume of tetrahedron.
6
[3011710777]
             
Q.27 Let ( p  q)  r  (q · r ) q  ( x 2  y 2 )q  (14  4x  6 y)p and ( r · r ) p  r where p and q are
two non-zero non-collinear vectors and x and y are scalars. Find the value of (x + y).
[3011710828]
   
Q.28 Let OA = a ; OB = 100 a  2 b and OC = b where O, A, and C are non collinear points. Let P
denotes the area of the parallelogram with OA and OC as adjacent sides and Q denotes the area of the
quadrilateral OABC. If Q = P. Find the value of . [3011710736]

Q.29 In ABC, a point P is chosen on side A B so that AP : PB = 1 : 4 and a point Q is chosen on the side
MC
B C so that CQ : QB = 1 : 3. Segment C P and A Q intersect at M. If the ratio PC is expressed as a

a
rational number in the lowest term as , find (a + b). [3011711041]
b

Q.30 Given f 2(x) + g2(x) + h2(x)  9 and U(x) = 3 f (x) + 4 g (x) + 10 h(x),where f (x), g (x) and h (x) are
continuous  x  R. If maximum value of U(x) is N , then find N. [3011711470]

  2  
Q.31 Let two non-collinear vectors a and b inclined at an angle be such that | a | 3 and | b | 4 .
3
A point P moves so that at any time t the position vector OP (where O is the origin) is given as
 
OP = (et + e–t) a + (et – e–t) b . If the least distance of P from origin is 2 a  b
where a, b  N then find the value of (a + b). [3011711572]
 
Q.32 If x , y are two non-zero and non-collinear vectors satisfying [3011711725]
   
[(a – 2)2 + (b – 3) + c] x + [(a – 2)2 + (b – 3) + c] y + [(a – 2)2 + (b – 3) + c] (x  y) = 0
where , ,  are three distinct real numbers, then find the value of (a2 + b2 + c2).

[ SUBJECT IVE T YPE]

             
Q.33 Find the scalars  &  if ax (bx c)  (a. b) b  ( 4  2  sin ) b  ( 2  1) c & ( c. c) a  c while b & c
are non zero non collinear vectors. [3011711623]

Q.34 The position vectors of the points A, B, C are respectively (1, 1, 1) ; (1, 1, 2); (0, 2, 1). Find a unit
vector parallel to the plane determined by ABC & perpendicular to the vector (1, 0, 1) .
[3011711040]
      
Q.35 Given that a,b,p,q are four vectors such that a  b   p , b .q  0 & ( b ) 2  1 , where µ is a scalar then
       
prove that ( a .q ) p  ( p .q ) a  p .q . [3011710981]

88
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

3 - D I M E N S I O N A L G E O M E T RY
[ SINGLE CORRECT CHOICE T Y PE]
Q.1 Equation of the line in the plane x + 3y – z = 9, which is perpendicular to the line

 
r  î  ˆj  k̂   2î  ˆj  k̂ and passing through a point where the plane P meets the given line, is

x 3 y2 z x 3 y2 z
(A)   (B)  
2 1 5 2 1 5

x 3 y2 z x 3 y2 z
(C)   (D)  
5 1 2 1 5 2
1 t 
Q.2 The distance between the line x= 2 + t, y = 1 + t, z = – and the plane r · ( î  2ˆj  6k̂ )  10 , is
2 2
1 1 1 9
(A) (B) (C) (D)
6 41 7 41
b
Q.3 If the lines x = 1 + a, y = –3 – a, z = 1 + a and x = , y = 1 + b, z = 2 – b are coplanar,,
2
then  is equal to
(A) –3 (B) 2 (C) 1 (D) –2

Q.4 If the equation of the plane passing through (1, 2, 0) and which contains the line
x  3 y 1 z2
= = is 6x + y + z = k, then the value of (2 – 5 – k) equals
3 4 2
(A) 3 (B) 4 (C) 5 (D) 6

x  1 y  1 z  10
Q.5 Consider line L  = = . Point P(1, 0, 0) and Q are such that PQ is perpendicular to
2 3 8
line L and the mid-point of PQ lies on the line L then Q is
(A) (3, –4, –2) (B) (5, –8, –4) (C) (1, –1, –10) (D) (2, –3, 8)

Q.6 The value of m for which straight line 3x – 2y + z + 3 = 0 = 4x – 3y + 4z + 1 is parallel to the plane
2x – y + mz – 2 = 0 is
(A) – 2 (B) 8 (C) 11 (D) – 18

Q.7 Let OA, OB, OC be coterminous edges of a cube. If l, m, n be the shortest distance between the sides
OA, OB, OC and their respective skew body diagonals to them respectively then
1 1 1   1 1 1 
 2  2  2 : 2
   is equal to
l m n   OA OB OC 2 
2

1 1
(A) 2 (B) 3 (C) (D)
2 3
Q.8 The distance of the z-axis from the image of the point M (2, –3, 3) in the plane x – 2y – z + 1 = 0, is
(A) 1 (B) 2 (C) 2 (D) 4

89
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ PARAGRAPH T Y PE]
Paragraph for question nos. 9 and 10
Consider a plane
x + y – z = 1 and the point A(1, 2, –3)
A line L has the equation
x = 1 + 3r
y=2–r
z = 3 + 4r [3011712346]
Q.9 The co-ordinate of a point B of line L, such that AB is parallel to the plane, is
(A) 10, –1, 15 (B) –5, 4, –5 (C) 4, 1, 7 (D) –8, 5, –9
Q.10 Equation of the plane containing the line L and the point A has the equation
(A) x – 3y + 5 = 0 (B) x + 3y – 7 = 0 (C) 3x – y – 1 = 0 (D) 3x + y – 5 = 0

Paragraph for question nos. 11 to 13

Consider a plane P passing through the point (1, 1, 1) and perpendicular to the vector î  ˆj  k̂ .
If A (1, 2, 3) and B (3, 1, 2) are any two points in space, then

Q.11 If the coordinates of point C on plane P such that AC + BC is minimum are (, , ), then value of ( +
 + ) equals
(A) 1 (B) 2 (C) 3 (D) 4

Q.12 Image of AB in plane P is

x 1 y z 1 x 1 y z  1
(A)   (B)  
2 1 1 2 1 1
x 1 y z  3 x 1 y  3 z  1
(C)   (D)  
2  1 1 2 1 1

Q.13 Coordinates of point D on plane P such that | AD – BD | is minimum are (k1, k2, k3), then the value of
(k1 + k2 + k3) equals
(A) 1 (B) 2 (C) 3 (D) 4

Paragraph for question nos. 14 to 16

Let A(1, –2, 3) and B(5, 4, 7) be two points. If P is a point situated such that it is at minimum possible
distance from both the points A and B.
Q.14 Equation of plane  passing through P and perpendicular to line joining A and B will be
(A) 4x + 6y + 4z = 7 (B) 2x + 3y + 2z = 19
(C) 2x – 3y + 2z = 19 (D) 4x – 6y – 4z = 17
1
 V 3
Q.15 If volume of tetrahedron formed by above plane  with coordinate planes is V and   = k, then
3
[k] is equal to
[Note: [k] denotes greatest integer function less than or equal to k.]
(A) 1 (B) 2 (C) 3 (D) 4
Q.16 Distance of point O(0, 0, 0) from the line joining A and B, measured perpendicular to the plane
3x + y + 5z = 10 will be
(A) 5 (B) 15 (C) 2 5 (D) 35

90
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ REASONING T Y PE]
x  4 y  5 z 1 x  2 y 1 z
Q.17 Given lines   and  
2 4 3 1 3 2
Statement-1: The lines intersect.
Statement-2: They are not parallel.
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true. [3011712120]

[ M ULT IPL E CORRECT CHOICE T YPE]

Q.18 Consider the family of planes x + y + z = c where c is a parameter intersecting the coordinate axes at P,
Q, R and , ,  are the angles made by each member of this family with positive x, y and z axis. Which
of the following interpretations hold good for this family.
(A) each member of this family is equally inclined with the coordinate axes.
(B) sin2 + sin2 + sin2 = 1
(C) cos2 + cos2 + cos2 = 2
(D) for c = 3 area of the triangle PQR is 3 3 sq. units. [3011712324]

Q.19 If a line passing through (–2, 1, p) and (4, 1, 2) is perpendicular to the vector 2î  7ˆj  6k̂ and

parallel to the plane containing the vectors î  qk̂ and qˆj  pk̂ then the ordered pair (p, q) is

 1  1
(A)   1,   (B) (0, 0) (C) (–1, 0) (D)  0, 
 2  3

Q.20 In 3-D space, let three lines L1, L2 and L3 be such that
L1 : intersecting the z-axis at P(0, 0, 2) and does not meet the x-y plane
L2 : passing through the origin and through the point P.
L3 : passing through the origin and making positive angles (, , ) with co-ordinate axes and 45° angle
with line L1
Identify the which of the following statement(s) is(are) correct?
(A) area of the triangle formed by the lines L1 , L2 and L3 is 2 square units.
(B) area of the triangle formed by the lines L1, L2 and L3 is 8 square units.
(C) If  = 60°, then equation of L1 is x = y; z = 2.
(D) If  = 60°, then equation of L1 is x + y = 0 and z = 2.

91
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ MAT RIX T YPE]

Q.21 Consider the following four pairs of lines in column-I and match them with one or more entries in
column-II.
Column-I Column-II
(A) L1 : x = 1 + t, y = t, z = 2 – 5t (P) non coplanar lines

L2 : r  ( 2,1,3) + (2, 2, – 10)
x 1 y  3 z  2
(B) L1 : = = (Q) lines lie in a unique plane
2 2 1
x2 y6 z2
L2 : = =
1 1 3
(C) L1 : x = – 6t, y = 1 + 9t, z = – 3t (R) infinite planes containing both the lines
L2 : x = 1 + 2s, y = 4 – 3s, z = s
x y 1 z2
(D) L1 : = = (S) lines do not intersect
1 2 3
x 3 y2 z 1
L2 : = = [3011712177]
4 3 2
[ INT EGER T YPE]
Q.22 Let A1, A2, A3, A4 be the areas of the triangular faces of a tetrahedron, and h1, h2, h3, h4 be the
corresponding altitudes of the tetrahedron. If volume of tetrahedron is 5 cubic units, then find the minimum
value of (A1 + A2 + A3 + A4) (h1 + h2 + h3 + h4) (in cubic units). [3011711718]
x 6 y7 z7
Q.23 Find the length of the perpendicular from point (1, 2, 3) to the line = = . [3011711087]
3 2 2
Q.24 The line which contains all points (x, y, z) which are of the form (x, y, z) = (2, –2, 5) + (1, –3, 2)
intersects the plane 2x – 3y + 4z = 163 at P and intersects the YZ plane at Q. If the distance
PQ is a b where a, b  N and a > 3 then find (a + b). [3011711138]

Q.25 A vector V = v1î  v 2 ĵ  v 3k̂ satisfies the following conditions: [3011711394]
 
(i) magnitude of V is 7 2 (ii) V is parallel to the plane x – 2y + z = 6
 
(iii) V is orthogonal to the vector 2î  3ˆj  6k̂ (iv) V · î > 0.
Find the value of (v1 + v2 + v3).
[ SUBJECT IVE T YPE]
x 1 y  1 z 1
Q.26 If a plane passes through the point (1, 1, 1) and is perpendicular to the line   , then
3 0 4
find its perpendicular distance from the origin. [3011711189]
Q.27 If the plane x + y + z = 1 is rotated through 90º about its line of intersection with the plane
x – 2y + 3z = 0, Find the equation of the new position of the plane. [3011711343]
x 1 y  2 z
Q.28 Find the equation of the plane containing the straight line   and perpendicular to the
2 3 5
plane x – y + z + 2 = 0. [3011711514]
Q.29 Find the equations to the line of greatest slope through the point (7, 2 , –1) in the plane x – 2y + 3z = 0
assuming that the axes are so placed that the plane 2x + 3y – 4z = 0 is horizontal. [3011711565]
Q.30 The position vectors of the four angular points of a tetrahedron OABC are (0, 0, 0); (0, 0, 2); (0, 4, 0)
and (6, 0, 0) respectively. A point P inside the tetrahedron is at the same distance 'r' from the four plane
faces of the tetrahedron. Find the value of 'r'. [3011711616]

92
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[JEE(MAIN) PREVIOUS YEAR]

Q.1 The non-zero vectors a , b and c are related by a = 8 b and c = –7 b . Then the angle between
a and c is [3011712799]
 
(A) (B) (C)  (D) 0 [AIEEE 2008]
4 2
Q.2 The vector a =  î + 2 ĵ +  k̂ lies in the plane of the vectors b = î + ĵ & c = ĵ + k̂ & bisects the angle
between b & c . Then which one of the following gives possible values of  & ? [3011712748]
(A)  = 1,  = 2 (B)  = 2,  = 1 (C)  = 1,  = 1 (D)  = 2,  = 2
[AIEEE 2008]
x 1 y  2 z  3 x  2 y  3 z 1
Q.3 If the straight lines = = and = = intersect at a point, then the integer
k 2 3 3 k 2
k is equal to [3011712697]
(A) 5 (B) 2 (C) –2 (D) –5 [AIEEE-2008]
 17  13 
Q.4 The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point  0, 2 , 2  .
 
Then [3011712646]
(A) a = 4, b = 6 (B) a = 6, b = 4 (C) a = 8, b = 2 (D) a = 2, b = 8
[AIEEE-2008]
  
Q.5 If u, v, w are non-coplanar vectors and p, q are real numbers, then the equality
3u pv pw   pv w qu   2w qv qu   0 holds for : [AIEEE 2009]
(A) exactly two values of (p,q) (B) more than two but not all values of (p, q)
(C) all values of (p, q) (D) exactly one value of (p, q) [3011712789]
x  2 y 1 z  2
Q.6 Let the line = = lie in the plane x + 3y – z +  = 0. then () equals
3 5 2
(A) (–6, 7) (B) (5, –15) (C) (–5, 5) (D) (6, –17) [3011712535]
[AIEEE-2009]
Q.7 The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines
of the vector are : [3011712484]
6 3 2 6 3 2 6 3 2
(A) , , (B) , , (C) , , (D) 6, –3, 2
5 5 5 7 7 7 7 7 7
[AIEEE-2009]
 ˆ       
Q.8 Let a  j  k̂ and c  î  ĵ  k̂ . Then the vector b satisfying a  b  c = 0 and a · b = 3, is
(A)  î  ˆj  2 k̂ (B) 2î  ˆj  2 k̂ (C) î  ĵ  2 k̂ (D) î  ĵ  2 k̂ [3011712586]
[AIEEE-2010]
  
Q.9 If the vectors a  î  ĵ  2 k̂ , b  2î  4 ĵ  4k̂ and c   î  ĵ  k̂ are mutually orthogonal, then (, µ)
is equal to [3011712637]
(A) (– 3, 2) (B) (2, – 3) (C) (– 2, 3) (D) (3, – 2)
[AIEEE-2010]

93
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.10 A line AB in three-dimensional space makes angles 45° and 120° with the positive x -axis and the
positive y-axis respectively. If AB makes an acute angle  with the positive z-axis, then  equals
(A) 30° (B) 45° (C) 60° (D) 75° [3011712739]
[AIEEE-2010]
Q.11 Statement-1 : The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x – y + z = 5.
Statement-2 : The plane x – y + z= 5 bisects the line segment joining A(3, 1, 6) and B(1, 3, 4).
(A) Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation of Statement-1.
(B) Statement-1 is true, Statement-2 is false and Statement-2 is not the correct explanation of Statement-1
(C) Statement-1 is true, Statement-2 is false
(D) Statement-1 is false, Statement-2 is true. [3011712790]
[AIEEE-2010]
  1      
  7
      
Q.12 If a  1 3î  k̂ and b  2î  3ˆj  6k̂ , then the value of 2a  b · a  b  a  2b is 
10
(A) –5 (B) –3 (C) 5 (D) 3 [3011712485]
[AIEEE-2011]
   
Q.13 The vectors a and b are not perpendicular and c and d are two vector satisfying
      
b  c  b  d and a · d  0 . Then the vector d is equal to [3011712536]
       
 b · c  a · c    b · c  a · c
(A) b      c (B) c      b (C) b      c (D) c      b
a · b a · b
 a · b      a · b
[AIEEE-2011]
y 1 z  3  5 
Q.14 If the angle between the line x =  and the plane x + 2y + 3z = 4 is cos–1   , then

2   14 
 equals [3011712638]
(A) 2/3 (B) 3/2 (C) 2/5 (D) 5/2 [AIEEE-2011]

Q.15 Statement-1 : The point A(1, 0, 7) is the mirror image of the point B (1, 6, 3) in the line
x y 1 z  2
  . [3011712689]
1 2 3
x y 1 z  2
Statement 2 : The line :   bisects the line segment joining A(1, 0, 7) and B(1, 6, 3).
1 2 3
(A) Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation of Statement-1.
(B) Statement-1 is true, Statement-2 is false and Statement-2 is not the correct explanation of Statement-1
(C) Statement-1 is true, Statement-2 is false
(D) Statement-1 is false, Statement-2 is true. [AIEEE-2011]
 
Q.16 Let â and b̂ be two unit vectors. If the vectors c  â  2b̂ and d  5â  4b̂ are perpendicular to each
other, then the angle between â and b̂ [3011712740]
   
(A) (B) (C) (D) [AIEEE-2012]
3 4 6 2
  
Q.17 Let ABCD be a parallelogram such that AB  q , AD  p and BAD be an acute angle. If r is the

vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by
   
  p·q    3 p · q  
(A) r  q      p (B) r  3q    p [3011712791]
p·p p ·p 
   
  3 p · q     p·q 
(C) r  3q    p (D) r  q      p [AIEEE-2012]
p ·p  p·p
94
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.18 An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is
(A) x – 2y + 2z – 1 = 0 (B) x – 2y + 2z + 5 = 0 [3011712490]
(C) x – 2y + 2z – 3 = 0 (D) x – 2y + 2z + 1 = 0 [AIEEE-2012]

x 1 y  1 z 1 x 3 yk z
Q.19 If the line   and   intersect, then k is equal to [3011712541]
2 3 4 1 2 1
9 2
(A) (B) 0 (C) – 1 (D) [AIEEE-2012]
2 9

x2 y 3 z 4 x 1 y  4 z  5
Q.20 If the lines = = and = = are coplanar, then k can have
1 1 k k 2 1
(A) exactly one value (B) exactly two values [3011712592]
(C) exactly three values (D) any value [JEE Main 2013]

Q.21 If the vectors AB  3î  4k̂ and AC  5î  2ˆj  4k̂ are the sides of a triangle ABC, then the length of
the median through A is [3011712643]
(A) 72 (B) 33 (C) 45 (D) 18 [JEE Main 2013]

Q.22 Distance between two parallel planes

2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is [3011712694]
5 7 9 3
(A) (B) (C) (D) [JEE Main 2013]
2 2 2 2
x 1 y  3 z  4
Q.23 The image of the line   in the plane 2x – y + z + 3 = 0 is the line
3 1 5
x 3 y 5 z 2 x 3 y5 z2
(A)   (B)  
3 1 5 3 1 5
x 3 y5 z2 x 3 y 5 z  2
(C)   (D)   [JEE Main 2014]
3 1 5 3 1 5
      
 
Q.24 If a  b b  c c  a  [a b c]2 then  is equal to
(A) 1 (B) 2 (C) 3 (D) 0 [JEE Main 2014]

Q.25 The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and
l2 = m2 + n2 is
   
(A) (B) (C) (D) [JEE Main 2014]
2 3 4 6
  
Q.26 Let a , b and c be three non-zero vectors such that no two of them are collinear and
   1     
(a  b)  c = b c a . If  is the angle between vectors b and c , then a value of sin  is
3
2 2 3 2 2  2
(A) (B) (C) (D)
3 3 3 3
[JEE Main 2015]

95
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.27 The equation of the plane containing the line 2x – 5y + z = 3; x + y + 4z = 5, and parallel to the plane,
x + 3y + 6z = 1, is
(A) x + 3y + 6z = 7 (B) 2x + 6y + 12z = –13
(C) 2x + 6y + 12z = 13 (D) x + 3y + 6z = –7 [JEE Main 2015]

x  2 y 1 z  2
Q.28 The distance of the point (1, 0, 2) from the point of intersection of the line = = and the
3 4 12
plane x – y + z = 16, is [JEE Main 2015]
(A) 3 21 (B) 13 (C) 2 14 (D) 8

Q.29 The distance of the point (1, – 5, 9) from the plane x – y + z = 5 measured along the line
x = y = z is [JEE Main 2016]
20 10
(A) (B) 3 10 (C) 10 3 (D)
3 3

x 3 y  2 z  4
Q.30 If the line,   lies in the plane, lx + my – z = 9, then l2 + m2 is equal to
2 1 3
(A) 2 (B) 26 (C) 18 (D) 5 [JEE Main 2016]

      3    
Q.31 Let a , b and c be three unit vectors such that a  ( b  c) = (b  c ) . If b is not parallel to c , then
2
 
the angle between a and b is [JEE Main 2016]
5 3  2
(A) (B) (C) (D)
6 4 2 3

Q.32 If the image of the point P (1, – 2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,
x y z
= = is Q, then PQ is equal to [JEE (Main) 2017]
1 4 5
(A) 3 5 (B) 2 42 (C) 42 (D) 6 5

Q.33 The distance of the point (1, 3, – 7) from the plane passing through the point (1, – 1, – 1) having normal
x 1 y  2 z  4 x  2 y 1 z  7
perpendicular to both the lines = = and = = is
1 2 3 2 1 1
20 10 5 10
(A) (B) (C) (D)
74 83 83 74
[JEE (Main) 2017]
       
Q.34 Let a = 2î  ˆj  2k̂ and b = î  ˆj . Let c be a vector such that | c  a | = 3, (a  b)  c = 3 and the
    
angle between c and a  b be 30°. Then a · c is equal to
25 1
(A) (B) 2 (C) 5 (D) [JEE (Main) 2017]
8 8

96
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[JEE(ADVANCED) PREVIOUS YEAR]

Q.1 Show by vector methods, that the angular bisectors of a triangle are concurrent and find an expression
for the position vector of the point of concurrency in terms of the position vectors of the vertices.
[JEE '2001 (Mains)]
  
Q.2 Find 3–dimensional vectors v 1, v 2 , v 3 satisfying
           
v 1  v 1 = 4, v 1  v 2 = –2, v 1  v 3 = 6, v 2  v 2 = 2, v 2  v 3 = –5, v 3  v 3 = 29.
[JEE '2001 (Mains)]

    2  2  2
Q.3 If a, b and c are unit vectors, then a  b  b  c  c  a does NOT exceed
(A) 4 (B) 9 (C) 8 (D) 6
[JEE '2001 (Screening)]
   
Q.4 Let a  î  k̂ , b  x î  ĵ  (1  x )k̂ and c  yî  x ˆj  (1  x  y)k̂ . Then [ a, b, c] depends on
(A) only x (B) only y (C) NEITHER x NOR y (D) both x and y
[JEE '2001 (Screening)]

 
Q.5 Let A (t ) = f 1 (t )i  f 2 (t )j and B( t )  g1 ( t ) i  g2 ( t ) j , t  [0, 1], where f1, f2, g1, g2 are
  
continuous functions. If A (t ) and B( t ) are nonzero vectors for all t and A(0) = 2i  3j ,
    
A(1) = 6i  2j , B(0) = 3i  2j and B(1) = 2i  6j , then show that A (t ) and B(t ) are parallel for
some t. [JEE '2001 (Mains)]

     
Q.6 If a and b are two unit vectors such that a + 2 b and 5 a – 4 b are perpendicular to each other then
 
the angle between a and b is [3011711479]
(A) 450 (B) 600 (C) cos–1 1 3 (D) cos–1 2 7 
[JEE 2002(Screening)]

  
Q.7 Let V  2î  ˆj  k̂ and W  î  3k̂ . If U is a unit vector, then the maximum value of the scalar triple
  

product U V W is  [3011711530]

(A) –1 (B) 10  6 (C) 59 (D) 60

[JEE 2002(Screening)]


Q.8 Let V be the volume of the parallelopiped formed by the vectors a  a1î  a 2 ˆj  a 3k̂ ,
 
b  b1î  b 2 ˆj  b 3k̂ , c  c1î  c 2 ĵ  c 3k̂ . If ar , br , cr , where r = 1, 2, 3, are non-negative real
3
numbers and  a r  b r  c r  = 3L, show that V  L3. [3011711581]
r 1
[JEE 2002(Mains), 5]

9 7
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

  
Q.9 If a = î  aˆj  k̂ , b = ˆj  ak̂ , c = aî  k̂ , then find the value of ‘a’ for which volume of
parallelopiped formed by three vectors as coterminous edges, is minimum, is [3011711632]
1 1 1
(A) (B) – (C) ± (D) none
3 3 3
[JEE 2003(Scr.)]

Q.10 Find the equation of the plane passing through the points (2, 1, 0) , (5, 0, 1) and (4, 1, 1).
[3011711683]
[ JEE 2003]

Q.11 If P is the point (2, 1, 6) then find the point Q such that PQ is perpendicular to the plane in (i) and the mid
point of PQ lies on it. [3011711683]
[ JEE 2003]

    
Q.12 If u , v , w are three non-coplanar unit vectors and , ,  are the angles between u and v ,
      
v and w , w and u respectively and x , y, z are unit vectors along the bisectors of the angles , , 
      1    2   
respectively. Prove that x  y y  z z  x   u v w  sec 2 sec 2 sec 2 .
16 2 2 2
[3011711734]
[JEE 2003]

x 1 y  1 z 1 x 3 y k z
Q.13 If the lines   and   intersect, then k = [3011711414]
2 3 4 1 2 1
(A) 2/9 (B) 9/2 (C) 0 (D) – 1
[ JEE 2004 (screening)]

Q.14 A unit vector in the plane of the vectors 2î  ˆj  k̂ , î  ĵ  k̂ and orthogonal to 5î  2 ĵ  6k̂
6î  5k̂ 3 ĵ  k̂ 2î  5k̂ 2 î  ĵ  2k̂
(A) (B) (C) (D)
61 10 29 3
[3011711414]
[ JEE 2004 (screening)]
      
Q.15 If a  î  j  k̂ , a · b  1 and a  b  ˆj  k̂ , then b =
(A) î (B) î  ˆj  k̂ (C) 2ˆj  k̂ (D) 2î [3011711414]
[ JEE 2004 (screening)]

           
Q.16 Let a , b , c , d are four distinct vectors satisfying a  b = c  d and a  c  b  d . Show that
       
a ·b  c ·d  a ·c  b ·d . [3011711465]
[JEE 2004]
Q.17 Let P be the plane passing through (1, 1, 1) and parallel to the lines L1 and L2 having direction ratios
1, 0, –1 and –1, 1, 0 respectively. If A, B and C are the points at which P intersects the coordinate axes,
find the volume of the tetrahedron whose vertices are A, B, C and the origin. [3011711465]
[JEE 2004]

9 8
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.18 A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at A, B and C. If the
1 1 1
centroid D (x, y, z) of triangle ABC satisfies the relation 2  2  2 = k, then the value of k is
x y z
(A) 3 (B) 1 (C) 1/3 (D) 9 [3011711516]
[JEE 2005 (Scr)]
Q.19 Find the equation of the plane containing the line 2x – y + z – 3 = 0, 3x + y + z = 5 and at a distance of
1 6 from the point (2, 1, – 1). [3011711516]
[ JEE 2005 (Mains)]
Q.20 Incident ray is along the unit vector v̂ and the reflected ray is along the unit vector ŵ . The normal is
along unit vector â outwards. Express ŵ in terms of â and v̂ .
[3011711516]

[ JEE 2005 (Mains)]

Q.21 A plane passes through (1, –2, 1) and is perpendicular to two planes 2x – 2y + z = 0 and
x – y + 2z = 4. The distance of the plane from the point (1, 2, 2) is [3011711567]
(A) 0 (B) 1 (C) 2 (D) 2 2 [JEE 2006]
    
Q.22 Let a  î  2ˆj  k̂ , b  î  ˆj  k̂ and c  î  ˆj  k̂ . A vector in the plane of a and b whose projection

on c has the magnitude equal to 1 3 , is [3011711618]
(A) 4î  ˆj  4k̂ (B) 3î  ĵ  3k̂ (C) 2î  ˆj  2k̂ (D) 4î  ˆj  4k̂ [JEE 2006]

Q.23 Let A be vector parallel to line of intersection of planes P1 and P2 through origin. P1 is parallel to the
vectors 2 ˆj + 3 k̂ and 4 ˆj – 3 k̂ and P2 is parallel to ˆj – k̂ and 3 î + 3 ˆj , then the angle between

vector A and 2 î + ˆj – 2 k̂ is [3011711669]
   
(A) (B) (C) (D) [JEE 2006]
2 4 6 3

Q.24 Match the following

Column-I Column-II [3011711720]
1 1
2 2
(A)  ( y  1) dy +  (1  y ) dy (P) 2 [JEE 2006]
0 0
(B) A point (, , ) lies on the plane x + y + z = 2.
0 1

The value of  such that the vector a   î   ˆj   k̂ (Q)  1  x dx +  1  x dx
1 0

satisfies k̂  ( k̂  a )  0
4
(C) In a triangle ABC, if cos A cosB + sinA sinB sinC = 1, (R)
3
then the value of sin C is
(D) The set of values of a for which the lines (S) 1
x + y = | a |, ax – y = 4
intersect in the region x > 0, y > 0, is the interval (a0, ).
Then the value of a0 is

9 9
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.25 Match the following [3011711436]

[JEE 2006, 6]
Column I Column II
5
(A) Let P be the plane passing through the point (2, 1, – 1) and perpendicular (P)
3
to the line of intersection of the planes 2x + y – z = 3 and x + 2y + z = 2.
Then the distance from the point  
3 , 2, 2 to the plane P is
n
1  1 
(B) If Lim  tan  2   t , then tan t is (Q) 1
n
i 1  2i 
a 2
(C) The sides a, b, c of a triangle ABC are in A.P. If cos 1  , (R)
bc 3
b c  
cos  2  , cos 3  , then tan 2 1  tan 2 3 = (S) 0
ac ab 2 2
(D) Let L be the line passing through the point (0, 1, 0) and perpendicular to the plane
x + 2y + 2z = 0. Then the distance from the point (0, 0, 0) to the line L is

Q.26 The number of distinct real values of , for which the vectors  2 î  ĵ  k̂ , î  2 ĵ  k̂ and î  ĵ  2 k̂
are coplanar, is [3011711487]
(A) zero (B) one (C) two (D) three [JEE 2007]
      
Q.27 Let a , b, c be unit vectors such that a  b  c  0 . Which one of the following is correct?
             
(A) a  b  b  c  c  a  0 (B) a  b  b  c  c  a  0 [3011711538]
            
(C) a  b  b  c  a  c  0 (D) a  b, b  c, c  a are mutually perpendicular..
[JEE 2007]

Q.28 Let the vectors P Q , Q R , R S , S T , T U and U P represent the sides of a regular hexagon.


Statement-1: P Q × R S  S T  0  [3011711538]
because [JEE 2007]
 
Statement-2: P Q  R S = 0 and P Q  S T  0
(A) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true.
Q.29 Consider the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5.
Statement-1: The parametric equations of the line of intersection of the given planes are
x = 3 + 14t, y = 1 + 2t, z = 15t. [3011711589]
[JEE 2007]
Statement-2: The vector 14î  2ˆj  15k̂ is parallel to the line of intersection of given planes.
(A) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true.

100
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

MATCH THE COLUMN:

Q.30 Consider the following linear equations [3011711640]
[JEE 2007]
ax + by + cz = 0 ; bx + cy + az = 0 ; cx + ay + bz = 0
Match the conditions/ expressions in Column I with statements in Column II.
Column I Column II
(A) a + b + c  0 and (P) the equation represent planes
2 2 2
a + b + c = ab + bc + ca meeting only at a single point.
(B) a + b + c = 0 and (Q) the equation represent the line
a2 + b2 + c2  ab + bc + ca x=y=z
(C) a + b + c  0 and (R) the equation represent identical planes
a2 + b2 + c2  ab + bc + ca
(D) a + b + c = 0 and (S) the equation represent the whole of
2 2 2 =
a + b + c ab + bc + ca the three dimensional space.

Q.31 The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors â , b̂, ĉ
1
such that â · b̂  b̂ · ĉ  ĉ · â  . Then the volume of the parallelopiped is [3011711691]
2
[JEE 2008]
1 1 3 1
(A) (B) (C) (D)
2 2 2 2 3

Q.32 Let two non-collinear unit vector â and b̂ form an acute angle. A point P moves so that at any time t the
position vector O P (where O is the origin) is given by â cos t  b̂ sin t . When P is farthest from origin

O, let M be the length of O P and û be the unit vector along O P . Then, [3011711742]
[JEE 2008]
1 1
â  b̂ â  b̂
(A) û  and M  (1  â · b̂) 2 (B) û  and M  (1  â · b̂) 2
| â  b̂ | | â  b̂ |
1 1
â  b̂ â  b̂
(C) û  and M  (1  2â · b̂) 2 (D) û  and M  (1  2â · b̂) 2
| â  b̂ | | â  b̂ |

Q.33 Consider three planes

P1 : x – y + z = 1 ; P2 : x + y – z = –1 ; P3 : x – 3y + 3z = 2
Let L1, L2, L3 be the lines of intersection of the planes P2 and P3, P3 and P1, and P1 and P2, respectively.
Statement-1 : At least two of the lines L1, L2 and L3 are non-parallel.
Statement-2 : The three planes do not have a common point.
(A) Statement-1 is True, Statement-2 is True; statement-2 is a correct explanation for statement-1
(B) Statement-1 is True, Statement-2 is True; statement-2 is NOT a correct explanation for statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True [3011711759]
[JEE 2008]

101
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Paragraph for Question Nos. 34 to 36

x 1 y  2 z 1 x 2 y2 z3
Consider the lines L1 :   ; L2 :   [3011711810]
3 1 2 1 2 3
Q.34 The unit vector perpendicular to both L1 and L2 is

 î  7ˆj  7 k̂  î  7ˆj  5k̂  î  7 ĵ  5k̂ 7 î  7 ĵ  k̂

(A) (B) (C) (D)
99 5 3 5 3 99

Q.35 The shortest distance between L1 and L2 is

17 41 17
(A) 0 (B) (C) (D)
3 5 3 5 3

Q.36 The distance of the point (1, 1, 1) from the plane passing through the point (–1, –2, –1) and whose
normal is perpendicular to both the lines L1 and L2 is
2 7 13 23
(A) (B) (C) (D) [JEE 2008]
75 75 75 75


Q.37 Let P(3, 2, 6) be a point in space and Q be a point on the line r  ( î  ĵ  2k̂ )   ( 3î  ˆj  5k̂ )

Then the value of  for which the vector PQ is parallel to the plane x – 4y + 3z =1 is [3011711861]
(A) 1/4 (B) – 1/4 (C) 1/8 (D) – 1/8
[JEE 2009]
          1
  
Q.38 If a , b, c and d are unit vectors such that a  b · c  d  1 and a · c  , then [3011711912]
2
     
(A) a , b, c are non-coplanar (B) b, c, d are non-coplanar
     
(C) b, d are non-parallel (D) a , d are parallel and b, c are parallel
[JEE 2009]

Q.39 A line with positive direction cosines passes through the point P(2, – 1, 2) and makes equal angles with
the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment
PQ equals [3011711963]
(A) 1 (B) 2 (C) 3 (D) 2
[JEE 2009]

102
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.40 Match the statements/expressions given in Column I with the values given in Column II.
Column I Column II

(A) Root(s) of the equation 2 sin2 + sin22 = 2 (P)
6
 6x   3x  
(B) Points of discontinuity of the function f(x) =   cos   , (Q)
  4
where [y] denotes the largest integer less than or equal to y

(C) Volume of the parallelopiped with its edges represented by the vectors (R)
3
î  ˆj, î  2ˆj and î  ˆj  k̂

     
(D) Angle between vectors a and b where a , b and c are unit (S)
2
   
vectors satisfying a  b  3 c  0 (T) 
[3011712014]
[JEE 2009, 8 ]

Q.41 Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations :
3x – y – z = 0 ; – 3x + z = 0 ; – 3x + 2y + z = 0 [3011712065]
2 2 2
Then the number of such points for which x + y + z  100 is [JEE 2009, 4]

x y z
Q.42 Equation of the plane containing the straight line   and perpendicular to the plane containing
2 3 4
x y z x y z
the straight lines   and   , is [3011711771]
3 4 2 4 2 3
(A) x + 2y – 2z = 0 (B) 3x + 2y – 2z = 0 (C) x – 2y + z = 0 (D) 5x + 2y – 4z = 0
[JEE 2010, 3]

Q.43 Two adjacent sides of a parallelogram ABCD are given by [3011711822]

AB  2î  10 ĵ  11k̂ and AD   î  2ˆj  2k̂
The side AD is rotated by an acute angle  in the plane of the parallelogram so that AD becomes AD.
If AD makes a right angle with the side AB, then the cosine of the angle  is given by
8 17 1 4 5
(A) (B) (C) (D) [JEE 2010, 5]
9 9 9 9

Q.44 If the distance of the point P(l, – 2, 1) from the plane x + 2y - 2z =  , where  > 0. is 5, then the foot
of the perpendicular from P to the plane is [3011711873]
8 4 7 4 4 1  1 2 10  2 1 5
(A)  , ,   (B)  ,  ,  (C)  , ,  (D)  ,  , 
3 3 3 3 3 3 3 3 3  3 3 2
[JEE 2010, 5]

103
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

   î  2 ĵ  2î  ˆj  3k̂
Q.45 If a and b are vectors in space given by a  and b  , then the value of
5 14
     
   
2a  b · a  b  a  2b , is  [3011711924]
[JEE 2010, 3]
x 1 y  2 z  3
Q.46 If the distance between the plane Ax – 2y + z = d and the plane containing the lines  
2 3 4
x 2 y 3 z  4
and   is 6 , then | d | is [3011711975]
3 4 5
[JEE 2010, 3]

Q.47 Match the statements in Column-I with the values in Column-II. [3011712026]
Column I Column II
(A) A line from the origin meets the lines (p) – 4
8
x
x  2 y 1 z 1 3  y  3  z 1
  and
1 2 1 2 1 1
at P and Q respectively. If length PQ = d, then d2 is
1  3
(B) The values of x satisfying tan–1 (x +3) – tan–1 (x – 3) = sin   are (q) 0
5
    
(C) Non-zero vectors a , b and c satisfy a · b  0
       
  
b  a  b  c  0 and 2 | b  c |  | b  a | (r) 4
  
If a  b  4c , then the possible values of  are
(D) Let f be the function on  ,  given by (s) 5
 9x  x
f(0) = 9 and f(x) = sin   / sin   for x  0
 2  2

2
The value of
  f x  dx is

(t) 6
[JEE 2010, (2+2+2+2)]

   
Q.48 Let a  î  ĵ  k̂ , b  î  ˆj  k̂ and c  î  ˆj  k̂ be three vectors. A vector v in the plane
   1
of a and b , whose projection on c is , is given by [3011712077]
3
(A) î  3 ĵ  3k̂ (B)  3î  3 ĵ  k̂ (C) 3î  ĵ  3k̂ (D) î  3 ĵ  3k̂
[JEE 2011, 3]

Q.49 The vector(s) which is/are coplanar with vectors î  ĵ  2k̂ and î  2ˆj  k̂ and perpendicular to the
vector î  ˆj  k̂ is/are [3011711776]
(A) ĵ  k̂ (B)  î  ĵ (C) î  ˆj (D)  ĵ  k̂
[JEE 2011, 4]

104
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

   
Q.50 Let a   î  k̂ , b   î  ĵ and c  î  2 ĵ  3k̂ be three given vectors. If r is a vector such that
       
r  b  c  b and r · a  0 , then the value of r · b , is [3011711827]
[JEE 2011, 4]
Q.51 The point P is the intersection of the straight line joining the points Q (2, 3, 5) and R (1, – 1, 4) with the
plane 5x – 4y – z = 1. If S is the foot of the perpendicular drawn from the point T (2, 1, 4) to QR, then
the length of the line segment PS is [3011712751]
1
(A) (B) 2 (C) 2 (D) 2 2
2
[JEE 2012, 3]
Q.52 The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and
2
x – y + z = 3 and at a distance from the point (3, 1, – 1) is [3011712753]
3
(A) 5x – 11y + z = 17 (B) 2 x + y = 3 2 – 1
(C) x + y + z = 3 (D) x – 2y=1 – 2 [JEE 2012, 3]

     
Q.53 If a and b are vectors such that a  b  29 and a  ( 2î  3ˆj  4k̂ ) = ( 2î  3ˆj  4k̂ )  b , then a
 
  
possible value of a  b ·  7 î  2 ĵ  3k̂ is [3011712702]
(A) 0 (B) 3 (C) 4 (D) 8 [JEE 2012, 3]

x 1 y  1 z x 1 y 1 z
Q.54 If the straight lines   and   are coplanar, then the plane(s) containing
2 k 2 5 2 k
these two lines is(are) [3011712752]
(A) y + 2z = – 1 (B) y + z = – 1 (C) y – z = – 1 (D) y – 2z = – 1
[JEE 2012, 4]
    2  2  2   
Q.55 If a , b and c are unit vectors satisfying a  b  b  c  c  a  9 , then 2a  5b  5c is
[3011712701]
[JEE 2012, 4]
x  2 y 1 z
Q.56 Perpendiculars are drawn from points on the line   to the plane x + y + z = 3. The feet
2 1 3
of perpendiculars lie on the line [3011712649]
x y 1 z  2 x y 1 z  2
(A)   (B)  
5 8  13 2 3 5
x y 1 z  2 x y 1 z  2
(C)   (D)   [JEE Advance 2013, 2]
4 3 7 2 7 5

105
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.57 Let PR = 3î  ˆj  2k̂ and SQ = î  3ˆj  4k̂ determine diagonals of a parallelogram PQRS and

PT = î  2 ĵ  3k̂ be another vector. Then the volume of the parallelepiped determined by the vectors
PT, PQ and PS is
(A) 5 (B) 20 (C) 10 (D) 30 [3011712549]
[JEE (Advanced) 2013, 2]
Q.58 A line l passing through the origin is perpendicular to the lines
l1 : (3  t ) î  ( 1  2 t ) ĵ  ( 4  2t ) k̂ , –  < t <  [3011712551]
l2 : (3  2s ) î  (3  2s) ˆj  ( 2  s) k̂ , –  < s < 
Then, the coordinate(s) of the point(s) on l2 at a distance of 17 from the point of intersection of l and
l1 is (are)
7 7 5 7 7 8
(A)  , ,  (B) (– 1, – 1, 0) (C) (1, 1, 1) (D)  , , 
 3 3 3 9 9 9
[JEE Advance 2013, 4]
y z y z
Q.59 Two lines L1 : x = 5, = and L2 : x = , = are coplanar. Then  can take value(s)
3 2 1 2  
(A) 1 (B) 2 (C) 3 (D) 4 [3011712599]
[JEE Advance 2013, 3]

 
Q.60 Consider the set of eight vectors V  a î  bĵ  ck̂ : a , b, c{1,1} . Three non-coplanar vectors can
be chosen from V in 2p ways. Then p, is [3011712601]
[JEE Advance 2013, 5]
Q.61 Match List-I with List-II and select the correct answer using the code given below the lists:
Column-I Column-II
  
(A) Volume of parallelepiped determined by vector a , b and c is 2. (P) 100
Then the volume of the parallelepiped determined by vectors
     
   
2 a  b , 3 b  c and c  a  is
  
(B) Volume of parallelepiped determined by vectors a , b and c is 5. (Q) 30
Then the volume of the parallelepiped determined by vectors
     
   
3 a  b , b  c and 2 c  a  is

(C) Area of a triangle with adjacent sides determined by vectors (R) 24

 
a and b is 20. Then the area of the triangle with adjacent sides
   

determined by vectors 2a  3b and a  b is  
(D) Area of a parallelogram with adjacent sides determined by vectors (S) 60
 
a and b is 30. Then the area of the parallelogram with adjacent
  

sides determined by vectors a  b and a is 
[JEE (Advanced) 2013, 3]

106
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

x 1 y z3 x 4 y3 z3

Q.62 Consider the lines L 1 :   , L2 :   and the planes
2 1 1 1 1 2
P1 : 7x + y + 2z = 3, P2 : 3x + 5y – 6z = 4. Let ax + by + cz = d be the equation of the plane passing
through the point of intersection of lines L1 and L2, and perpendicular to planes P1 and P2.
Match List-I with List-II and select the correct answer using the code given below the lists:
List I List II
(A) a= (P) 13
(B) b= (Q) –3
(C) c= (R) 1
(D) d= (S) –2 [JEE (Advanced) 2013, 3]

Q.63 From a point P (, , ), perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1
and y = – x, z = – 1. If P is such that QPR is a right angle, then the possible value(s) of  is(are)
(A) 2 (B) 1 (C) – 1 (D) – 2
[JEE (Advanced) 2014, 3]
   
Q.64 Let x , y and z be three vectors each of magnitude 2 and the angle between each pair of them is .
3
     
If a is a nonzero vector perpendicular to x and y  z and b is a nonzero vector perpendicular to y
 
and z  x , then
         
(A) b  ( b · z ) ( z  x ) (B) a  ( a · y) ( y  z )
          
(C) a · b   ( a · y) ( b · z) (D) a  ( a · y) ( z  y) [JEE (Advanced) 2014, 3]

  
Q.65 Let a , b and c be three non-coplanar unit vectors such that the angle between every pair of them is

        p 2  2q 2  r 2
. If a  b  b  c = pa  qb  rc , where p, q and r are scalars, then the value of is
3 q2
[JEE (Advanced) 2014, 3]

Q.66 In R3, consider the planes P1 : y = 0 and P2 : x + z = 1. Let P3 be a plane, different from P1 and P2,
which passes through the intersection of P1 and P2. If the distance of the point (0, 1, 0) from P3 is 1 and
the distance of a point (, , ) from P3 is 2, then which of the following relations is(are) true?
(A) 2 +  + 2 + 2 = 0 (B) 2 –  + 2 + 4 = 0
(C) 2 +  – 2 – 10 = 0 (D) 2 –  + 2 – 8 = 0
[JEE Adv. 2015, 4]

107
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.67 In R3, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant
distance from the two planes P1 : x + 2y – z + 1 = 0 and P2 : 2x – y + z – 1 = 0. Let M be the locus
of the feet of the perpendiculars drawn from the points on L to the plane P1. Which of the following
points lie(s) on M?

 5 2   1 1 1   5 1  1 2
(A)  0, ,  (B)  , ,  (C)  , 0,  (D)  , 0, 
 6 3   6 3 6  6 6  3 3
[JEE Adv. 2015, 4]

      
Q.68 Let PQR be a triangle. Let a  QR , b  RP and c  PQ . If | a | = 12, | b | = 4 3 and b · c = 24,
then which of the following is (are) true?
 
| c |2  | c |2 
(A)  | a | = 12 (B)  | a | = 30
2 2
     
(C) | a  b  c  a |  48 3 (D) a · b = – 72
[JEE Adv. 2015, 4]

Q.69 Column-I Column-II

(A) In a triangle XYZ, let a, b and c be the lengths of the sides (P) 1
opposite to the angles X, Y and Z respectively. If 2 (a2 – b2) = c2
sin (X  Y )
and  = , then possible values of n for which
sin Z
cos (n) = 0 is (are)
(B) In a triangle XYZ, let a, b and c be the lengths of the sides opposite (Q) 2
to the angles X, Y and Z respectively.
a
If 1 + cos 2X – 2cos 2Y = 2sin X sin Y, then possible value(s) of is (are)
b
(C) In R2, let 3 î  ˆj , î  3 ĵ and  î  (1  ) ĵ be the position vectors (R) 3
of X, Y and Z with respect to the origin O, respectively. If the distance of
3
Z from the bisector of the acute angle of OX with OY is ,
2
then possible value(s) of |  | is (are)
(D) Suppose that F() denotes the area of the region bounded by x = 0, (S) 5
x = 2, y2 = 4x and y = | x – 1 | + | x – 2 | + x, where  {0, 1}.
8 2
Then the value(s) of F() + , when  = 0 and  = 1, is (are) (T) 6
3
[JEE Adv. 2015, 2 + 2 + 2 + 2]

108
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

   
Q.70 Suppose that p, q and r are three non-coplanar vectors in R3. Let the components of a vector s along
   
p, q and r be 4, 3 and 5 respectively. If the components of this vector s along
        
(  p  q  r ), ( p  q  r ) and (  p,  q  r ) are x, y and z respectively, then the value of 2x + y + z is
[JEE Adv. 2015, 4]

Q.71 In R2, if the magnitude of the projection vector of the vector  î   ĵ on 3 î  ˆj is 3 and if
 =2 + 3 , then possible value(s) of |  | is (are)
(A) 1 (B) 2 (C) 3 (D) 4
[JEE Adv. 2015, MTC, 2]

Q.72 Consider a pyramid OPQRS located in the first octant (x  0, y  0, z  0) with O as origin, and OP and
OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with
OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then

(A) the acute angle between OQ and OS is [JEE (Advanced) 2016, 4]
3
(B) the equation of the plane containing the triangle OQS is x – y = 0
3
(C) the length of the perpendicular from P to the plane containing the triangle OQS is
2

15
(D) the perpendicular distance from O to the straight line containing RS is
2

Q.73 Let P be the image of the point (3, 1, 7) with respect to the plane x – y + z = 3. Then the equation of the
x y z
plane passing through P and containing the straight line   is [JEE (Advanced) 2016, 3]
1 2 1
(A) x + y – 3z = 0 (B) 3x + z = 0 (C) x – 4y + 7z = 0 (D) 2x – y = 0

1
Q.74 Let û  u1î  u 2 ĵ  u 3k̂ be a unit vector in R3 and ŵ  (î  ˆj  2k̂ ) . Given that there exists a vector
6
  
v in R3 such that | û  v | 1 and ŵ · (û  v)  1 . Which of the following statement(s) is (are) correct ?

(A) There is exactly one choice for such v

(B) There are infinitely many choices for such v
(C) If û lies in the xy-plane then | u1 | = | u2 |
(D) If û lies in the xz-plane then 2| u1 | = | u3 | [JEE (Advanced) 2016, 4]

Q.75 The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes
2x + y – 2z = 5 and 3x – 6y – 2z = 7, is
(A) – 14x + 2y + 15z = 3 (B) 14x + 2y – 15z = 1
(C) 14x + 2y + 15z = 31 (D) 14x – 2y + 15z = 27
[JEE (Advanced) 2017, 3]

109
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.76 Let O be the origin and let PQR be an arbitrary triangle. The point S is such that
OP · OQ  OR · OS = OR · OP  OQ · OS = OQ · OR  OP · OS .
Then the triangle PQR has S as its
(A) circumcentre (B) orthocenter (C) incentre (D) centroid
[JEE (Advanced) 2017, 3]

Paragraph for question 77 & 78

Let O be the origin, and OX , OY , OZ be three unit vectors in the directions of the sides QR , RP ,

PQ , respectively, of a triangle PQR.

Q.77 OX OY = [JEE (Advanced) 2017, 3 ]

(A) sin (P + R) (B) sin 2R (C) sin (Q + R) (D) sin (P + Q)

Q.78 If the triangle PQR varies, then the minimum value of cos (P + Q) + cos (Q + R) + cos (R + P) is
3 5 3 5
(A) (B) (C) (D)
2 3 2 3
[JEE (Advanced) 2017, 3]

110
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[RANK BOOSTER]
VECTOR
[ SINGLE CORRECT CHOICE T Y PE]

           
Q.1 Four vectors a , b, c and x satisfy the relation (a · x ) b  c  x where b · a  1 . The value of x in terms
  
of a , b and c is equal to
      
c
(a · c )b  c(a · b  1)
(A)   (B)  
(a · b  1) a · b 1
       
2( a · c ) b  c 2( a · c ) c  c
(C)   (D)  
a · b 1 a · b 1
       
Q.2 If a and b are two non-zero vectors, then the value of scalar [(a  b)  a ] · [( b  a )  b] equals
             
(A) | b | 2 | a  b | 2 (B)  (a · b) | a  b | 2 (C) a 2 | a  b | 2 (D) (a · b) | a  b | 2
        
Q.3 a , b, c are three coplanar unit vectors such that a  b  c = 0. If three vectors p, q, r parallel to
  
a , b, c respectively and having integral but different magnitudes, then among the following options
  
p  q  r can take a value equal to

(A) 0 (B) 1 (C) 3 (D) 2

     
Q.4 If the acute angle that the vector  i   j  k makes with the plane of the two vectors 2 i  3 j  k and
  
i  j  2k is cot–1 2 , then
(A) ( + ) =  (B) ( + ) = 
(C) ( + ) =  (D) +  +  = 0

[ PARAGRAPH T Y PE]
Paragraph for question nos. 5 to 7
Let OABC be a regular tetrahedron with side length unity. Angle between skew lines OA & BC be ,
angle between skew lines OB & CA be  and angle between skew lines OC & AB be .
Q.5 sin2  + sin2  + sin2  is equal to
3 9
(A) 0 (B) (C) (D) 3
4 4
Q.6 Volume of tetrahedron is equal to
1 1 1 1
(A) (B) (C) (D)
2 3 2 6 2 12
Q.7 Let foot of perpendicular from vertex O on plane face ABC be N then AN is equal to
1 2 2 1
(A) (B) (C) (D)
3 3 3 6

111
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Paragraph for questions nos. 8 to 11

  
Suppose the three vectors a , b, c on a plane satisfy the condition that [3011712448]
        
| a |  | b |  | c | = | a  b | = 1; c is perpendicular to a and b · c > 0, then
  
Q.8 The angle formed by 2a  b and b is equal to
  
(A) (B) (C) (D) 0
4 3 2
  
Q.9 If the vector c is expressed as a linear combination a  b then the ordered pair (, ) is equal to

 1 1   1 2   2 1   2 2 
(A)  ,  (B)  ,  (C)  ,  (D)  , 
 3 3  3 3  3 3  3 3

      
Q.10 For real numbers x, y the vector p  xa  yc satisfies the condition 0  p · a  1 and 0  p · b  1. The
 
maximum value of p · c is equal to
1
(A) 1 (B) 3 (C) 3 (D)
3
  
Q.11 For the maximum value of x and y, the linear combination of p in terms of a and b is equal to
       
(A) a  2b (B) 2a  b (C) 2 (a  b) (D) 3a  2b

[ M ULT IPL E CORRECT CHOICE T YPE]

    
Q.12 If a , b are non-parallel unit vectors and c is a vector in a plane perpendicular to a and b such that
  2  
c = 5 and a  b = a  b then

  3   3  
(A) a · b = (B) a · b = (C) [a b c] = – 4 (D) [a b c] = 4
5 5

  3 cos  sin 
Q.13 If a = 3 sin  î  cos  ˆj and b = î  ˆj represents diagonals of a parallelogram
3  2 cos 2 3  2 cos 2
3
such that area of the parallelogram is less than or equal to then
10
  3  
(A) a · b = (B) a · b = 0
5
    9     9
(C) [a b a  b] = (D) [a b a  b] =
100 25

112
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ MAT RIX T YPE]

Q.14 Let â , b̂ and ĉ are three non coplanar unit vectors mutually inclined to each other at an angle . d̂ is
also a unit vector such that d̂ · â  d̂ · b̂  d̂ · ĉ  cos  .
Match the entries for  given in column-I with their corresponding values for  given in column-II.
Column-I Column-II
() ()
 
(A) (P)
2 4
  1 
(B) (Q) cos–1  
3  3
1 2
(C) cos–1   (R) cos1
 4 3
5 
(D) cos–1   (S)
8 6
[3011712279]

     
Q.15 Let a and b be unit vectors and angle between them is . If a · b and a  b are the two roots of

 1 1
the cubic equation x3 –  
2  1 x2 +  2   x – = 0, then
 2 2
List-I List-II
1 1
(A)   2   2 is equal to (P) 0
(a · b ) ab
(B) if  is the area of the triangle whose sides are represented by (Q) 1
  1
a & b and 2 is the angle between them, then is equal to

(C) third root of the equation is (R) 2
 
| a b |
2
(D)  sin

x dx is equal to (S) 4
a·b

[ INT EGER
T YPE]
  
Q.16 Let a 3 dimensional vector V satisfies the condition, 2V  V  ( î  2 ĵ) = 2î  k̂ .

If 3 V = m where m  N, then find m. [3011711674]

   
Q.17 Let a  x 2 î  3 ĵ  ( x  3) k̂ and b  î  3 ĵ  ( x  3) k̂ be two vectors such that | a | = | b | .

   
3
   
 v1 

If v1  2a  3b & v 2  3a  2b and I = v 2 dx , then find the value of [I].
1
[Note: [k] denotes greatest integer function less than or equal to k.]

113
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

3 - D I M E N S I O N A L G E O M E T RY
[ SINGLE CORRECT CHOICE T Y PE]

Q.1 If acute angle between the line r = î  2 ĵ   ( 4î  3k̂ ) and xy plane is  and acute angle between the
planes x + 2y = 0 and 2x + y = 0 is  then (cos2 + sin2) equals
1 2 3
(A) 1 (B) (C) (D)
4 3 4

Q.2 The point of intersection of the plane r · (3î  5ˆj  2k̂ )  6 with the straight line passing through the
origin and perpendicular to the plane 2x – y – z = 4, is (x0, y0, z0). The value of (2x0 – 3y0 + z0), is
(A) 0 (B) 2 (C) 3 (D) 4

Q.3 Let  be the plane containing the line x + y – z – 1 = 0 = x + 4y + 3z and parallel to the line
6(x – 1) = 3y = 2(z + 1). The perpendicular distance of the plane from origin is
1 1 1
(A) 1 (B) (C) (D)
2 3 6

Q.4 Let the equation of the two planes P1 and P2 are


P1 : r · (3î  ĵ  2k̂ ) = 4 and

P2 : r · ( î  ˆj  k̂ ) = 2
If the equation of the plane containing the line of intersection of the planes P1 and P2 and also containing
the point with p.v. ( 2î  2ˆj  k̂ ) is ax + by + 2z = c, then the value of (a + b + c) is
(A) 5 (B) 7 (C) 11 (D) 13

[ PARAGRAPH T Y PE]
Paragraph for question nos. 5 to 7
  
Let r1  a 2 î  2b ˆj  6 k̂ and r2  2a î  a ĵ  a 2 k̂ be non-zero vectors and r3  î  2ˆj  a k̂ be a
position vector of a point where a, b  R. P1 and P2 are two planes for the largest integral value of b,
  
which containing vector r1 and r2 is the normal vector of both the planes and passing through point r3 .

Q.5 If  is the acute angle between the planes then cos  is

7 7
(A) 0 (B) (C) (D) 1
3 6 18

Q.6 If d1 and d2 are perpendicular distances of the planes from the origin then (d1 + d2) equals
5 7 3
(A) 0 (B) (C) (D)
6 6 6 6

Q.7 Equation of the plane containing the line P1 = 0 = P2 and passing through the origin is
(A) 2x – y + 2z = 0 (B) 2x + y – z = 0 (C) 2x + y – 2z = 0 (D) x + y – 2z = 0

114
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

[ M ULT IPL E CORRECT CHOICE T YPE]

x 1 y  2 z  3 x  1 y  2 3( z  3) x 1 y  2 z  3
Q.8 Let L1 :   , L2 :   and L3 :  
2 1 3 1 3 5  32  19 15
be three lines. A plane intersecting these lines at A, B and C respectively such that PA = 2, PB = 3
and PC = 6 where P  (1, 2, 3). If V is the volume of the tetrahedron PABC and d is the
perpendicular distance of the plane from the point P then-
(A) V = 18 cubic units (B) V = 6 cubic units
6
(C) d = units (D) d = 7 units
14
x 1 y  3 z 1
Q.9 Consider a plane P containing the line L : = = . If Q(1, 10, 1) be a point in space then
1 2 3
(A) Equation of plane P located at a maximum distance from Q is x – 5y + 3z + 11 = 0.
(B) Equation of plane P located at a minimum distance from Q is 3x – z = 2.
(C) Image of Q with respect to L is (3, 0, 7).
(D) Orthogonal projection of Q in L is (2, 5, 4).

x2 y 1 z  1
Q.10 The line = = intersects the curve x2 – y2 = a2, z = 0 if a is equal to
3 2 1
(A) 4 (B) 5 (C) –4 (D)  5

[ MAT RIX T YPE]

Q.11 List I List II
(A) Given four points A(2, 1, 0), B(1, 0, 1), C(3, 0, 1) and D(0, 0, 2). (P) 9
Point D lies on a line L orthogonal to the plane determined by
points A, B, C. If point of intersection of plane ABC and line L is
(x0, y0, z0), then (7x0 + 2y0 + 8z0) is equal to
   
(B) If volume of parallelopiped formed by vectors a  b , b  c and (Q) 10
 
c  a is 25 square units, then the volume of parallelopiped formed
     
by vectors a  b , b  c and c  a is equal to
3
(C) A variable plane at a distance of unit from the origin cuts the (R) 11
2
coordinate axes at P, Q and R.If the centroid G(u, v, w) of triangle
PQR satisfies the relation u–2 + v–2 + w–2 = , then  is equal to
 
(D) Let  =  î  ĵ  2k̂ and  =  î  2 ĵ  k̂ be two vectors. (S) 12
 
The area of parallelogram having diagonals 3  and 2 is equal to
Codes :
(A) (B) (C) (D)
(A) R Q P S
(B) R Q S P
(C) Q R S P
(D) Q R P S

115
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

x 1 y  2 z  3 x 2 y 3 z 4
Q.12 Let L1 : = = and L2 : = = be two lines.
2 3 4 3 4 6
List-I List-II
(A) If plane containing line L1 and parallel to line (P) 3
L2 is ax + by – z + d = 0, then (a + b + d) equals
(B) If  is the shortest distance between both the lines (Q) 2
1
then 2 equals

1
(C) If P1 is the plane containing L1 and parallel to L2 and P2 (R)
5
is the plane containing L2 and parallel to L1 , then distance
between both the planes, is
(D) If vector along the shortest distance of both the lines is (S) 5
pî  q ˆj  k̂ , then (p + q) equals
Code :
(A) (B) (C) (D)
(A) Q P S R
(B) P S P Q
(C) Q S R P
(D) P S R Q

[ INT EGER T YPE]

Q.13 Consider a plane passing through three points A(a, 0, 0), B(0, b, 0), C(0, 0, c) with a > 0, b > 0, c > 0.
Let d be the distance between the origin O and the plane and m be the distance between the
origin O and the point M(a, b, c). If a, b, c vary in the range of any positive numbers, then find the
2
m
minimum value of   . [3011711667]
d
x y z
Q.14 Line L meets lines L1 :   and L2 : x  2  y  1  z  4 orthogonally at points P and Q.
1 2 3 2 4 5
5D
(PQ)2 is D. DRs of line L are (a, b, c) {a, b, c  I}, then least value of  a  b  c .
3
Q.15 Let equation of plane be x + 2y + z – 3 = 0. An insect starts flying from point P(1, 3, 2) in straight line.
It touches the plane at point R(a, b, c) and then goes to point Q(3, 5, 2) in straight line. If distance
travelled PR + QR is minimum then find the value of (a + b + c).
Q.16 Let the lines

L1 : r = (7 î + 6 ˆj + 2 k̂ ) + (–3 î + 2 ˆj + 4 k̂ )

L2 : r = (5 î + 3 ˆj + 4 k̂ ) + (2 î + ˆj + 3 k̂ )

be intersected by a line parallel to the vector 2 î – 2 ˆj – k̂ at P, Q respectively, find | PQ | .

116
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

ANSWER KEY
EXERCISE-1
SECTION-A
5  1 2 27
Q.1  Q.3 cos–1    Q.4 Q.5
2  3 3 2
 1
Q.7 5î  10ˆj  10k̂ Q.8
1
  Q.9
4

d  î  ˆj  k̂ 
 2
1 1 1 2 2 1
Q.10 0 Q.11 ± ,± ,± Q.12  î  ˆj  k̂
3 3 3 3 3 3
 
Q.13 1 Q.14 
p  2 32î  ĵ  14k̂  Q.15 4 Q.16 5b
1
Q.17 p = – Q.18 –10
3
Q.19 Show that the four points A, B, C and D with position vectors 4iˆ  5jˆ  k,
ˆ ˆj  k,3
ˆ ˆi  9 ˆj 4 kˆ and

4( ˆi  ˆj  k)
ˆ respectively are coplanar..

Q.20  =1 Q.21 
 ˆi  11jˆ  7kˆ  Q.22  = – 9, µ = 27

1 1
Q.23
7

6î  3 ĵ  2k̂  Q.24
2
210 sq. unit Q.26 19 2

11 ˆ
Q.27 3î  j  5k̂ Q.28 1 Q.29 5 2
3
 21
Q.30 = Q.31 15 2 sq. units
2

Q.33 Show that the pointsA ( 2î  3 ĵ  5k̂ ) , B ( î  2ˆj  3k̂ ) and C (7î  k̂ ) are collinear..
   1 
Q.34 If a and b are unit vectors and  is the angle between them, then prove that cos = | a  b |.
2 2
4
Q.35 91 cubic units Q.36 7 ( î  ĵ  k̂ ) Q.37
5

Q.38 x = 2, y = 1 Q.39 0 Q.40 ± 3( î  2ˆj  2k̂ )

 8
Q.41 y = 3 or y = – 1 Q.42 75 Q.43 sin 1  
 21 
3 2 8  19  
Q.44 , , Q.45 cos 1  Q.46 A  (5î  4 ĵ  6k̂ )   (3î  7 ĵ  2k̂ )
77 77 77  21 

117
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

 13 23  
Q.47  , , 0
5 5 
Q.48 
r  3î  4 ĵ  3k̂    5î  7ˆj  2k̂  Q.49 90º

5 7
Q.50 Q.52 –1 Q.53 7x – 5y + 4z – 8 = 0 Q.54 1,
2 3

Q.55 r .( 20î  23 ĵ  26k̂ )  69 Q.57 17

3 2x 2 y z  3 
Q.58 (i) ; (ii)   = 1; (iii)  0, , 0  ; (iv) x = 2t + 2 ; y = 2t + 1 and z = – t + 3
2 3 3 3  2 

SECTION-B
VECTOR
Q.1 A Q.2 D Q.3 A Q.4 C Q.5 B
Q.6 A Q.7 B Q.8 D Q.9 A Q.10 B
Q.11 A Q.12 D Q.13 C Q.14 C Q.15 C
Q.16 D Q.17 C Q.18 C Q.19 A Q.20 C
Q.21 C Q.22 A Q.23 A Q.24 D Q.25 C

3-DIMEN SIONAL G E O M E T RY
Q.1 C Q.2 A Q.3 B Q.4 B Q.5 C
Q.6 D Q.7 D Q.8 B Q.9 B Q.10 B
Q.11 C Q.12 B Q.13 D Q.14 C Q.15 B
Q.16 A Q.17 A Q.18 A Q.19 B Q.20 A

EXERCISE-2
VECTOR
Q.1 A Q.2 B Q.3 C Q.4 D Q.5 D
Q.6 B Q.7 D Q.8 C Q.9 C Q.10 B
Q.11 A Q.12 B Q.13 C Q.14 A Q.15 A
Q.16 D Q.17 ABD Q.18 ABC Q.19 ABD
Q.20 (A) T; (B) U ; (C) P ; (D) R ; (E) Q; (F) S; (G) W; (H) V Q.21 (A) S; (B) P; (C) R; (D) Q
Q.22 (A) Q; (B) S; (C) R; (D) Q; (E) P Q.23 35 Q.24 5 Q.25 34
Q.26 48 Q.27 5 Q.28 51 Q.29 13 Q.30 1125

( 1) n 
Q.31 488 Q.32 13 Q.33  n   , n  I &  1
2
1
Q.34  (î  5 ĵ  k̂ )
3 3

118
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

3-DIMEN SIONAL G E O M E T RY
Q.1 A Q.2 D Q.3 D Q.4 D Q.5 B
Q.6 A Q.7 A Q.8 A Q.9 D Q.10 B
Q.11 C Q.12 A Q.13 C Q.14 B Q.15 C
Q.16 D Q.17 D Q.18 ABC Q.19 BD Q.20 AC
Q.21 (A) R ; (B) Q, (C) Q, S, (D) P, S Q.22 240 Q.23 7 Q.24 23
Q.25 12 Q.26 7/5 Q.27 x – 8y + 7z = 2

x 7 y2 z 1 2
Q.28 2x + 3y + z + 4 = 0 Q.29   Q.30
22 5 4 3
EXERCISE-3

Q.1 C Q.2 C Q.3 D Q.4 B Q.5 D

Q.6 A Q.7 B Q.8 A Q.9 A Q.10 C
Q.11 A Q.12 A Q.13 D Q.14 A Q.15 B
Q.16 A Q.17 D Q.18 C Q.19 A Q.20 B
Q.21 B Q.22 B Q.23 B Q.24 A Q.25 B
Q.26 C Q.27 A Q.28 B Q.29 C Q.30 A
Q.31 A Q.32 B Q.33 B Q.34 B

EXERCISE-4
  
Q.2 v 1  2i , v 2   i  j, v 3  3i  2j  4k Q.3 B Q.4 C

Q.6 B Q.7 C Q.9 D Q.10 x + y – 2z = 3

Q.11 (6, 5, –2) Q.13 B Q.14 B Q.15 A Q.17 9/2
Q.18 D Q.19 2x – y + z – 3 = 0 and 62x + 29y + 19z – 105 = 0

Q.20 ŵ = v̂ – 2( â · v̂ ) â Q.21 D Q.22 A Q.23 B

Q.24 (A) Q, R; (B) P, (C) S, (D) P Q.25 (A) Q, (B) Q, (C) R, (D) P Q.26 C
Q.27 B Q.28 C Q.29 D Q.30 (A) R; (B) Q; (C) P; (D) S
Q.31 A Q.32 AC Q.33 D Q.34 B Q.35 D
Q.36 C Q.37 A Q.38 C Q.39 C
Q.40 (A) Q, S ; (B) P, R, S, T ; (C) T ; (D) R Q.41 7 Q.42 C
Q.43 B Q.44 A Q.45 5 Q.46 6

119
VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.47 (A) T (B) P, R (C) Q (D) R Q.48 C Q.49 AD Q.50 9

Q.51 A Q.52 A Q.53 C Q.54 C Q.55 3
Q.56 D Q.57 C Q.58 D Q.59 A Q.60 5
Q.61 (A) R ; (B) S ; (C) P ; (D) Q Q.62 (A) R :;(B) Q ; (C) S ; (D) P Q.63 C
Q.64 ABC Q.65 4 Q.66 BD Q.67 AB Q.68 ACD
Q.69 (A) P, R, S ; (B) P ; (C) P, Q ; (D) S, T Q.70 BONUS Q.71 AB
Q.72 BCD Q.73 C Q.74 BC Q.75 C Q.76 B
Q.77 D Q.78 A

EXERCISE-5
VECTOR
Q.1 A Q.2 B Q.3 C Q.4 A Q.5 D
Q.6 C Q.7 A Q.8 C Q.9 B Q.10 B
Q.11 C Q.12 BCD Q.13 BD
Q.14 (A) Q ; (B) R, S ; (C) R, S ; (D) P Q.15 (A) S ; (B) R ; (C) Q ; (D) P Q.16 6
Q.17 3

3 - D I M E N S I O N A L G E O M E T RY
Q.1 A Q.2 D Q.3 C Q.4 A Q.5 B
Q.6 D Q.7 A Q.8 BC Q.9 ABCD Q.10 AC
Q.11 (A) R ; (B) Q ; (C) S ; (D) P Q.12 (A) ; P (B) ; S (C) ; R (D) ; Q Q.13 9
Q.14 6 Q.15 2 Q.16 9

120

As-17 - Vectors and 3D - Math Assignment - JEE Dropper AIR - Mahesh Jain Sir - Prarit
No ratings yet
As-17 - Vectors and 3D - Math Assignment - JEE Dropper AIR - Mahesh Jain Sir - Prarit
26 pages
Vector Algebra SSSVV 2324
No ratings yet
Vector Algebra SSSVV 2324
4 pages
Vector-03 - Exercise 1
No ratings yet
Vector-03 - Exercise 1
20 pages
Categorised Questions - Vector-3-D NJ - 247
No ratings yet
Categorised Questions - Vector-3-D NJ - 247
18 pages
Chapter 07
No ratings yet
Chapter 07
4 pages
Vector Objective Questions Exercise
100% (1)
Vector Objective Questions Exercise
5 pages
JEE Main Vector Algebra Important Questions (2022)
No ratings yet
JEE Main Vector Algebra Important Questions (2022)
22 pages
XII Math Vector Algebra Questions
No ratings yet
XII Math Vector Algebra Questions
7 pages
15 XII-M1-05 Vectors - Solutions - 64dccc28abbdb
No ratings yet
15 XII-M1-05 Vectors - Solutions - 64dccc28abbdb
45 pages
JEE Advanced Vector Algebra Important Questions
No ratings yet
JEE Advanced Vector Algebra Important Questions
19 pages
Kp9ozl24
No ratings yet
Kp9ozl24
10 pages
PDF 1715404421
No ratings yet
PDF 1715404421
3 pages
Vectors and 3D Questions
No ratings yet
Vectors and 3D Questions
2 pages
Vectors
No ratings yet
Vectors
5 pages
Question 1663827
No ratings yet
Question 1663827
7 pages
Vector
No ratings yet
Vector
22 pages
Revision Test 7
No ratings yet
Revision Test 7
4 pages
Questionbank Xii Mathematics-1
0% (1)
Questionbank Xii Mathematics-1
10 pages
X9 G 0 e 9 NC 8 MUexkzzgj 5
No ratings yet
X9 G 0 e 9 NC 8 MUexkzzgj 5
37 pages
Vector - 3D - One Page
No ratings yet
Vector - 3D - One Page
270 pages
Vector-04 - Exercise
100% (1)
Vector-04 - Exercise
28 pages
Chapter 5
No ratings yet
Chapter 5
5 pages
12TH Impulse Mathematics Test (Vector - 1) (Answerkey) Date - 22-11-2024
No ratings yet
12TH Impulse Mathematics Test (Vector - 1) (Answerkey) Date - 22-11-2024
3 pages
Vector and 3-d by Mohit Tyagi Sit
No ratings yet
Vector and 3-d by Mohit Tyagi Sit
24 pages
Entrance (2020-21) - Mathematics - (Practice - MCQS) : Vectors, 3-Di, Line & Plane
No ratings yet
Entrance (2020-21) - Mathematics - (Practice - MCQS) : Vectors, 3-Di, Line & Plane
10 pages
(22285) Sheet 5 Vector and 3d B
No ratings yet
(22285) Sheet 5 Vector and 3d B
75 pages
Untitled 1
No ratings yet
Untitled 1
37 pages
Vectors
No ratings yet
Vectors
6 pages
Vectors 1
No ratings yet
Vectors 1
18 pages
Vectors PDF
No ratings yet
Vectors PDF
12 pages
Vectors 1
No ratings yet
Vectors 1
4 pages
Vectors Work Sheets
No ratings yet
Vectors Work Sheets
5 pages
15 XII-M1-05 Vectors - Solution - 1 - 668b962925eb2
No ratings yet
15 XII-M1-05 Vectors - Solution - 1 - 668b962925eb2
48 pages
Vectors and 3D Question Bank
No ratings yet
Vectors and 3D Question Bank
7 pages
Class 12th Maths Chapter 10 (Vector Algebra)
No ratings yet
Class 12th Maths Chapter 10 (Vector Algebra)
7 pages
Class 12th Maths Chapter 10 (Vector Algebra) Unsolved PDF
No ratings yet
Class 12th Maths Chapter 10 (Vector Algebra) Unsolved PDF
7 pages
Vector 3-D by Om Sir PDF
50% (2)
Vector 3-D by Om Sir PDF
38 pages
Vectors
No ratings yet
Vectors
7 pages
Vectors Dot Product
No ratings yet
Vectors Dot Product
2 pages
Vectors MCQ Questions and Answers
No ratings yet
Vectors MCQ Questions and Answers
13 pages
Vector
100% (1)
Vector
29 pages
Hsslive-Xii-Maths-Qb-2023-10 VECTORS
No ratings yet
Hsslive-Xii-Maths-Qb-2023-10 VECTORS
7 pages
Junior Maths Vectors Guide
No ratings yet
Junior Maths Vectors Guide
11 pages
Question 1815785
No ratings yet
Question 1815785
3 pages
Vectors & 3D Geometry Problems
No ratings yet
Vectors & 3D Geometry Problems
7 pages
Chennai Public School Chapter10: Vector Algebra Class Xii
100% (1)
Chennai Public School Chapter10: Vector Algebra Class Xii
2 pages
Class 12 Vectors and 3D Geometry Exam
No ratings yet
Class 12 Vectors and 3D Geometry Exam
8 pages
2 Mark Type (Vectors)
No ratings yet
2 Mark Type (Vectors)
6 pages
Vector Test 2
No ratings yet
Vector Test 2
2 pages
ATS 10 Vector Algebra
No ratings yet
ATS 10 Vector Algebra
2 pages
DPP 5 (Vectors)
No ratings yet
DPP 5 (Vectors)
2 pages
Class 12th Maths Chapter 10 (Vector Algebra) Unsolved
No ratings yet
Class 12th Maths Chapter 10 (Vector Algebra) Unsolved
7 pages
CPP Vector
100% (1)
CPP Vector
14 pages
12 Maths Vec - 3D
No ratings yet
12 Maths Vec - 3D
2 pages
Xii Vector SPM
No ratings yet
Xii Vector SPM
1 page
Product Presentation GreenSpace Surface-Mounted BC201605
No ratings yet
Product Presentation GreenSpace Surface-Mounted BC201605
13 pages
HVAC
No ratings yet
HVAC
12 pages
Quantum Computing and Cryptography Today PDF
No ratings yet
Quantum Computing and Cryptography Today PDF
22 pages
Characteristics and Nature of Matter WAEC Standard
100% (1)
Characteristics and Nature of Matter WAEC Standard
3 pages
Ds-Tl2002ai 20191231
No ratings yet
Ds-Tl2002ai 20191231
3 pages
The Erotic Codex of DR Lyra Nocturne - Maths Papers - Gallery
No ratings yet
The Erotic Codex of DR Lyra Nocturne - Maths Papers - Gallery
81 pages
CHERRY MX Low Profile RGB Switch
No ratings yet
CHERRY MX Low Profile RGB Switch
1 page
1.1 Solids Liquids and Gases QP
No ratings yet
1.1 Solids Liquids and Gases QP
7 pages
SOAL PTS II B.ING KELAS 5 New
No ratings yet
SOAL PTS II B.ING KELAS 5 New
4 pages
Class Xi Unit Test-1 2022-23
No ratings yet
Class Xi Unit Test-1 2022-23
3 pages
General Science Grade 9
No ratings yet
General Science Grade 9
8 pages
What Are Conductors and Insulators?
No ratings yet
What Are Conductors and Insulators?
14 pages
Structural Beam Specifications
No ratings yet
Structural Beam Specifications
1 page
Types of Electrical Relays
No ratings yet
Types of Electrical Relays
17 pages
4 Power Transmission
No ratings yet
4 Power Transmission
27 pages
Harmonic 1-Forms On Conformally at Riemannian Manifolds: Research Open Access
No ratings yet
Harmonic 1-Forms On Conformally at Riemannian Manifolds: Research Open Access
13 pages
Stanley Meyer Water Fuel Patents
No ratings yet
Stanley Meyer Water Fuel Patents
442 pages
UPL12500W Ver.18
No ratings yet
UPL12500W Ver.18
2 pages
Construction Materials Cost List
No ratings yet
Construction Materials Cost List
2 pages
38KUT - 42KTD - KZL - KSM Service Manual Carrier
100% (1)
38KUT - 42KTD - KZL - KSM Service Manual Carrier
99 pages
00 - TES - SD - 005.01 Typical Data Sheets (Water)
No ratings yet
00 - TES - SD - 005.01 Typical Data Sheets (Water)
286 pages
Exam Study Materials Blaster Part 4.explosives Handbook
No ratings yet
Exam Study Materials Blaster Part 4.explosives Handbook
52 pages
Hge 8
No ratings yet
Hge 8
2 pages
Outdoor Cooling in Qatar Stadiums
100% (1)
Outdoor Cooling in Qatar Stadiums
69 pages
Scientific Review: EM Radiation Effects
No ratings yet
Scientific Review: EM Radiation Effects
14 pages
Math 202B Solutions: Assignment 7 D. Sarason
No ratings yet
Math 202B Solutions: Assignment 7 D. Sarason
2 pages
Design Assumptions and Specifications: Max B
No ratings yet
Design Assumptions and Specifications: Max B
4 pages
Math G10 (PT-3)
No ratings yet
Math G10 (PT-3)
4 pages
Subjective Mock Test 02 - Test Paper (Physics) - Hope (Backlog Series)
No ratings yet
Subjective Mock Test 02 - Test Paper (Physics) - Hope (Backlog Series)
2 pages
IC Manufacturing
No ratings yet
IC Manufacturing
54 pages

Vect&3d 1

Uploaded by

Vect&3d 1

Uploaded by

VE CT OR & 3 -DIM ENS IONA L GE OMET RY

Q.10 Find the projection of the vector î  ˆj on the vector î  ˆj .

Q.22 Find  and  if

Q.28 Write thevalue of î.(ˆj  k̂ )  ĵ.( k̂  î )  k̂.( ĵ  î ) .

Q.40 Determine vector of magnitude 9 which is perpendicular to both the vectors:

THREE DIMENSION GEOMETRY

Q.45 Find theanglebetween the pair of linesgiven by

Q.46 Writethevector equation of thefollowing line.

Q.49 Find the angle between the lines 2x = 3y = –z and 6x = – y = –4z.

Q.58 Let P = (1, 0, – 1) ; Q = (1, 1, 1) and R = (2, 1, 3) are three points.

SECTION-B [JEE (MAIN)]

Q.5 If E is the intersection point of diagonals of parallelogram ABCD and

(A) 2 (B) 7 (C) 14 (D) 14 [3011712154]

(A) 0 (B) 1 [3011712103]

The area of the shaded region in the adjacent figure, is D

(A) 2 19 (B) 4 19 (C) 2 38 19 (D) none [3011712163]

(A) [ a b c] (B) 2[ a b c] (C) 3[ a b c] (D) 0

(C) x2 + y2 + z2 = 4K2 (D) x–2 + y–2 + z–2 = 4K–2

(A) P2 & P3 (B) P2 & P4 (C) P1 & P3 (D) P1 & P4

Q.8 Given planes

Q.13 Image point of (1, 3, 4) in the plane 2x – y + z + 3 = 0 is - [3011712187]

Q.14 The graph of the equation x2 + y2 = 0 in three dimensional space is - [3011712136]

(A) c > 0 (B) c =  3 (C) 0 < c < 1 (D) c > 2

Paragraph for question nos. 12 to 14

[MULTIPLE CORRECT CHOICE TYPE]

[ MAT RIX T YPE]

Q.21 Column I Column II

Q.22 Column-I Column-II

[ INT EGER T YPE]

OP . î = 2 and OQ . î = – 2 and the magnitude of the vector OP  OQ = 2M, (where O is origin)

[ SUBJECT IVE T YPE]

Paragraph for question nos. 11 to 13

Q.12 Image of AB in plane P is

Paragraph for question nos. 14 to 16

[ M ULT IPL E CORRECT CHOICE T YPE]

[ MAT RIX T YPE]

[JEE(MAIN) PREVIOUS YEAR]

Q.22 Distance between two parallel planes

[JEE(ADVANCED) PREVIOUS YEAR]

(A) –1 (B) 10  6 (C) 59 (D) 60

[ JEE 2005 (Mains)]

Q.24 Match the following

Q.25 Match the following [3011711436]

MATCH THE COLUMN:

Q.33 Consider three planes

Paragraph for Question Nos. 34 to 36

 î  7ˆj  7 k̂  î  7ˆj  5k̂  î  7 ĵ  5k̂ 7 î  7 ĵ  k̂

Q.35 The shortest distance between L1 and L2 is

Q.43 Two adjacent sides of a parallelogram ABCD are given by [3011711822]

(C) Area of a triangle with adjacent sides determined by vectors (R) 24

x 1 y z3 x 4 y3 z3

Q.69 Column-I Column-II

Paragraph for question 77 & 78

PQ , respectively, of a triangle PQR.

Q.77 OX OY = [JEE (Advanced) 2017, 3 ]

(A) 0 (B) 1 (C) 3 (D) 2

Paragraph for questions nos. 8 to 11

[ M ULT IPL E CORRECT CHOICE T YPE]

[ MAT RIX T YPE]

Q.4 Let the equation of the two planes P1 and P2 are

Q.5 If  is the acute angle between the planes then cos  is

[ M ULT IPL E CORRECT CHOICE T YPE]

[ MAT RIX T YPE]

[ INT EGER T YPE]

be intersected by a line parallel to the vector 2 î – 2 ˆj – k̂ at P, Q respectively, find | PQ | .

Q.38 x = 2, y = 1 Q.39 0 Q.40 ± 3( î  2ˆj  2k̂ )

Q.1 C Q.2 C Q.3 D Q.4 B Q.5 D

Q.6 B Q.7 C Q.9 D Q.10 x + y – 2z = 3

Q.20 ŵ = v̂ – 2( â · v̂ ) â Q.21 D Q.22 A Q.23 B

Q.47 (A) T (B) P, R (C) Q (D) R Q.48 C Q.49 AD Q.50 9

You might also like