# VECTOR by SonuSarraf

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VECTOR
By:- Nishant Gupta
For any help contact:
9953168795, 9268789880
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VECTOR

  
1. The necessary and sufficient condition for three points with position vectors a, b, c to be collinear is that
      
there exist scalars x, y, z not all zero such that xa  yb  zc  0 where x + y + z = 0.
     
2. If A and B are two points with position vectors a and b respectively, then the position vector of a point
C dividing AB in the ratio m : n
                                        
mb  na                                    mb  na
(i) Internally is,                        (ii) Externally is,
mn                                       mn

3. If S is any point in plane of  ABC, then SA  SB  SC  3SG , where G is the centroid of ΔABC.
       
4. If a and b are two non zero vectors inclined at an angle θ, then
 
                                                                 ab  ˆ
(i) a  b | a | | b | cos θ                    (ii) Projection of a on b =   a  b
|b|
         
     a  b 
      ˆ  a  b 
       
(iii) Projection vector of a on b =   b    b
 |b| 
              2
| b | 

    2  2         2        
(iv) | a  b | | a |  | b | 2 (a  b)
 
                                                       ab
(v) (a  b)  (a  b) | a | 2  | b | 2            (vi) cos θ =   
|a| |b|
                                                                                      
(vii) a  b  | a | | b | sin  n , where n is a unit vector perpendicular to the plane of a and b
ˆ
                 
                      a  b
(ix) Unit vectors perpendicular to the plane of a and b is                  ±  
|a  b|
 
                                             1              1              1 |ab|
(x) If a , b are unit vectors at an angle θ, then sin  | a  b | , cos  | a  b | , tan     
2 2               2 2               2 2 |ab|

1              1             1
5. Area of ΔABC =          | AB  AC |  | BC  BA |  | BC  CA |
2              2             2
                                                                 1      
6. If a, b, c are the PV the vertices A, B, C of ΔABC, then Area of ΔABC = | a  b  b  c  c  a |
2
     
| a b  bc  ca |
Length of the perpendicular from C on AB =                  
| a  b|

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85
Contact: 9953168795, 9268789880
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                        
7. Scalar triple product : (a  b)  c & is denoted by [a b c]
                                                                            
(i) [a b c] = volume of the parallelepiped whose three coterminous edges are a, b and c

(ii) [a b c] = 0 if and only if they are coplanar.
                                                      
(iii) [a  b b  c c  a | =2 [a b c]      ,        [a  b b  c c  a | = [a b c] 2
                                            
8. Vector triple product { a  (b  c) and (a  b)  c } a  (b  c)  (a . c) b - (a . b) c                     ( 132 – 123 )
                                   
Also, a  (b  c)  b  (c  a)  c  (a  b)  0
  
9. The volume V of a tetrahedron whose three coterminous edges in the right handed system are a, b, c is
1 
given by V = [a b c]
6
                       
10. (i) A set of non-zero vectors a 1 , a 2 , a 3 , . . . . . . , a n is linearly independent, if
                                        
x1a 1  x2 a 2  . . . . . .  x n a n  0  x1 = x2 =…….. = xn = 0
                           
(ii) A set of vectors a 1 , a 2 , a 3 , . . . . . . , a n is linearly dependent, if there exist scalars x1, x2, ………… xn not
                                      
all zero such that x 1a 1  x2 a 2  . . . . . .  x n a n  0

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85
Contact: 9953168795, 9268789880
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ASSIGNMENT
VECTOR

1.
 
 
 

(axb) 2  a.b
2
is
7.   A particle acted upon by forces 3 i  2 j 5 k
      

2a 2 b 2                                                                 
and 2 i  j 3 k is displaced form a point P to
(a) 1/2                  (b) 3/2
a point Q whose respective position vectors
(c) 5/2                  (d) 4/2                                                                
are 2 i  j 3 k and 4 i  3 j 7 k . The work
2.   The area of the triangle determined by the
vectors 3i + 4j and - 5i + 7j is                           done by the force is
(a) 141                  (b) 132                           (a) 77 units            (b) 24 units
(c) 41 /2                (d) N/T                           (c) 63 units            (d) 48 units
3.   Points     whose        position     vectors    are   8.   A force F = 6i + λ j + 4k acting on a particle
ˆ  3ˆ, 40ˆ  8ˆ, aˆ  52ˆ are collinear if              displaces it from A (3,4,5)to B (1,1,1). If the
60i j i j i                j
work done is 2 units, then λ is
(a) a = 40               (b) a = -40                       (a) -10                 (b) –2
(c) a = 20               (d) N/T                           (c) 5                   (d) 2.
4.   Adjacent sides of ||gm are along                      9.   Length of longer diagonal of llgm constructed
            
a  ˆ  2ˆ & b  2ˆ  ˆ. s between diagonals
i    j        i j                                                                          
on 5a  2b & a  3b . Given | b | 3 & | a | 2 2 &
(a) 30o & 150o           (b) 45o & 135o                                    
angle between a & b is π/4
(c) 90o & 90o         (d) N/T                              (a) 15                  (b) √113
                                 
5.    a & b are unit vectors such that a  3b is ┴              (c) √593                (d) √369
                          
to 7 a  5b , then angle between a & b is                             i j ˆ
10. The vector ˆ  xˆ  3k is rotated through an
(a) π/2                  (b) π / 3                        angle θ and doubled in magnitude, then it
(c) π /4                 (d) N/T                                                j ˆ
becomes 4ˆ  (4x  2)ˆ  2k . The value of x is:
i

6.                        ˆ    ˆ
If the unit vectors A and B are inclined at π              (a) 2 / 3,2          (b) 1 / 3,2
and | A ˆ - B |/2 is
ˆ                                              (c) 2 / 3,0           (d) (2, 7)
(a) 0                    (b) π/2                      11. If P and Q Be two given points on the curve y
(c) 1                    (d) π/ 4                         = x + 1/x such that OP.I = 1 and OQ.I = -1
where I is a unit vector along the x-axis, then
the length of vector 2OP + 3OQ is
(a) 5 5                 (b) 3 5

(c) 2 5                 (d)       5

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85
Contact: 9953168795, 9268789880
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                                 
12. Let A, B, C be three vectors such that A (B + C)    20. If  a  b  c  0, | a | 3 , | b | 5 & | c | 7 , then
+ B. (C + A) + C. (A + B) = 0 And |A| = 1,                               
|B| = 4 , |C| = 8 ,then |A + B + C| equals                θ between a & b is
(a) 13                (b) 81                             (a) a = 40                (b) a = -40
(c) 9                 (d) 5                              (c) a = 20             (d) N/T
  
ˆ      ˆ
13. If the unit vectors A and B are inclined at an      21. If 2 out of 3 vectors a, b, c are unit vectors,
                  
ˆ    ˆ
angle 2θ and | A - B |<l then for θ  [0,], θ           a  b  c  0 & 2( a.b  b.c  c.a ) + 3 = 0, then
may lie in the interval                                 third vector is of length-
(a) ( /6 , /3)      (b) ( /6 , /2 ]                  (a) 3                     (b) 2
(c) ( 5/6 ,  ]      (d) [/2 ,5/6 ]                   (c) 1                        (d) N/T
  
ˆ     ˆ
14. If unit vectors A and B such that STP [ Aˆ          22. Let a, b, c be 3 vectors such that
        
ˆ ˆ      ˆ            ˆ      ˆ
B A x B ] = 1/4 then A and B are inclined              a.(b  c)  b.(c  a)  c.(a  b)  0                  and
                                 
(a) π/6               (b) π/2                           | a | 1, | b | 4, | c | 8 then | a  b  c | equals
(c) π/3               (d) π/ 4                           (a) 13                    (b) 81
ˆ       ˆ
15. If A and B unit vectors then greatest value             (c) 9                    (d) N/T
ˆ ˆ         ˆ   ˆ                                                                             
of | A - B | + | A + B | is                         23. Let a  b is orthogonal to b & a  2b is

(a) 2                 (b) 4                             orthogonal to a , then
                                
(c) 2√2               (d) √2                            (a) | a | = √2 | b |     (b) | a | = 2 | b |
16. Let a,b,c be three vectors such that | a | = | c                                               
(c) | a | = | b |        (d) 2 | a | = | b |
|= l; |b |=4 and | b x c | = 15. If b-2c = λa                                                          
Then a value of λ is                                24. Magnitude of projection of vector i  2 j  k
(a) 1                 (b) - l                                                j ˆ
on vector 4ˆ  4ˆ  7k is
i
(c) 2                 (d) - 4
(a) 3                     (b) 3 6
17. Moment of couple formed by forces 5 ˆ  k &  i ˆ
-5 ˆ  k acting at ( 9 ,-1 , 2 ) & ( 3 , -2 , 1 )        (c)     6 /3              (d) N/T
i ˆ
25. Position vectors of          A & B are
i j ˆ
(a)  ˆ  ˆ  5k          i j ˆ
(b) ˆ  ˆ  5k
j ˆ           j ˆ
2ˆ  2ˆ  k & 2ˆ  4ˆ  4k . Length of internal
i             i
(c) 2ˆ  2ˆ  10k
i    j     ˆ (d)  2ˆ  2ˆ  10k
i    j     ˆ                    bisector of  BOA of  AOB is

18. Vector r which is equally inclined to co-                        136                       136
(a)                       (b)
ordinate axes such that | r | = 15 3 is                           9                        9

i j ˆ
(a) ˆ  ˆ  k                   
i j ˆ
(b) 15 ˆ  ˆ  k                  (c)
20
(d) N/T
3


(c) 7 ˆ  ˆ  k
i j ˆ          (d) None                                                               j ˆ
26. Magnitude of moment of force - 2ˆ  6ˆ  8k
i
                                                        i j ˆ
acting at point 2ˆ  ˆ  3k about point
19. For 3 vectors u, v, w, which of the following
j ˆ
ˆ  2ˆ  k
i
expressions is  to any of remaining three ?
                      
(a) u.(v  w)        (b) (v  w).u                       (a)     211               (b) 0
                        
(c) v.(u  w)        (d) (w x u ).v                      (c)     54                (d) N/T

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Contact: 9953168795, 9268789880
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(a) 2/3                    (b) 3/2
                       
27. If a  2b  3c  0, then axb  bxc  cxa is equal                    (c) 2                (d) N/T
                             
to                                                                          i j ˆ    b  4ˆ  3ˆ  4k , c  ˆ  ˆ  k
35. If a  ˆ  ˆ  k,    i j ˆ             i     j ˆ
 
 
(a) 6 b  c
 
(b) 6 a  b                                 are linearly dependent vectors & | c |  3

 
(c) 6 c  a           (d) N/Ts                                      then
(a)  = 1, β = - 1         (b)  =1, β =  1
ˆ ˆ
28. If a & b are unit vectors represented by
                                                                   (c)  = -1, β =  1        (d)  =  1, β = 1
OA & OB , then unit vector along bisector of
 AOB is scalar multiple of                                              
36. If a, b & c are 3 non-coplanar vectors such
ˆ ˆ
(a) a  b                        ˆ ˆ
(b) a  b                                                  
   bc
ˆ ˆ
(c) b  a                  (d) N/T                                  that a  (b  c)      , then angle between
2
29. Value

of       
    
a  b, b  c, c  a           where
 
a & b is
                   
| a | 1, | b | 2 &| c | 3 is                                       (a) 3π/4                   (b) π/4
(a) 1                        (b) 6                                   (c) π/2                (d) N/T

(c) 0               (d) N/T                                     37. Vector c directed along internal bisectors of
                                                                                     
30. If [ 2a  4b, c, d] = [a c d]  [b c d] , then                      between         vectors                 j ˆ
a  7ˆ  4ˆ  4k
i           &
                        
  =                                                                     i j ˆ
b   2ˆ  ˆ  2k with | c |  5 6 is
(a) 6                        (b) – 6                                       5 ˆ ˆ          ˆ                  5 ˆ ˆ ˆ
(a)     ( i  7 j  2k )          (b)     (5i  5 j  2k)
(c) 10                    (d) N/T                                           3                                 3
 ˆ ˆ                       
31. Let a  i  k,                                      ˆ
b  xˆ  ˆ  (1  x)k &
i j                                    5 ˆ ˆ ˆ
 ˆ ˆ                                                             (c)     (i  7 j  2k)     (d) N/T
ˆ
c  yi  xj  (1  x  y)k. Then [a b c] depends                            3
on                                                              38. If                        ˆ
( 2ˆ  6ˆ  27k )
i    j                       i j ˆ
x (ˆ  bˆ  ck)  0
(a) only x                   (b) only y                             then a & b are
(c) neither x nor y      (d) N/T                                     (a) 3 &27/2                (b) 3 & 25/2
              ˆ                          (c) -3 & 27/2             (d) N/T
j ˆ
32. Vector ˆ  (a  ˆ)  ˆ  (a  ˆ)  k  (c  k) is
i ˆ i j
 ˆ2  ˆ2                ˆ
(a) 0                     (b) a
                                 39. Value of | a  i |  | a  j | + | a  k |2 is
                                                               (a) a2                     (b) 2a2
(c) 2 a                   (d) N/T
 ˆ ˆ ˆ  ˆ ˆ ˆ                                                  (c) 3a2                         (d) N/T
33. Let a  2i  j  k , b  i  2 j  k & unit vector                                                            
                                                            40. For non-coplanar vectors a, b & c the relation
c be coplanar. If c is  to a, then c =
  
| (a  b). c |  | a | | b | | c | if
1                            1
(a)           j ˆ
(ˆ  k)         (b)         (ˆ  ˆ  k)
i j ˆ                                                            
2                            3                              (a) a.b  b.c  c.a              (b) a.b  0  b.c
                               
(c)
1
(ˆ  2ˆ)
i      j          (d) N/T                                 (c) a.b  0  c.a                (d) a.b  b.c  c.a =0
5
                                                            ˆ ˆ ˆ
41. a , b & c          are      unit         vectors        then
i j ˆ
34. Let a  ˆ  ˆ  2k & b  ˆ  ˆ. If c is vector such
i j                                                2           2
ˆ ˆ    ˆ ˆ                   2
                                                          a b  bc  ca
ˆ ˆ                  does not exceed
that a . c  | c | , | c  a |  2 2 & angle between
                                                               (a) 4                      (b) 9
a  b & c is 30o, then | (a  b )  c | =

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(c) 8                 (d) 6                        44. Vector OP = i- 3j -2k and OQ = -3i + j -2k.
42. A vector a = ( α ,β, γ) makes an obtuse angle          Then OM, the positive vector of bisector of
with y- axis, equal angles with with b = ( β ,         angle POQ, is
-2γ , 3 α ) & c = (2γ , 3α ,-β ) and is                 (a) i - j - k        (b) 2 ( i + j –k )
perpendicular to d = (1, -1, 2). If |a | = 2√3 ,       (c) i + j + k        (d) – i + j + k
then the vector a is                                          
45. Let a, b, c be P.V. of vertices of a ∆ ABC
(a) (2, 2,-2)         (b) (-2. -2, -2)
whose circumcenter            is origin then
(c) (-2,-2, 2)            (d) (2,-2,-2).               orthocenter is equals
                                                                         
43. If a , b, c are non coplanar vectors such that         (a) a  b  c        (b) ( a  b  c ) /3
                      
[ k(a  b), k 2 b, k c] = [a, b  c, c ] , k has               
(c) ( a  b  c ) /2 (d) N/T
(a) no value          (b) exactly one value        46. Vectors     a, b & c are related by a = 8b &
(c) exactly two values                                 c = - 7c then angle between a & c is
(d) exactly three values                                (a) 0                (b) π /4
(c) π/2              (d) π

1           2          3            4          5    6             7      8               9        10
a           c          b            c          b    c             d      a               b         b
11          12         13           14         15   16            17     18              19        20
d           c          c            a          c    d             b      b               c         b
21          22         23           24         25   26            27     28              29        30
c           c          a            d          a    b             a      a               c         a
31          32         33           34         35   36            37     38              39        40
c           c          a            b          d    a             d      a               b         d

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85
Contact: 9953168795, 9268789880
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41      42      43      44      45      46
b       d       a       b       a       d

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85
Contact: 9953168795, 9268789880

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