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VECTOR ALGEBRAVECTOR ALGEBRA
1. Let aˆ, bˆ be unit vectors. If c be a vector such that the 6. Let a iˆ ˆj kˆ and c 2iˆ 3iˆ 2k . Then the
number of vectors b such that b c a and
angle between â and c is , and bˆ c 2 c aˆ ,
12
b 1, 2,........,10 is :
2
then 6c is equal to
(a) 0 (b) 1
(a) 6 3 3 (b) 3 3 (c) 2
(d) 3
7. Let a and b be the vectors along the diagonal of a
(c) 6 3 3 (d) 6 3 1 parallelogram having area 2 2. let the angle between
2. Let â and b̂ be two unit vectors such that
a and b be acute. a 1 and a b a b . If
aˆ bˆ 2 aˆ bˆ 2. If 0, is the angle
c 2 2 a b 2b , then the angle between b and c
between â and bˆ, then among the statements :
is :
S1 : 2 aˆ bˆ aˆ bˆ
(a) (b)
4 4
S 2 : The projection of
â on â bˆ is 1
2 (c)
5
(d)
3
(a) Only (S1) is true 6 4
(b) Only (S2) is true 8. Let a iˆ 2 ˆj kˆ and b 2iˆ ˆj kˆ, where
(c) Both (S1) and (S2) are true R. If the area of the parallelogram whose adjacent
(d) Both (S1) and (S2) are false
sides are represented by the vectors a and b is
3. Let a a1iˆ a2 ˆj a3 kˆ ai 0, i 1, 2,3 be a vector 2 2
which makes equal angles with the coordinates axes
15 2 4 , then the value of 2 a a b b is
OX , OY and OZ . Also, let the projection of a on the equal to
(a) 10 (b) 7
vector 3iˆ 4 ˆj be 7. Let b be a vector obtained by
(c) 9 (d) 14
rotating a with 90 . If a , b and x-axis are coplanar, 9.
Let a be a vector which is perpendicular to the vector
then projection of a vector b on 3iˆ 4 ˆj is equal to
3iˆ
1ˆ
2
j 2kˆ. If a 2iˆ kˆ 2iˆ 13 ˆj 4kˆ, then the
(a) 7 (b) 2
(c) 2 (d) 7 projection of the vector a on the vector 2iˆ 2 ˆj kˆ is
4. If a b 1, b 1, b c 2 and c a 3, then the value 1
(a) (b) 1
3
of a b c , b c a , c b a is :
5 7
(c) (d)
(a) 0 (b) 6a b c 3
3
10. Let a iˆ 3 ˆj kˆ, b 3iˆ ˆj 4k and
(c) 12c a b (d) 12b c a
ˆ
c i 2 ˆj 2k , , R , be three vectors. If the
5. Let a iˆ ˆj 2kˆ, b 2iˆ 3 ˆj kˆ are c iˆ ˆj k be
10
three given vectors. Let v be a vector in the plane of a projection of a on c is and
3
2
and b whose projection on c is . If v ˆj 7, then b c 6iˆ 10 ˆj 7 kˆ, then the value of equal
3
to :
ˆ ˆ
v i k is equal to (a) 3 (b) 4
(a) 6 (b) 7 (c) 5 (d) 6
(c) 8 (d) 9 11. Let A, B, C be three points whose position vectors
respectively are :
, VECTOR ALGEBRA
a iˆ 4 ˆj 3kˆ 4 5
(c) (d)
5 6
b 2iˆ ˆj 4kˆ, R
16. Let a iˆ ˆj kˆand b 3i 5 ˆj 4kˆ be two
ˆ
c 3iˆ 2 ˆj 5kˆ
vectors, such that a b iˆ 9iˆ 12kˆ. Then the
If is the smallest positive integer for which a, b , c
projection of b 2a on b a is equal to
ar e non-collinear, then the length of the median, in
ABC , through A is : 39
(a) 2 (b)
5
82 62
(a) (b) 46
2 2 (c) 9 (d)
5
69 66
(c) (d) 17. Let a 2iˆ ˆj 5kˆ and b iˆ ˆj 2kˆ. If
2 2
23
12. Let ABC be
a triangle
such that
a b iˆ kˆ , then b 2 ˆj is equal to
2
BC a , CA b , AB c , a 6 2, b 6 3, and
(a) 4 (b) 5
b c 12 consider the statements : (c) 21 (d) 17
S1 : a b c b c 6 2 2 1
18. Let vector a has a magnitude 9. Let a vector b be
such that for every x, y R R 0, 0 , the vector
2
S 2 : ABC cos . Then
1
3
xa yb is perpendicular to the vector 6 ya 18 xb .
(a) Both (S1) and (S2) are true
Then the value of a b is equal to :
(b) Only (S1) is true
(c) Only (S2) is true (a) 9 3 (b) 27 3
(d) Both (S1) and (S2) are false (c) 9 (d) 81
13. Let a iˆ ˆj 2kˆ and b be a vector such that 19. Let S be the set of all a R for which the angle
between the vectors u a log e b iˆ 6 j 3kˆ and
a b 2iˆ kˆ and a b 3. Then the projection of b
on the vector a b is :- v log e b iˆ 2 j 2a log e b kˆ, b 1 is acute.
2 3 Then S is equal to:
(a) (b) 2
21 7 4
(a) – , (b)
2 7 2 3
(c) (d)
3 3 3 4 12
(c) , 0 (d) ,
14. Let a iˆ ˆj k and b 2iˆ ˆj kˆ, and 0. If
ˆ 3 7
the projection of a b on the vector i 2 ˆj 2kˆ is 20. Let a 3iˆ ˆj and b iˆ 2 ˆj kˆ. Let c be a vector
30, then is equal to
satisfying a b c b c. If b c are non-parallel,
15 then the value of is:
(a) (b) 8
2 (a) 5 (b) 5
13 (c) 1 (d) –1
(c) (d) 7
2 21. Let â and b̂ be two unit vectors such that the angle
15. A vector a is parallel to the line of intersection of the
between them is . If is the angle between the
plane determined by the vectors iˆ, iˆ ˆj and the plane 4
determined by the vectors iˆ ˆj , iˆ kˆ. The obtuse
vectors â bˆ
and aˆ 2bˆ 22 aˆ bˆ , then the
angle between a and the vector b iˆ 2 ˆj 2kˆ is value of 164 cos 2 is equal to :
3 2 (a) 90 27 2 (b) 45 18 2
(a) (b)
4 3
(c) 90 3 2 (d) 54 90 2
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