13 - VECTOR ALGEBRA
Page 1
( Answers at the end of all questions ) (1)
If C is the midpoint of AB and P is any point outside AB, then
→
→
→
→
→
→
( a ) PA + PB = 2 PC →
( c ) PA + PB + 2 PC = 0
(2)
→2
→
→
→
→
→2
→
^ 2
→2
(b) a
→
^
+ ( a × k )
2
[ AIEEE 2005 ]
is equal to
→2
(c) 2a
(d) 4a
[ AIEEE 2005 ]
^
^
^
Let a, b and c be distinct non-negative numbers. If the vectors a i + a j + k , ^
^
i + k
^
^
^
and c i + c j + b k
lie in a plane, then c is
( a ) the Geometric Mean of a and b ( c ) equal to zero
(4)
→
a , the value of ( a × i )
(a) 3 a
(3)
→
( d ) PA + PB + PC = 0
→
For any vector
→
( b ) PA + PB = PC
If
→
→
→
a , b , c
→
→
( b ) the Arithmetic Mean of a and b ( d ) the Harmonic Mean of a and b [ AIEEE 2005 ]
are non-coplanar vectors and λ is a real number , then 2
→
→
→
→
→
→
λ ( a + b ) · [ λ b × λ c ] = a · [ ( b + c ) × b ] for ( a ) exactly one value of λ ( c ) exactly three values of λ
(5)
Let
→
^
^
a = i - k,
→
^
( b ) no value of λ ( d ) exactly two values of λ
^
^
b = xi + j + (1 - x) k
and
→
^
[ AIEEE 2005 ]
^
^
c = yi + x j + (1 + x - y) k.
→→→
Then [ a b c ] depends on ( a ) only y
(6)
Let a, b,
( b ) only x
c,
( c ) both x and y
( d ) neither x nor y
be three non-zero vectors such that no two of these are collinear. If the
vector a + 2b is collinear with c , and b + 3c a + 2b + 6c , for some non-zero scalar λ equals ( a ) λa
[ AIEEE 2005 ]
( b ) λb
( c ) λc
(d) 0
is collinear with a , then
[ AIEEE 2004 ]
13 - VECTOR ALGEBRA
Page 2
( Answers at the end of all questions )
(7)
^
^
^
displace it from a point i + 2 j + 3 k standard units by the forces is given by ( a ) 40
(8)
If
( b ) 30
( c ) 25
v,
λb + 4c
to the point
^
and
^
^
5i + 4 j + k.
^
^
^
3i + j - k
which
The work done in
[ AIEEE 2004 ]
and ( 2λ - 1 ) c
w
are non-coplanar for
( b ) all except one value of λ ( d ) no value of λ
[ AIEEE 2004 ]
be such that l u l = 1, l v l = 2 and l w l = 3. If the projection of v w along u
along u is equal to that of then l u - v + w l equals (a) 2
^
( d ) 15
( a ) all values of λ ( c ) all except two values of λ
Let u,
^
are non-coplanar vectors and λ is a real number, then the vectors
a, b, c
a + 2b + 3c ,
(9)
^
4 i + j - 3k
A particle is acted upon by constant forces
(b)
( 10 ) Let a, b, c
7
(c)
14
v, w
and
are perpendicular to each other,
( d ) 14
[ AIEEE 2004 ]
(a × b) ×
be non-zero vectors such that
c =
1 l b l l c l a . If θ is 3
the acute angle between the vectors b and c , then sin θ equals 1 3
(a)
( 11 ) If
2 3
(b)
a
a2
1 + a2
b
b2
1 + b2
c
c2
1 + c2
(c)
2 3
(d)
2 2 3
[ AIEEE 2004 ]
2
= 0
2
2
and vectors ( 1, a, a ), ( 1, b, b ) and ( 1, c, c ) are
non-coplanar, then the product abc equals (b) -1
(a) 2
( 12 )
→
→
(c) 1
(d) 0 →
→
[ AIEEE 2003 ] →
→
→
→
→
a , b , c are three vectors, such that a + b + c = 0, l a l = 1, l b l = 2, l c l = 3,
then
→ →
(a) 0
a⋅b
+
→ →
b
⋅c
(b) -7
+
→ →
c
⋅a
(c) 7
is equal to (d) 1
[ AIEEE 2003 ]
13 - VECTOR ALGEBRA
Page 3
( Answers at the end of all questions ) →
→
displaced from the point ( a ) 20 units →
( 14 ) If
→
u,
→
v
→
w
→
→
→
and
3 i +
→
→
j - k
is
→
5 i + 4 j + k . The total work done by the forces is
( b ) 30 units
and
→
→
→
4 i + j - 3 k
( 13 ) A particle acted on by constant forces
(c)
40 units
( d ) 50 units
[ AIEEE 2003 ]
are three non-coplanar vectors, then
→
→
→
→
( u + v - w ). ( u - v ) × ( v - w ) equals (a) 0
→ →
(b)
→
u . v ×w
→
→ →
(c)
→
→ →
→
u .w × v
→
→
→
→
The vectors AB = 3 i + 4 k and AC = 5 i - 2 j triangle ABC. The length of a median through A is
( 15 )
(a)
(b)
18
72
(c)
(d)
33
→
→
→
→
→
→
- 6 j + 10 k , - i - 3 j + 4 k and 5 i -
( a ) square
( 17 ) Let
→
u
=
such that (a) 0
( b ) rhombus
→
→
i + j ,
→
v
→ ^
→
→
i - j
=
u ⋅ n = 0 and
(b) 1
( c ) rectangle
w =
→ ^
v ⋅ n = 0, then
(c) 2
are the sides of a
→
→
→
- 4 j + 7k ,
→
j
+ 5 k respectively. Then ABCD is a
( d ) parallelogram
→
and
→
+ 4k
[ AIEEE 2003 ] →
i
[ AIEEE 2003 ]
288
( 16 ) Consider points A, B, C and D with position vectors 7 i →
→
(d) 3 u . v ×w
→
→
i + 2 j
[ AIEEE 2003 ]
→
^
+ 3 k . If
n
is a unit vector
→ ^
w ⋅n
(d) 3
[ AIEEE 2003 ]
( 18 ) The angle between any two diagonals of a cube is ( a ) 45° ( 19 ) If vector value of (a)
→
( b ) 60° →
→
→
→
→
→
→
a ×( b × c)
i - j + k →
→
→
→
→
i + j - k ;
a =
→
(c) 3 i - j + k
( d ) tan -
( c ) 90° b =
→
→
1
→
i - j + k
2 2 and
[ AIEEE 2003 ] →
c
=
→
→
→
i - j - k , then the
is →
(b) 2 i
→
→
- 2 j
→
→
(d) 2 i +2 j - k
[ AIEEE 2002 ]
13 - VECTOR ALGEBRA
Page 4
( Answers at the end of all questions ) →
( 20 ) If
a
→
→
→
→
→
→
a
on
→
→
b
→
→
→
→
i - 2 j + 2 k , then a
→
→
→
→
2 i + j + k
[ AIEEE 2002 ]
6 and
→
b
=
→
→
→
5 i - 3 j + k , the orthogonal projection of
→
a
is →
→
→
→
→
→
(a) 5 i - 3 j + k (c)
=
6
(d) →
c
is
→
2 i + j + 2k
=
→
and
i - 2 j + k
→
3 →
→
(b)
i - j + k
(c)
( 21 ) If
→
→
→
→
3
→
= - i + 2 j + k
→
i + j + k
→
→
b
a + b + c
unit vector parallel to
(a)
→
i + j - 2k ;
=
→
→
→
→
→
→
( b ) 9 (5 i - 3 j + k )
5 i - 3 j + k 35
(d)
9(5 i - 3 j + k) 35 →
→
i + k
( 22 ) If the angle between two vectors
[ AIEEE 2002 ] →
i -
and
→
→
j + ak
is
π , then the value 3
of a is (a) 2 →
( 23 ) If
a =
→
→
(c) -2
(b) 4 →
→
i + 2 j - 3k
and
→
→
(d) 0 →
[ AIEEE 2002 ]
→
→
→
b = 3 i - j + 2 k , then angle between ( a + b ) and
→
( a - b ) is ( a ) 0°
( b ) 30°
( c ) 60°
( d ) 90°
[ AIEEE 2002 ]
→
→
→
( 24 ) The value of sine of the angle between the vectors i - 2 j + 3 k is 5 5 5 5 (b) (c) (d) (a) 21 7 14 2 7 →
→
→
( 25 ) If vectors a i + j + k , (a) a + b + c = 0 ( c ) a + b + c = abc + 2
→
→
→
i + b j + k
and
→
→
→
i + j + ck
( b ) abc = - 1 ( d ) ab + bc + ca = 0
and
→
→
→
2 i + j + k [ AIEEE 2002 ]
are coplanar, then
[ AIEEE 2002 ]
13 - VECTOR ALGEBRA
Page 5
( Answers at the end of all questions )
( 26 )
→
If
→
→
are three non-zero, noncoplanar vectors
a, b, c
→ b2
→ →
→
= b +
→ c2
⋅
b
a → a ,
→
→ → c1 = c
-
→
l a l2 → → → → → c ⋅ b1 → c⋅ a → c a b1 , → → 2 2 la l l b1 l
=
⋅
a → a -
⋅
c
=
b
-
b
⋅
a → a ,
→
l a l2
b → b ,
→
l a l2
l b l2 → c3 =
and
and
→ →
→
→ →
→ →
c
→ b1
→ →
→
⋅
c
c -
→
a → a -
l a l2
→ c ⋅ b2 → b2 , → 2 l b2 l
→
then the set of orthogonal vectors is →
→
→
→
→
→
( a ) ( a , b1 c1 ) ( c ) ( a , b1 c 3 )
( 27 ) If
→
^
^
^
→
→
→
→
→
( d ) ( a , b2 c2 ) →
^
a = i + j + k,
(a) 2 i
→
( b ) ( a , b1 c 2 )
(b)
a
^
^
⋅
→
b = 1
^
i - j + k
( 28 ) A unit vector is orthogonal to ^
^
[ IIT 2005 ]
→
and
(c)
^
i
^
→
^
(d)
^
^
5 i + 2 j + 6k
^
j - k,
a × b = ^
then
→
b
is equal to
^
2j - k
[ IIT 2004 ]
and is coplanar to
^
^
^
2i + j + k
and
^
i - j + k , then the vector is ^
(a )
^
3j - k 10
^
(b)
^
2i + 5j 29
^
(c)
^
6 i - 5k 61
^
^
^
2i + 2j - k (d) 3
[ IIT 2004 ]
( 29 ) The value of a so that the volume of parallelopiped formed by vectors ^
^
j + ak
(a)
( 30 )
If
^
(b) 2
3 →
a
^
and a i + k
and
→
b
(c)
1
(d) 3
3
are two unit vectors such that
^
[ IIT 2003 ] →
a
→
( b ) 60°
^
i + a j + k,
becomes minimum is
perpendicular to each other, then the angle between a ( a ) 45°
^
( c ) cos -
1
1 3
( d ) cos
-1
→
+ 2b and 2 7
→
b
and
→
5a
→
- 4 b are
is [ IIT 2002 ]
13 - VECTOR ALGEBRA
Page 6
( Answers at the end of all questions ) →
→
→
→
→
V = 2 i + j - k,
( 31 ) If
W =
→
→
i + 3k
→
and
U is a unit vector, then the maximum
→ → →
value of the scalar triple product [ U V W ] is (a) -1
( 32 )
→
→
( 33 ) If
→
^ ^ i - k,
^
b = xi
[a b c ]
( a ) only x
(c) 8
→
→ → →
then
(d)
59
[ IIT 2002 ]
60
→
→
→
→
→
- b l2 + l b - c l2 + l c - a l2
are unit vectors, then l a
(b) 9
a =
(c)
6
→
→
If a , b and c does not exceed (a) 4
( 34 )
10 +
(b)
(d) 6
[ IIT 2001 ]
^ ^ ^ ^ → ^ j + ( 1 - x ) k and c = y i + x j + ( 1 + x - y ) k ,
+
depends on
( b ) only y
( c ) neither x nor y
( d ) both x and y
[ IIT 2001 ]
Let the vectors a, b, c and d be such that ( a × b ) × ( c × d ) = 0. Let P1 and P2 be planes determined by the pairs of vectors a, b and c, d respectively, then the angle between P1 and P2 is (a) 0
(b)
π 4
(c)
( 35 ) If a, b and c are [ 2a - b, 2b - c, 2c - a ] = (a) 0
(c) -
→
→
π 2
(d)
unit
(b) 1
→
π 3
coplanar
[ IIT 2000 ]
vectors,
(d)
3
(c)
→ →
→ →
→ →
→ →
→ →
→ →
a⋅b + b⋅c + c⋅a = 0 a⋅b = b
⋅c = c⋅a
(b) (d)
the
scalar
triple
product
[ IIT 2000 ]
3
( 36 ) If the vectors a , b and c form the sides BC , CA (a)
then
and
→
→
→
→
→
→
→
→
→
AB of a triangle ABC, then →
a×b + b×c + c×a = 0 →
a×b + b×c ×a = 0
[ IIT 2000 ]
13 - VECTOR ALGEBRA
Page 7
( Answers at the end of all questions ) ( 37 )
→
^
→ →
→
→
a ⋅ c = l c l,
( 38 )
Let
→
→
l(a × b ) × 2 3
(b)
3 2
^
^
→
→
^
^
1 2
→
a , then
5
^
→
a and b
→
(b)
( i -2j )
→
Let
^
→
(d)
→
→
v
^
^
^
1 3
3
^
and
^
→
→
a unit vector
c
be coplanar. If
^
^
(i - j - k)
30°,
→
c
[ IIT 1999 ]
^
→
→
If
→
u
=
→
→ → →
a - (a⋅b)b
^
^
→
and
→ →
→
→
[ IIT 1999 ]
and
→
^
^
^
c = i + αj + β k
are linearly
3 , then α and β respectively are
( c ) - 1, ± 1
( b ) 1, ± 1
For three vectors u , v w of the remaining three ? →
is
^
b = 4 i + 3j + 4 k →
→
c
(- i - j - k) ^
1
→
^
a = i + j + k,
( a ) u ⋅( v × w )
→
c =
→
→
and
that
→
dependent vectors and l c l =
( 41 )
→
(a × b )
such
[ IIT 1999 ]
(d) l u l + u ⋅( a + b )
( a ) 1, - 1
→
vector
(d) 3
→ → → (c) l u l + l u⋅ b l
^
a
c l =
→ → → (b) l u l + l u⋅ a l
^
is
is
→
→
c
→
(a) l u l
If
→
If
be two non-collinear unit vectors.
v = a × b , then
( 40 )
^
and the angle between
2
b = i + 2j - k
(- j + k) ^
1
(c)
^
b = i + j.
(c) 2
a = 2i + j + k,
(a)
→
and
→
is perpendicular to
( 39 )
^
lc - al = 2
then (a)
^
a = 2 i + j - 2k
Let
( d ) ± 1, 1
[ IIT 1998 ]
which of the following expressions is not equal to any →
( b ) ( v × w )⋅ u
(c)
→
→
→
→
→
→
v ⋅ ( u × w ) ( d ) ( u × v )⋅ w
[ IIT 1998 ]
( 42 ) Which of the following expressions are meaningful questions ? →
→
→
(a) u ⋅( v ×w) (b)
→ →
→
( u ⋅ v )⋅ w
→ → →
(c) ( u ⋅ v )w
→
→ →
(d) u × ( v ⋅w)
[ IIT 1998 ]
13 - VECTOR ALGEBRA
Page 8
( Answers at the end of all questions ) ( 43 )
Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p × [ ( x - q ) × p ] + q × [ ( x - r ) × q ] + r × [ ( x - p ) × r ] = 0, then x is given by 1 ( p + q - 2r ) 2 1 (p + q + r ) 3
(a) (c)
→
^
→ →
(d) →
^
a = i - j,
( 44 ) Let
1 (p + q + r ) 2 1 ( 2p + q − r ) 3
(b)
^
^
b = j - k
→
^
^
c = k - i . If
and
→ → →
[ IIT 1997 ] ^
d
is a unit vector such that
^
a ⋅ d = 0 = [ b , c , d ] , then d equals ^
(a) ±
( 45 )
^
^
^
i + j - 2k
(b) ±
6
→
→
^
^
^
i + j - k
(c)
3
→
^
^
i + j + k 3
→
→
→
u , v and w be vectors such that u + v + w = 0 .
Let
→
→ →
→ →
→ →
and l w l = 5, then the value of u . v + v . w + w . u ( b ) - 25
( a ) 47
( 46 ) If
→
→
(c) 0
^
± k
(d)
[ IIT 1995 ]
→
If
lu l
→
= 3,
l v l = 4
is
( d ) 25
[ IIT 1995 ]
→
A , B and C are three non-coplanar vectors, then
→
→
→
→
→
→
→
( A + B + C ) ⋅ ( A + B ) × ( A + C ) equals → → →
(a) 0
( 47 )
If
→
→
→
a, b, c
→
3π 4
( c ) 2 [ A, B , C ]
and π 4
(b)
→
b
→ → →
( d ) - [ A, B , C ]
are non-coplanar unit vectors such that
angle between a (a)
→ → →
( b ) [ A, B , C ]
→
→
→
[ IIT 1995 ]
→
→
b×c
a × (b × c ) =
2
, then the
is (c)
π 2
(d)
π
[ IIT 1995 ]
^
^
^
( 48 ) Let a, b, c be distinct non-negative numbers. If the vectors a i + a j + c k, ^
^
^
and c i + c j + b k ( a ) AM of a and b
^
^
i + k
lie in a plane, then c is ( b ) GM of a and b
( c ) HM of a and b
(d) 0
[ IIT 1993 ]
13 - VECTOR ALGEBRA
Page 9
( Answers at the end of all questions ) ( 49 ) Let
→
^
^
→
^
a = 2 i - j + k,
A vector in the plane of ^
^
^
( a ) 2 i + j - 3k (c)
( 50 )
If
^
^
→
→
a, b, c
the relations
^
^
i + 2j - k
→
→
and
c =
→
b and c whose projection on ^
^
^
^
^
^
^
^
^
i + j - 2k →
a
be three vectors.
is of magnitude
2 3
is
( b ) 2 i + 3j + 3 k ^
- 2 i - j + 5k
→
^
b =
( d ) 2 i + j + 5k
[ IIT 1993 ]
→
be three non-coplanar vectors and → → b × c
→
p =
→ → →
→ → c × a
→
q =
;
[ a b c ]
→ → →
→
→
p, q, r →
r
;
→ → a × b
=
[ a b c ]
are vectors defined by
→ → →
,
[ a b c ]
then the value of the expression →
→
→
→
→
→
→
→
→
( a + b ). p + ( b + c ). q + ( c + a ). r (a) 0
(b) 1
(c) 2
is equal to
(d) 3
[ IIT 1988 ]
( 51 ) The number of vectors of unit length perpendicular to vectors →
→
a = ( 1, 1, 0 ) and b = ( 0, 1, 1 ) is
( a ) one
( 52 ) Let
( b ) two
→
^
( c ) three
^
^
a = a1 i + a 2 j + a 3 k ,
→
vectors
→
a2
a3
b1
b2
b3
c1
c2
c3
(d)
→
c
a and b . If the angle between
a1
(a) 0
^
(b) 1
^
( e ) none of these
^
b = b1 i + b 2 j + b 3 k
three non-zero vectors such that →
( d ) infinite
is a →
and
→
^
[ IIT 1987 ]
^
^
c = c 1 i + c 2 j + c 3 k be
unit vector perpendicular to both the →
a and b
is
π , then 6
is equal to
(c)
3 ( a 12 + a 2 2 + a 3 2 ) ( b 12 + b 2 2 + b 3 2 ) ( c 12 + c 2 2 + c 3 2 ) 4
1 ( a 12 + a 2 2 + a 3 2 ) ( b 12 + b 2 2 + b 3 2 ) 4
( e ) none of these
[ IIT 1986 ]
13 - VECTOR ALGEBRA
Page 10
( Answers at the end of all questions )
( 53 )
→
A vector a has components 2p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the →
counterclockwise sense. If, with respect to the new system, a and 1, then ( b ) p = 1 or p = -
(a) p = 0
1 3
( c ) p = - 1 or p =
( d ) p = 1 or p = - 1
( 54 )
4 13
1 3
( e ) none of these
The volume of the parallelepiped whose → → OB = i + j - k, OC = 3i - k, is (a)
has components p + 1
(b) 4
(c)
2 7
sides
are
[ IIT 1986 ]
given
by
→ OA
( d ) none of these
= 2i
- 3j,
[ IIT 1983 ]
( 55 ) The points with position vectors 60i + 3j, 40i - 8j, ai - 52j are collinear if ( a ) a = - 40
( b ) a = 40
→
( 56 ) For non-zero vectors only if (a) (c)
→
a
→
b
⋅ ⋅
( 57 ) The scalar (a) 0
→
→
→
→
b = 0, c = 0,
→
→
b c
→
( c ) a = 20
→
a, b, c ,
⋅ ⋅
→
→
c = 0
(b)
→
a = 0
→
→
l( a
(d)
→
( d ) none of these
→
× b ) →
c
→
a
⋅ ⋅
⋅
→
→
→
c l = l a l l b l l c l holds if and
→
→
a = 0,
→
→
[ IIT 1983 ]
→
b = b
a
⋅
→
⋅
→
b = 0 →
c = c
⋅
→
a = 0
[ IIT 1982 ]
→
A ⋅ ( B + C ) × ( A + B + C ) equals → → →
→ → →
(b) [ A B C] + [ B C A]
→ → →
(c) [ A B C]
( d ) none of these
[ IIT 1981 ]
13 - VECTOR ALGEBRA
Page 11
( Answers at the end of all questions )
Answers 1 a
2 c
3 a
4 b
5 d
6 d
7 a
8 c
9 c
21 d
22 d
23 d
24 d
25 c
26 b
27 c
28 a
29 c
41 c
42 a,c
43 b
44 a
45 b
46 d
47 a
48 b
49 c
10 d
11 b
12 b
13 c
14 c
15 c
16 b
17 d
18 d
19 b
20 a
30 b
31 c
32 b
33 c
34 a
35 a
36 b
37 b
38 a
39 b,c
40 d
50 d
51 b
52 d
53 b
54 b
55 a
56 d
57 a
58
59
60
