# Vector Algebra MCQs Part I

**1) If a=4i+6j and b=3j+4k, then the vector form of component of a along
b is **

**a)**

**b)**

**c)**

**d)**3j+4k

**2) The vector
is to be written as the sum of a vector parallel
to
and a vector
**

**a)**

**b)**

**c)**

**d)**

**3) If
**

**a)**u is a unit vector

**b)**u=a+i+j+k

**c)**u=2a

**d)**u=8(i+j+k)

**4) The volume of a parallelopiped whose sides are given by
is**

**a)**4/13

**b)**4

**c)**2/7

**d)**none of these

**5) If then
the value of **

**a)**60

**b)**64

**c)**70

**d)**-74

**6) The scalar
equals **

**a)**0

**b)**[ABC]+[BCA]

**c)**[ABC]

**d)**none of these

**7) Let a=2i-j+k ; b=i+2j-k and c=i+j+2k be three vectors. A vector in the plane
of b and c whose projection on a is of magnitude
is **

**a)**2i-3j-3k

**b)**2i+3j+3k

**c)**-2i-j+5k

**d)**2i+j+5k

**8) If the vectors c, a=xi+yj+zk and b=j are such that a, c, b form a right
handed systems then c is**

**a)**zi-xk

**b)**0

**c)**yj

**d)**-zi+xk

**9) ABCD is a quadrilateral with
If its area is times the area
of adjacent sides, then is equal
to
**

**a)**5

**b)**

**c)**1

**d)**

**10) ABCDEF is a regular hexagon where center O is the origin. If the position
vectors of A and B are respectively
the is equal to**

**a)**

**b)**

**c)**

**d)**none of these

**11) The position vectors of two vertices and the centroid of a triangle are
. The position vector of the
third vertex of the triangle is**

**a)**

**b)**

**c)**

**d)**none of these

**12) Let the position vectors of the point A,B,C be
respectively. Then the ABC is**

**a)**right angled

**b)**equilateral

**c)**isosceles

**d)**none of these

**13) . are three vectors of which
every pair is noncollinear. If the vector
are collinear with **

**a)**a unit vector

**b)**the null vector

**c)**

**d)**none of these

**14) ****a)** **b)** **c)** **d)**

**15) The position vectors of three points are
and are noncoplanar vectors. The
points are collinear when**

**a)**

**b)**

**c)**

**d)**None of these

**16) are linearly dependent
vectors and then**

**a)**

**b)**

**c)**

**d)**

**17) Let A vectors along one of the
bisectors of the angle <AOB is**

**a)**

**b)**

**c)**

**d)**none of these

**18) A vector has components 2p and 1 with respect to a rectangular Cartesian
system. The axes are rotated through an angle
about the origin in the anticlockwise sense. If the vector has components
p+1 and 1 with respect to the new system then**

**a)**

**b)**p=0

**c)**

**d)**p=1,-1

**19) If are two vectors of magnitude
2 inclined at an angle then
the angle between is**

**a)**

**b)**

**c)**

**d)**None of these

**20) Let Then the angle between
**

**a)**

**b)**

**c)**

**d)**

**21) A vector of magnitude 4 which is equally inclined to the vectors ****a)** **b)** **c)** **d)** none of these

**22) If ****a)** **b)** **c)** **d)** none of these

**23) Let is****a)** **b)** 6**c)** **d)** none of these

**24) ****a)** **b)** **c)** **d)** none of these

**25) If
are unit vectors such that
is also a unit vector then the angle between the vectors**

**a)**

**b)**

**c)**

**d)**

**26) ****a)** **b)** **c)** **d)**

**27) is
equal to**

**a)**

**b)**3

**c)**

**d)**none of these

**28) is
equal to**

**a)**

**b)**0

**c)**

**d)**none of these

**29) If a,b,c are the pth, qth, rth terms of an HP andthen**

**a)**

**b)**

**c)**

**d)**

**30) If ****a)** **b)** **c)** **d)**

**31) Let **

**a)**

**b)**

**c)**

**d)**

**32) Let **

**a)**

**b)**

**c)**

**d)**none of these

**33) If are
three vectors of equal magnitude and the angle between each pair of vectors
is
such that
is equal to**

**a)**2

**b)**-1

**c)**1

**d)**

**34) If ****a)** 1**b)** **c)** 3**d)** none of these

**35) If
equal to**

**a)**

**b)**

**c)**

**d)**none of these

**36) If ****a)** **b)** **c)** **d)** none of these

**37) Two vectors ****a)** perpendicular to each other**b)** parallel to each other**c)** **d)**

**38) ABC is an equilateral triangle of side a. the value of ****a)** **b)** **c)** **d)** none of these

**39) If **

**a)**

**b)**

**c)**

**d)**none of these

**40) ****a)** 0**b)** **c)** **d)** 1

**41) If
are two noncollinear and nonzero vectors such that where,
a,b,c are the lengths of the sides of a triangle, then the triangle is**

**a)**right angled

**b)**obtuse angled

**c)**equilateral

**d)**isosceles

**42) If
are any three vectors such that is**

**a)**

**b)**

**c)**

**d)**none of these

**43) Let ****a)** 1**b)** 0**c)** -1**d)** 2

**44) The vectors are
the sides of a triangle which is **

**a)**equilateral

**b)**isosceles

**c)**right angled

**d)**both b and c

**45) ****a)** **b)** **c)** **d)**

**46) ****a)** an equilateral triangle**b)** a right angled triangle**c)** an isosceles triangle**d)** collinear vectors

**47) ****a)** -3/2**b)** 0**c)** -1**d)** 1

**48) Let ****a)** **b)** **c)** **d)**

**49) What is the value of****a)** 0**b)** **c)** **d)**

**50) The position vectors of the points A and B are
respectively. P divides AB in the ratio 3 : 1. Q is the mid-point of AP. The
position vector of Q is **

**a)**

**b)**

**c)**

**d)**

**51) ****a)** -1**b)** 0**c)** 2**d)** -3

**52) ****a)** **b)** **c)** **d)** none of these

**53) ****a)** **b)** **c)** **d)** none of these

**54) ****a)** 0**b)** 1**c)** **d)**

**55) A vector
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,
has components p+1 and 1, then
**

**a)**p = 0

**b)**

**c)**

**d)**p = 1 or p = -1

**56) ****a)** 0**b)** 1**c)** 2**d)** -2

**57) The angle between vectors ****a)** **b)** **c)** **d)** 0

**58) **

are

vectors reciprocal to the non-coplanar vectors

then

**a)** **b)** 1**c)** 4**d)** 0

**59) Let be
three non-coplanar vectors and
are vectors defined by the relations
Then the value of the expression **

**a)**0

**b)**1

**c)**2

**d)**3

**60) The value of ****a)** **b)** **c)** **d)** 0

**61) Which one of the following is not a vector?****a)** Momentum**b)** Velocity**c)** Mass**d)** Angular velocity

**62) What is needed to represent a scalar?****a)** a real number only**b)** a real number and a unit of measurement**c)** a unit of measurement only**d)** none of these

**63) The magnitude of a vector is****a)** unique**b)** not unique**c)** a unique positive integer**d)** not unique but can have only finite number of values

**64) If is a unit vector ^
r to , then the second unit vector
^ r to
is **

**a)**

**b)**

**c)**

**d)**

**65) The projection of on OX, OY,
OZ are respectively 12, 3 and 4, then the magnitude of
is**

**a)**13

**b)**169

**c)**19

**d)**16

**66) The angle between the straight lines
**

**a)**0

**b)**

**c)**

**d)**

**67) The volume ( in cubic units) of the parallelopiped whose edges are represented
by the vectors **

**a)**2

**b)**0

**c)**

**d)**

**68) A unit vector normal to the plane through the points ****a)** **b)** **c)** **d)** none

**69) The work done by the force
in moving a particle along a straight line from the point
(3,2,-1) to (2, -1,4) is**

**a)**0

**b)**4

**c)**15

**d)**19

**70) Two like parallel forces P and 3P act on a rigid body at points A and B
respectively. If the forces are interchanged in position, the resultant will
be displaced through a distance of**

**a)**

**b)**

**c)**

**d)**

**71) Let
If the point of P on the line segment BC is equidistant from AB and AC then
is**

**a)**

**b)**

**c)**

**d)**none of these

**72) Two vectors are said to be equal if ****a)** their magnitudes are same**b)** direction same**c)** originate from the same point**d)** they have same magnitude and same sense of direction

**73) Two vectors a and b are parallel and have equal magnitudes, then****a)** they are equal**b)** they are not equal**c)** they may or not be equal**d)** they do not have the same direction

**74) If a is non-zero vector of modulus a and m is a non-zero scalar,**

then m a is a unit vector if

**a)**

**b)**

**c)**

**d)**none

**75) a and b are two unit vectors and
is the angle between them. Then a+b is a unit vector if**

**a)**

**b)**

**c)**

**d)**

**76) The position vectors of A and B are a and b respectively,**

then the position vector of a point P which divides AB in the ratio

1:2 is

**a)**

**b)**

**c)**

**d)**

**77) Point A is a+2b, P is a and P divides AB in the ratio 2:3. The position
vector of B is**

**a)**2a-b

**b)**b-2a

**c)**a-3b

**d)**b

**78)
is the angle between the two vectors a and b then **

**a)**

**b)**

**c)**

**d)**

**79) If a be a non-zero vector then which of the following is correct?****a)** a . a = 0**b)** a . a > 0**c)** **d)**

**80) a and b are two non-zero vectors, then (a+b).(a-b) is equal to ****a)** a + b**b)** **c)** **d)**

**81) a.b=0 implies only****a)** a=0**b)** b=0**c)** **d)**

**82) If a,b,c be three non-zero vectors then the equation a.b=a.c implies****a)** b=c**b)** a is orthogonal to both b and c**c)** a is orthogonal to b-c**d)** either b=c or a is orthogonal to b-c

**83) If a and b include an angle of
and their magnitudes are 2 and ,
then a.b is equal to**

**a)**3

**b)**

**c)**

**d)**-3

**84) If (i, j, k) be a set of orthogonal unit vectors, then****a)** i.i+j.j+k.k=0**b)** i.j+j.k+k.i=3**c)** i.i=j.j=k.k=1**d)** i.j=j.k=k.i=1

**85) If
be the angle between the vectors 4(i-k) and i+j+k, then
is **

**a)**

**b)**

**c)**

**d)**

**86) The angle between the vectors 2i+ 3j+ k and 2i-j-k is****a)** **b)** **c)** **d)** 0

**87) If a and b are two vectors, then
is a unit vector if **

**a)**

**b)**

**c)**

**d)**none

**88) [a b c] is the scalar triple product of three vectors, a, b and c, then
[a b c] is equal to**

**a)**[b a c]

**b)**[c b a]

**c)**[b c a]

**d)**[a c b]

**89) If
is the angle between vectors a and b, then
is equal to **

**a)**0

**b)**

**c)**

**d)**

**90)
is equal to **

**a)**(a.b)c-(b.c)b

**b)**(a.b)a+(a.b)c

**c)**(b.c)a-(b.c)b

**d)**(a.c)b-(a.b)c

**91) ,
then **

**a)**u is a unit vector

**b)**u=a+b+c

**c)**u=0

**d)**

**92) If a= 4i+2j-5k, b=-12i-6j+15k, then the vectors a, b are****a)** orthogonal**b)** parallel**c)** non-coplanar**d)** none of these

**93) If [i, j, k] be orthogonal set of unit vectors, then ****a)** **b)** **c)** **d)**

**94) If the position vectors of three points are, a-2b+3c, 2a+3b-4c, -7b+10c,
then the three points are**

**a)**collinear

**b)**coplanar

**c)**non-collinear

**d)**neither

**95) If a+b+c=0, then
the angle between a and b is**

**a)**

**b)**

**c)**

**d)**

**96) If a, b, c are any three coplanar unit vectors, then****a)** **b)** **c)** **d)**

**97) If a.b=a.c and
then**

**a)**a is perpendicular to b-c

**b)**a is parellel to b-c

**c)**either a=0 or b=c

**d)**none of these

**98) If
then (a+b).(a-b) is **

**a)**+ive

**b)**-ive

**c)**zero

**d)**none of these

**99) The vector 2i+j-k is perpendicular to i-4j+,
if
is equal to **

**a)**0

**b)**-1

**c)**-2

**d)**-3

**100) The area of parallelogram having diagonals a=3i+j-2k and b=i-3j+4k is****a)** **b)** **c)** 8**d)** 4

**Answers** **Ans 1)** b*Note that the component of vector a along the vector b is***Ans Desc 1)** Note that the component of vector a along the vector b is

**Ans 2)** a

*Any vector perpendicular to A
*

Ans Desc 2)

Ans Desc 2)

Any vector perpendicular to A

**Ans 3)** c

**Ans 4)** b

**Ans 5)** d

**Ans 6)** a

**Ans 7)** c*Any vector r in the plane of b and c is r=b+tc
or r(1+t)i+(2+t)j-(1+2t)k
*

**Any vector r in the plane of b and c is r=b+tc**

Ans Desc 7)

Ans Desc 7)

or r(1+t)i+(2+t)j-(1+2t)k

**Ans 8)** a

*Since a, c, b form a right handed system
*

Ans Desc 8)

Ans Desc 8)

Since a, c, b form a right handed system

**Ans 9)** b

Area of quad. ABCD

= Area of

Also, Area of parallelogram with AB and AD as adjacent sides,

Hence, Area of quad. ABCD

Ans Desc 9)

Area of quad. ABCD

= Area of

Also, Area of parallelogram with AB and AD as adjacent sides,

Hence, Area of quad. ABCD

**Ans 10)** b

=.

Ans Desc 10)

=.

**Ans 11)** a*The positive vector of the centroid =
are the position vectors ofo the vertices.
*

**The positive vector of the centroid =**

Ans Desc 11)

Ans Desc 11)

are the position vectors ofo the vertices.

**Ans 12)** b

Similarly, find

Ans Desc 12)

Similarly, find

**Ans 13)** b*Here, Subtracting,
or
But are noncollinear.
*

**Here, Subtracting,**

Ans Desc 13)

Ans Desc 13)

or

But are noncollinear.

**Ans 14)** a

Solving these, are in AP.

Ans Desc 14)

Solving these, are in AP.

**Ans 15)** c

*If the point be A, B and C respectively then
The points are collinear if*

Ans Desc 15)

Ans Desc 15)

If the point be A, B and C respectively then

The points are collinear if** **

**Ans 16)** d*Here,
Now
and
*

**Here,**

Ans Desc 16)

Ans Desc 16)

Now

and

**Ans 17)** c

**Ans 18)** a*Here, . After rotation, let
the vector be
Then But the magnitude of
a vector does nto change with rotation of axes.
*

**Here, . After rotation, let**

Ans Desc 18)

Ans Desc 18)

the vector be

Then But the magnitude of

a vector does nto change with rotation of axes.

**Ans 19)** a*Here,
If the angle between
or
or
Now,*

**=4+4+2.2=12**

**Here,**

Ans Desc 19)

Ans Desc 19)

If the angle between

or

or

Now,

=4+4+2.2=12

**Ans 20)** b*1=
or*

**1=**

Ans Desc 20)

Ans Desc 20)

or

**Ans 21)** c*Let
Now, (from the question)
or x+y=y+z=z+x=t (say).
Adding, 2(x+y+z)=3tor x+y+z=
*

**Let**

Ans Desc 21)

Ans Desc 21)

Now, (from the question)

or x+y=y+z=z+x=t (say).

Adding, 2(x+y+z)=3tor x+y+z=

**Ans 22)** b

So,

Ans Desc 22)

So,

**Ans 23)** a

Similarly, we get

Adding these.

=1+2+3+0=6

Ans Desc 23)

Similarly, we get

Adding these.

=1+2+3+0=6

**Ans 24)** b*Let ***Ans Desc 24)** Let

**Ans 25)** d

and are

unit vectors.

Angle

between a and b is 120^{0.}

Ans Desc 25)

and are

unit vectors.

Angle

between a and b is 120^{0.}

**Ans 26)** b

Ans Desc 26)

**Ans 27)** a*See the hint to Q. NO. 15 As
*

**See the hint to Q. NO. 15 As**

Ans Desc 27)

Ans Desc 27)

**Ans 28)** a*a ^{2} + b^{2} + 2ab – (a^{2} + b^{2} –*

2ab) = 4ab = 4 a.b

**a**

Ans Desc 28)

Ans Desc 28)

^{2}+ b

^{2}+ 2ab – (a

^{2}+ b

^{2}–

2ab) = 4ab = 4 a.b

**Ans 29)** b

From the question,

Putting these in (1) and simplifying,

Clearly,

Ans Desc 29)

From the question,

Putting these in (1) and simplifying,

Clearly,

**Ans 30)** b*Here
or
*

**Here**

Ans Desc 30)

Ans Desc 30)

or

**Ans 31)** a

(given)

Now,

and

or 6t+3s=0or2t+s=0.

or

Ans Desc 31)

(given)

Now,

and

or 6t+3s=0or2t+s=0.

or

**Ans 32)** b*Adding and simplifying,
therefore,
they are coplanar.*

**Adding and simplifying,**

Ans Desc 32)

Ans Desc 32)

therefore,

they are coplanar.

**Ans 33)** c

Then

Similarly,

Now, 6=

Ans Desc 33)

Then

Similarly,

Now, 6=

**Ans 34)** b

Ans Desc 34)

**Ans 35)** a

or

Ans Desc 35)

or

**Ans 36)** b*By condition.***Ans Desc 36)** By condition.

**Ans 37)** d

Or

Ans Desc 37)

Or

**Ans 38)** c

or

Ans Desc 38)

or

**Ans 39)** a*Here,
*

**Here,**

Ans Desc 39)

Ans Desc 39)

**Ans 40)** b

Add

these.

Ans Desc 40)

Add

these.

**Ans 41)** c* are
noncoplanar. Therefore, the given linear relation is possible only when b-c=0,
c-a=0, a-b=0. So, a=b=c.*

**are**

Ans Desc 41)

Ans Desc 41)

noncoplanar. Therefore, the given linear relation is possible only when b-c=0,

c-a=0, a-b=0. So, a=b=c.

**Ans 42)** a

Ans Desc 42)

**Ans 43)** b

or

or

or

Ans Desc 43)

or

or

or

**Ans 44)** d

Ans Desc 44)

**Ans 45)** a

Ans Desc 45)

**Ans 46)** b

Ans Desc 46)

**Ans 47)** a

Ans Desc 47)

**Ans 48)** c

Ans Desc 48)

**Ans 49)** c

Ans Desc 49)

**Ans 50)** a

Ans Desc 50)

**Ans 51)** c

Ans Desc 51)

**Ans 52)** b

Ans Desc 52)

**Ans 53)** c

Ans Desc 53)

**Ans 54)** c

Ans Desc 54)

**Ans 55)** b*Let
make angle
with the original axes. Let the original co-ordinate system be rotated through
angle
*

**Let**

Ans Desc 55)

Ans Desc 55)

make angle

with the original axes. Let the original co-ordinate system be rotated through

angle

**Ans 56)** d

Ans Desc 56)

**Ans 57)** a

Ans Desc 57)

**Ans 58)** b

Ans Desc 58)

**Ans 59)** d

Ans Desc 59)

**Ans 60)** c

*The value of *

Ans Desc 60)

Ans Desc 60)

The value of

**Ans 61)** c

**Ans 62)** b

**Ans 63)** c

**Ans 64)** c

**Ans 65)** a*Projections of the line joining
*

**Projections of the line joining**

Ans Desc 65)

Ans Desc 65)

**Ans 66)** c

Ans Desc 66)

**Ans 67)** a

Ans Desc 67)

**Ans 68)** c

Ans Desc 68)

**Ans 69)** c*Let be the force and A and B
be the given points
*

**Let be the force and A and B**

Ans Desc 69)

Ans Desc 69)

be the given points

**Ans 70)** a

In the first case, if the resultant acts at C, then

In the second case, if the resultant act at D, then

Ans Desc 70)

Ans Desc 70)

In the first case, if the resultant acts at C, then

In the second case, if the resultant act at D, then

**Ans 71)** c

Ans Desc 71)

**Ans 72)** d

**Ans 73)** d*Parallel does not imply that the same sense of direction. Hence they are
equal if the sense of direction is same and not equal if the sense of the
direction is opposite. Hence (d) is the correct answer.*

**Parallel does not imply that the same sense of direction. Hence they are**

Ans Desc 73)

Ans Desc 73)

equal if the sense of direction is same and not equal if the sense of the

direction is opposite. Hence (d) is the correct answer.

**Ans 74)** c*a is the modulus of vector a. Modulus is ma is*

**a is the modulus of vector a. Modulus is**

Ans Desc 74)

Ans Desc 74)

*m*a is

**Ans 75)** d*a+b is unit vector if
*

**a+b is unit vector if**

Ans Desc 75)

Ans Desc 75)

**Ans 76)** b*Since by
section formula.*

**Since by**

Ans Desc 76)

Ans Desc 76)

section formula.

**Ans 77)** c

Ans Desc 77)

**Ans 78)** c

Ans Desc 78)

**Ans 79)** b*Square of vector is square of its modulus ***Ans Desc 79)** Square of vector is square of its modulus

**Ans 80)** d

Ans Desc 80)

**Ans 81)** d

**Ans 82)** d*. It
follows that a is orthogonal to b-c or b-c =0 i.e., b=c or a=0. But a is non-zero
vector. Hence the correct answer is (d) which includes all the above cases.*

**. It**

Ans Desc 82)

Ans Desc 82)

follows that a is orthogonal to b-c or b-c =0 i.e., b=c or a=0. But a is non-zero

vector. Hence the correct answer is (d) which includes all the above cases.

**Ans 83)** b* *

Ans Desc 83)

**Ans 84)** c*Since i.i=1=j.j=k.k***Ans Desc 84)** Since i.i=1=j.j=k.k

**Ans 85)** c

Ans Desc 85)

**Ans 86)** a

**Ans 87)** b

Ans Desc 87)

**Ans 88)** c*Scalar triple product remains unchanged if cyclic order is maintained.***Ans Desc 88)** Scalar triple product remains unchanged if cyclic order is maintained.

**Ans 89)** d

Ans Desc 89)

**Ans 90)** d

**Ans 91)** c*Check yourself. Since***Ans Desc 91)** Check yourself. Since

**Ans 92)** b

**Ans 93)** b*Since ***Ans Desc 93)** Since

**Ans 94)** a*If A, B, C are the two points, then
and so A, B, C are collinear.*

**If A, B, C are the two points, then**

Ans Desc 94)

Ans Desc 94)

and so A, B, C are collinear.

**Ans 95)** d

Ans Desc 95)

**Ans 96)** c

**Ans 97)** c

**Ans 98)** c

**Ans 99)** c

**Ans 100)** b