Vectors (1) Units Vectors Units Vectors Magnitude of Vectors Magnitude of Vectors.
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05Vectors
Syllabus
Representation of a Vector Addition of Vectors
Subtraction of Vectors Multiplication of a
Vector by a Scalar. Linear Combinations.
Collinearity and Coplanarity of Vectors.Product
of Two Vectors. Scalar Triple Product. Vector
Triple Product
Those quantities which have only magnitude and
as well as direction are called vector quantities or
vectors e.g.Afootball player hit the ball to give a
pass to another player of his team.
Hence, he apply a quantity (called force) which
involves muscular strength (magnitude) and
(direction in which another player is positioned).
Scalar Quantities
The quantities which have only magnitude are
known as scalar quantities e.g. Mass, volume,
work, etc.
Vector Quantities
The quantities which have both magnitude and
direction are known as vector quantities e.g.
Force, velcoity, etc.
Representation of a Vector
A vector is represented by a line segment e.g.
a = AB Here, A is called the initial point and B is
called the terminal point.
Magnitude or modulus of a is | a | = | AB |= AB.
Types of Vector
Vectors can be defined into following types:
i. Zero or Null Vector A vector whose
magnitude is zero and has arbitrary direction
is known as zero or null vector.
ii. Unit Vector A vector whose modulus is unity
is known as unit vector a is denoted by
ˆ ˆa, thus a 1
aa
a
iii. Like and Unlike Vectors The vectors which
have same direction are called like vectors and
which have opposite directions are called unlike
vectors.
a
b
iv. Collinear or Parallel Vectors :
Vectros having the same or parallel support
are called collinear vectros.
v. Coinitial Vectors Two or more vectors having
the same initial point are called coinitial vectors.
vi. Coplanar Vectors
A system of vectors is said to be coplanar, if
their support is parallel to the same plane.
vii.Coterminous Vectors Vectors which have same
terminal points are called coterminous vectors.
e.g.
Here, a, band c are coterminous vectors.
viii. egative of a Vector
Avector is said to be negative of a given vector,
if its magnitude is the same as that of the given
vector but direction is opposite.
e.g. a aix. Reciprocal of a Vector Avector having same
direction as that of a given vector a but
magnitude equal to the reciprocal of the given
vector is known as the reciprocal of a and it is
denoted by a–1.
1 1a
a
x. Localised and Free Vectors Those vectors
which have not fixed initial point are called
free vectors and a vector which is drawn
parallel to a given vector through a specified
point in space is called localised vector.
xi. Position Vector The vector OA which
represents the position of the point A with
respect to a fixed point 0 is called position
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vector.
(b) they have the same or parallel support.
(c) they have same direction.
xiii.Orthogonal Vectors Two vectors are said to
be orthogonal, if the angle between them is a
right angle.
Components of a Vector in Two and Three
Dimensional System
i. Any vector r can be expressed as a linear
combination of two unit vectors ˆ ˆi and j and} at
right angle.
i.e.
The vectors x ˆ ˆi and j are vector components of
vector r. The scalars x and yare called the scalar
components of r in the direction of X-axis and
Y-axis respectively.
| r | = 2 2x y = tan–1
y
x
ii. The position vector of r = ˆ ˆ ˆxi yj zk
The vectors ˆxi , yj and ˆzk are vector
components of r.
The scalars x,y and z are scalar components ofr
in the direction of X -axis, Y-axis and Z-axis.
| r | 2 2 2x y z
Direction cosines of r are cos .cos and cos such that
cos = l = x
r , cos = m = y
r cos = n z
r
Addition of Vectors
The addition of two vectors a and b is denoted by
a + b and it is known as resultant of a and b.
There are following three methods of addition of
vectors:
i. Triangle Law of Addition
If two vectors a and b lie along the two sides of a
triangle in consecutive order (as shown in the
figure), then third side represents the sum
(resultant) a + b.
i.e. c = a+ b
ii. Parallelogram Law of Addition
If two vectors are represented by two adjacent
sides of a parallelogram, then their sum is
represented by the diagonal of the parallelogram.
OQ = OP + PQ c = a + b
iii. Addition in Component Form
If the vectors are defined in terms of ˆ ˆ ˆi, j,k i.e. if
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a = 1 2 3ˆ ˆ ˆa i a j a k
b = 1 2 3ˆ ˆ ˆb i b j b k
Then, a + b =(al +b
l) i + (a
2+ b
2) j + (a
3+ b
3)k
Properties of Vectors Addition
There are following properties of vectors addition:
• Closure property The sum of two vectors is
always a vector.
• Commutativity For any two vectors a and b,
we have
a + b = b + a
• Associativity For any three vectors a, band c,
we have a + (b + c) = (a + b) + c
• Additive identity For any vector a, we have
0 + a = a + 0
• Additive inverse For every vector a, (-a) is the
additive inverse of the vector a.
i.e. a + (–a) = (– a) + a = 0
• Example 1
If A, Band C are the vertices of a ABC, then
what is the value of AB + BC + CA?
a. 0 b. 1
c. 2 d. 4
Sol (a) By triangle law of vectors addition,
we get AB + BC = AC
Now, AB + BC + CA = AC + CA
(adding CA on both sides)
AB + BC + CA = AC – AC
[ AC = – CA or CA = – AC)
AB + BC + CA = O
Subtraction of Vectors
If a and b are two vectors, then
a – b = a + (–b)
If a = 1 2 3ˆ ˆ ˆa i a j a k
b = 1 2 3ˆ ˆ ˆb i b j b k
a – b = (a1 – b
1 ) i + (a
1 –b
2) j + (a
3–b
3) k
Properties of Vectors Subtraction
There are following properties of vectors
subtraction:
• a = b b c a
• (a– b) – c a – (b – c)
• |a + b| < | a | + | b |
• |a + b| > | a | – | b |
• |a – b| < | a | + | b |
• |a – b| > | a | – | b |
• Example 2
Vectors drawn from the origin 0 to the points
A, B and C are respectively a, b and 4a – 3b.
Find AC and Be.
a. 3(a – b), 4(a – b)
b. 4(a – b), 2(a – b)
c. 3(a – 2b ),4(a – b)
d. (a – b), (a + b)
Sol (a) It is given that OA = a, OB = b and OC = 4a -
3b
In AC. we have
OA+ AC = OC AC = OC – OA
AC = 4a – 3b – a = 3a – 3b = 3(a – b)
In OBC BC. we have
OB + BC = OC
BC = OC – OB
BC = 4a – 3b – b = 4(a–b)
Multiplication of a Vector by a Scalar
If a is a vector and m is a scalar. then m a is a
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vector whose magnitude is m times the magnitude
of a.
Properties of Scalar Multiplication
There are following properties of scalar
multiplication:
• m(–a) = – ma
• (–m) (–a) = ma
• m(na) = (mn)a =n(ma)
• (m + n) a = ma + na.
• m(a + b) =ma + mb
Example 3
Find a vector in the direction of vector a = ˆ ˆi 2 j
that has magnitude 7 units.
a.2 3ˆ ˆi j3 8
b.5 7ˆ ˆi j
85
c.4 2ˆ ˆi j6 8
d.7 14ˆ ˆi j5 5
Sol (d) We have. a = ˆ ˆi 2 j , then
|a| = 2 2(1) ( 2) = 1 4 = 5
The unit vector in the direction gf the given vector
a is
a = 1 1 ˆ ˆ.a (i 2 j)a 5
= i 1
j5 5
Now. the vector having magnitude 7 units and in
the direction ofa is 7 a
= 7 i 2
j5 5
=
7 14ˆ ˆi j5 5
• Example 4The position vector of the vertices P, Q and R of
a triangle are ˆ ˆ ˆ ˆ ˆ ˆi j 3k,2i j 2k and
ˆ ˆ ˆ5i 2 j 6k respectively. The length of the
bisector PS of the QPR, where S is on the
segment RQ. is
a.5
103
b.2
3
c.3
104
d.3
105
Sol (c) ˆ ˆ ˆ ˆ ˆ ˆPQ (2i j 2k) (i j 3k) |
ˆ ˆ ˆPQ i 2 j k = 2 2 21 2 1 = 6
| PR| ˆ ˆ ˆ6i 3j 3k 3 6
Now. QS : SR = PQ : PR = 6
3 6 =
1
3
Position vector of S - Position vector of P
PS = 3
4 (–i ˆ ˆ( i 3j) |PS| =
310
4
Linear CombinationsGiven a finite set of vectors a. b. c. ... . the vector
r = x a + y b + z c +... is called a linear combination
of a,b,c ..... for any x, y. z .... R We have the
following results:
i If a and b are non-zero. non-collinear vectors.
then x a + y b = x' a + y' b
x = x' ; y = y'.
ii. Fundamental Theorem Let a, b be non-zero,
non-collinear vectors. Then. any vector r
coplanar with a and b can be expressed
uniquely as a linear combination of a, b i.e.
there exists some unique x, y E R such that x
a + y b = r.
iii. If a, band c are non-zero. non-coplanar vectors.
then
xa + y b + z c = x'a + y' b + z' c
x = x', y = y', z = z'
iv. Fundamental Theorem in Space Let a, band c
be non-zero, non-coplanar vectors in space.
Then, any vector r, can be uniquely expressed
as a linear combination of a, b and c i.e. there
exists some unique x, y, z R such that
x a + y b + z c = r.
v. If x1' x
2, ... , x
n are n non-zero vectors and
k1' k
2, .... , kn are n scalars and if the linear
combination k1x
l + k
2 x
2+ ...+ k
n x
n = 0
kl = 0, k
2= 0 ...k
n = 0, then we say that vectors
x1' x
2, ... , x
n are linearly independent vectors.
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vi. If x1' x
2, ... , x
n are not linearly independent,
then they are said to be linearly dependent
vectors. i.e.if k1x
1 + k
2x
2 +..... + k
nx
n= 0, if
there exists atleast one
Kr 0(r = 1,2,....n), then x
1' X
2 ,..... , X
n are
said to be linearly dependent.
Linearly Dependent
If Kr, 0; k
1x
1+ k
2x
2 + k
3x
3 +···+ k
rx
r +···+ k
nx
n=
0 – Krx
r =k
1x
l + k
2x
2 + ...+ k
r–1 x
r–1+... + k
r+1
xr+1
+... + knx
n
– K, r
1
Kx
r = K
1– x
1 + k
2 – x
2+ ··· ,
+ Kr–1
rK
X
r–1 +...+ K
n r
1
Kx
n
xr = c
1x
1 + c
2x
2 +....+ c
r–1 + x
r –1 + cr x
r–1 +...+c
nx
n
i.e., xr is expressed as a linear combination of
vectors.
xr = c
1,x
r .....x
r+1 ..... x
n
Hence, x, with x1' x
2,.... x,
r – 1' x
n forms a linearly
dependent set of vectors.
If a = ˆ ˆ ˆ3i 2 j 5k , then a is expressed as a linear
combination of vectors i, j k. Also, a,i,j,k form a
lineary dependent set of vectors.
In general, every set of four vectors is a linearly
dependent system.
ˆ ˆi, j and k are linearly independent set of vectors
for k 1 2 3 1 1 2 3ˆ ˆ ˆk i k j k k 0 k 0 k 0 k k
Collinearity and Coplanarity of Vectors
Test of Collinearity of Three Points
The three points A, Band Cwith position vectors
a, band c, respectively are collinear, if and only if
there exist scalars x, y and Z not all zero such that
i. x a.+ y b + z c = 0 ii. x + y + z = 0
The vectors ABand AC are collinear, if there exists
a linear relation between the two, such that
Example 5
The three points which have the position vectors
ˆ ˆ ˆ ˆ60i 3j,40i 8 j and ˆ ˆai 52 j are collinear. If a
is equal to
a. 30 b. – 40
c. – 30 d. 25
Sol (b) The three points are collinear, if
ˆ ˆ ˆ ˆ ˆ ˆx(60i 2 j) y(40i 8j) z(ai 52 j) 0 Such that, x,y and z are not all zero and x+y+z = 0
(60x + 40y + az) i + (3x – 5y – 52z) j = 0
and x + y = z = 0
i.e.60x + 40y + az = 0
3x – 5y – 52z =0
and x + y + z =0
Then, points will be collinear, if
60 40 a
3 8 52
1 1 1
= 0
a= – 40
Coplanarity of Three Points
Three points A, Band C represented by position
vectors a, b and c respectively represent two
vectors AB and AC. From the figure, two vectors
are always coplanar i.e. two vectors always form
their own plane.
Thus, a, band cwill be coplanar, if we can find
two scalars A and J..L such that
a = b = c.
Three vectors 1 2 3 1 2 3ˆ ˆ ˆ ˆ ˆ ˆa i a j a k,b i b j b k
and 1 2 3ˆ ˆ ˆc i c j a k are coplanar, if
1 2 3
1 2 3
1 2 3
a a a
b b b
c c c = 0
Coplanarity of Four Points
The necessary and sufficient condition that four
points with position vectors a, b, c and d should
be coplanar is that there exist four scalars x, y, z
and t not all zero, such that
xa + yb + zc + td = 0, x + y + z + t = 0
Then prove that the four points A,B,C and D
having position vectors as a, b, c and d are
coplanar.
Step I Find the vector AB, AC and AD having the
reference point as A.
Step II Express one of these vectors as the linear
combination of the other two
AB = AC + AD
Step III Now, compare the coefficients on LHS and
RHS in respective manner and thus find the
respective value of and .
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Step IV If real values of the scalars", and fl exist, then
the three vectors representing four points are
coplanar otherwise not.
Example 6The vectors 5a + 6b + 7c, 7a – 8b + 9c
and 3a + 20b + 5c
are (where a, b, c are three non-coplanar vectors)
a. collinear b. coplanar
c. non-coplanar d. one of these
Sol (b) Let A = Sa + 6b + 7c, B = 7a – 8b + 9c
and C = 3a + 20b + Sc
A, B and C are coplanar.
xA + yB + zC = 0
must have a real solution for x, y and z other than
(0, 0, 0).
Now, x (5a + 6b + 7c)+ y(7a - 8b + 9c) + z
(3a + 20b + Sc) = 0
(5x+7y+ 3z)a+(6x–8y+20z)b+(7x+9y+ Sz)c=0
5x + 7y + 3z = 0
6x – 8y + 20z = 0
7x + 9y + 5z = 0
(as a, band c are non-coplanar vectors)
Now, D =
3 7 3
6 8 20
7 9 5
= 0
So, the three linear simultaneous equation in x, y
and z have a non-trivial solution.
Hence, A,B and Care coplanar vectors.
Important Formulae
i. Section Formula Let a and b be two vectors
represented by OA and OB and the point P divides
AB in the ratio m : n.
If P divides AB in the ratio m :n internally, then
r = mb na
m n
If P divides AB in the ratio m : n externally, then
r = mb na
m n
ii. Mid-point Formula If C(c) is the mid-point of AB,
then
c = a b
2
iii. Centroid of a Triangle Centroid of
ABC = a b c
3
where, a, band c are the position vectors of the
vertices with respect to origin 0.
Product of Two Vectors
There are two types of product of two vectors:
i. Scalar or Dot Product of Two Vectors
The scalar product of two vectors a and b is
expressed as
(a.b = |a| ||b| cos )
where, 0 < q p
a.b. < |a| |b|
a.b > 0 angle between a and b is acute.
angle between a and b is obtuse.
Geometrical Interpretation
OL is the projection of vector b in the direction of
vector a.
OL = b cos [ |a| = a and |b| – b ]
a.b = a (bcos) = (ab) cos = b(acos)
cos = a.b
ab
Projection of b in the direction of OA = OL =
a.b
a
OL = 2
a.b a.ba
a | a |
Projection of a in the direction of OB = OM =
a.b
b
OM = 2
a.b.b
b
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Properties of Scalar Product
There are following properties of scalar product:
• a· b = b· a [Commutativity)
• a . (b + c) = a . b + a.c [Distributivity)
• ˆ ˆ ˆ ˆ ˆ ˆi.i j. j k.k 1
• ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆi. j j.i 0, j.k k. j 0,k.i i.k 0
• For any two vectors a and b
(a) |a + b | = |a |+ |b| a || b
(b) |a + b|2 = |a|2 + |b|2 a b
(c) |a + b| = |a– b| a b
• If dot product of two vectors is zero, then
atleast one of the vectors is a zero vector or
they are perpendicular.
• Example 7
Two vectors = ˆ ˆ4i 3j and yare perpendicular
to each other in the xy-plane. The vector in the
same plane having the projection 1, 2 along and
is
a. ˆ ˆ2i j b. ˆ ˆ3i j
c. ˆ ˆ2i j d. ˆ ˆ2i j
Sol (c) Given, = ˆ ˆ4i 3j and i.e. . = 0
Let = ˆ ˆ2 i 4 j for all values of x,
Suppose required vector be ˆ ˆli mj
Projection of ex along = . ; 1. =
4l 3m
5
4 l + 3m = 5 ...(i)
Similarly, projection of along
= . ,2 =
3 l 4 m
5
3l – 4 m = 10 ...(ii)
From Eqs. (i) and (ii), we get l = Z and m = –1,
ˆ ˆ2 j j Application of Dot Product
Let a particle be placed atO and a force f
represented by OB be acting on the particle at O.
Then, Work done = (Force)· (Displacement)
i.e. (W = f·d = fd cos )
ii. Vector Product of Two Vectors
The vector product of two non-null and non-
parallel vectors a and b is expressed as
(a × b = ab sin n )
where, | a | = a, |b| = b
where, is the angle between a, band ii is a unit
vector perpendicular to the plane of a and b such
that a, b and ii form a right handed system.
(|a × b| 1 = |a| |b| sin )
Geometrical Interpretation of Vector
Product
Modulus of ax b is the area of the parallelogram
whose adjacent sides are represented by a and b.
|a × b| = Area of parallelogram OACE.
Properties of Vector Product
There are following properties of vector product:
• a × b b × a
a × b = – (b a a)
• a × b = 0
a || b or collinear or a = 0 or b = 0
• ˆ ˆ ˆ ˆ ˆ ˆi i j j k k = 0
• ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆi j k, j k k i j
• ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆj i k,k j i, i k j
• Lagrange's identity | a×b|2 =|a|2 |b|2 (a – b)2
• (ma) × b = m(a × b) = a × (mb)
• a × [b + c) = a × b + a × c
Vector Product in Terms of Component
If 1 2 3ˆ ˆ ˆa a i a j a k
and 1 2 3ˆ ˆ ˆb b i b j b k
Then, a × b = 1 2 3
1 2 3
ˆ ˆ ˆi j k
a a a
b b b
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Example 8
Find the unit vector perpendicular to the plane
ABC, where the position vectors of A, B and Care
ˆ ˆ ˆ ˆ ˆ ˆ2i j k, i j 2k and ˆ ˆ2i 3k respectively
a.ˆ ˆ ˆ3i 2 j k
14
b.
ˆ ˆ ˆ3i j 5k
18
c.ˆ ˆ ˆ2i 2 j 3k
15
d.
ˆ ˆ ˆi 2 j 2k
17
Sol (a) We have,
OA = ˆ ˆ ˆ2i j k OB = ˆ ˆ ˆi j k
OC = ˆ ˆ2i 3kNow, AB = OB – OA
= ˆ ˆ ˆ ˆ ˆ ˆ(i j 2k) (2i j k) = ˆ ˆ ˆi 2 j k AC = OC – OA
= ˆ ˆ ˆ ˆ ˆ ˆ ˆ(2i 3k) (2i j k) j 2k
Now, AB × AC =
ˆ ˆ ˆi j k
1 2 1
0 1 2
= ˆ ˆ ˆ ˆ ˆ ˆi (4 1) j( 2 0) k( 1 0) 3i 2 j k The required unit vector perpendicular to the
plane ABC
= AB AC
AB AC
= 2 2 2
ˆ ˆ ˆ3i 2 j k
3 2 ( 1)
= ˆ ˆ ˆ3i 2 j k
9 4 1
=
ˆ ˆ ˆ3i 2 j k
14
Angle between Two Vectors
If is the angle between two vectors a and b,
then
sin a b
ab
, if
a = 1 2 3 1 2 3ˆ ˆ ˆ ˆ ˆ ˆa i a j a k,b b i b j b k
sin2 =
2 2 2
2 3 3 2 1 3 3 1 1 2 2 1
2 2 2 2 2 2
1 2 3 1 2 3
(a b a b ) (a b a b ) (a b a b )
(a a a )(b b b )
Vector Normal to the Plane of Two Given
Vectors
The vectors of magnitude' A' normal to the plane
of a and b
+ (a b)
a b
Condition for Vectors to be Parallel
If 1 2 3 1 2 2ˆ ˆ ˆ ˆ ˆ ˆa i a j a kand b b i b j b k are
parallel, then
a × b = 0 or 1 2 3
1 2 3
a a a
b b b
Condition for Three Points A, B, C to be
Collinear
Determine AB and BC and show that
AB × BC = 0
or AB = kBC
where, k is any scalar.
Area of Parallelogram and Triangle
These formulae are given below:
i. The area of a parallelogram with adjacent sides a
and b is | a × b |
ii. The area of a parallelogram with diagonals
a and b is 1
2 |a × b|
iii. The area of a plane quadrilateral ABCD is
1
2 |AC×BD| ,where, AC and BD are diagonals.
iv. The area of a triangle with adjacent sides a and b
is | a × b |
v. The area of a ABC is 1
2 |AB×AC|
vi. If a, band c are position vectors of vertices of
MBC, then area = 1
2 (a × b) + (b × c) + (c × a)|.
If (a × b) + (b × c) + (cx a) =0, then three points
with position vectors a, band c are collinear.
Example 9
If a = ˆ ˆ ˆ ˆ ˆ2i 3j k,b i k and ˆ ˆc 2 j k three
vectors, find the area of the parallelogram having
diagonals (a + b) and (b + c).
a.11
5 sq. units b.
13
6 sq units
c.21
2 sq units d.
23
3 sq units
Sol (c) We have, ˆ ˆ ˆ ˆ ˆa 2i 3j k,b i k and
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c = ˆ ˆ2 j k
Now, a+b= ˆ ˆ ˆ ˆ ˆ(2i 3j k, b i k and c = ˆ ˆ2 j k
= ˆ ˆ ˆi 3j 2k
and (a + b) × (b + c) =
ˆ ˆ ˆi j k
1 3 2
1 2 0
= ˆ ˆ ˆi(0 4) j(0 2) k(2 3)
= ˆ ˆ ˆ4i 2 j k Area of parallelogram having diagonals (a + b)
and (b + c)
=
ˆ ˆ ˆ4i 2 j k(a b) (b c)
2 2
= 16 4 1
2
=
21
2 =
1
2 21 sq. units
Scalar Triple ProductThe scalar triple product of three vectors a, band
c is defined as
(a b).c a b csin cos
where, is the angle between a and b and is the
angle between a × b and c. It is also defined as
[a b c].
Geometrical Interpretation of a Scalar
Triple Product
The scalar triple product [a b c] represents the
volume of the parallelopiped whose coterminous
edges a, b, c form a right handed system of
vectors.
Properties of Scalar Triple Product
There are following properties of scalar triple
product
• If a = 1 2 3ˆ ˆ ˆa i a j a k ,
1 2 3ˆ ˆ ˆb i b j b k
and 1 2 3ˆ ˆ ˆc i c j c k
Then,
1 2 3
1 2 3
1 2 3
a a a
(a b).c b b b
c c c
• (a × b) . c = a .(b xc)
• [a b cl = [b c a] = [c a b]
• [abc] = – [bac]
• [k abc] = k [abc]
• [a + b c d] = [a c d] + [b c d]
• If [a b c] = 0, then, a, band c are coplanar.
• Example 10If a.b and c are non -coplanar vectors and is a
real number, then the vectors a +2 b + 3c,
b + 4c and (2 –1) are non-coplanar for
a. all except one value of I!
b. no value of c. all except two values of d. all values of
Sol (c) Given that, a, b and c are non-coplanar vectors.
i.e. [a b c] 0
Now, a + 2b + 3c, ub + 4c and (2 – 1)c
will be non-coplanar if
(a + 2b 3c)' [b] × (2m –1) c ]c) 0
(a + 2b 3c). [(2–1) (abc)]0 (2 –1) [abc] 0
0,1/2
Hence, given vectors will be non-coplanar for all
values of 0 and 1
2.
Vector Triple Product
Let a, b, ebe any three vectors, then the
expression a × [b × c) is a vector and is called a
vector triple product.
Geometrical Interpretation of ax (bx c)
Consider the expression a x [b × c) which itself is
a vector, since it is a cross product of two vectors
a and [bx e).Now, a × (b × c) is a vector
perpendicular to the plane containing a and (b ×
c) but bx e is a vector perpendicular to the plane
b and c, therefore a x [b × c)is a vector lies in the
plane of band c and perpendicular to a.
Hence, we can express a × [b × c) in terms of
band c
i.e. a × (b × c) = x b+y c, where x and yare
scalars.
• a × (b × c) = (a·c) b – (a·b) c
• (a × b) × c = (a·e) b – (b·c) a
• (a × b × c a × (b × c)
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Example 11
If ˆ ˆ ˆ ˆ ˆ ˆi (r i) j (r j) k (r k) (a b)
where
a 0,b 0, then a. r = a × b
b. r = a b
2
c. r = 0
d. None of the above
Sol (b) Given, ˆ ˆ ˆ ˆ ˆ ˆi (r i) j (r j) k(r k) a b
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ(i.i) r (i.r) i ( j. j)r ( j.r) j (k.k)r (k.r) k a b
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3r (i.r) i ( j.r) j (k.k) r (k.r)(k.r)k} a b
Now ˆ ˆ ˆr xi yj zk
r = ˆ ˆ ˆxi7 yj zk
ˆ ˆi.r x, j.r y
and k.r zThen, Eq. (i) becomes
ˆ ˆ ˆ3r {xi yj zk} a5b 3r – r = 3 × b
r = a b
2
Applications of Vectors in Geometry
i) 'The points A, Band C are collinear' means
(a) area of MBC is zero
(b) b – a and c – a are collinear vectors
(c) b – a and c – a are parallel
(d) (b – a) × (c – a) = 0
(e) There exist a, 13and y not all zero such that
+ 13 + c = 0 and a + 13+ y = O.
Otherwise, A, B and C are not collinear.
ii. 'A, B, C and D are coplanar' means
(a) volume of tetrahedron ABCD is zero.
(b) b – a, c – a and d – a are coplanar.
(c) [b – a, c–a, d – a) = O
(d) there exist a, 13,yand 0 not all zero such that
a + 13+ c + d = 0 and a + 13 + y + 0 = 0.
Otherwise A, B, Cand D are not coplanar.
iii: If a and b are the position vectors of A and B and
r be the position vector of the point P which
divides the join of A and B in the ratio m : n, then
r = mb na
m n
'+' sign takes for internal ratio and
'–' sign takes for external ratio.
iv. If a, b and c be the PV of three vertices of ABC
and r be the PV of the centroid of ABC, then
a + b + c
r = a b c
3
v. Equation of straight line in vector form
(a) Vector equation of the straight line passing
through origin and parallel to b is given by r =
t b, where t is scalar.
(b) Vector equation of the straight line passing
through a and parallel to b is given by
r × b = a × b or r = a + t b, where t is scalar.
(c) Vector equation of the straight line passing
through a and b is given by (r×a) × (b× a) = 0
or r = a + t (b × a), where t is scalar.
(d) Equation of straight line passing through the
point a perpendicular to two non-parallel
vectors c and d is (r – a) × (c × d) = 0.
• Example 12
The vector equation of a line-passing through a
point with position vector ˆ ˆ ˆ2i j k and parallel
to the line joining the points with position
vector ˆ ˆ ˆi 4 j k and ˆ ˆ ˆi 2 j 2k is
a. ˆ ˆ ˆ2i j k
b. ˆ ˆ ˆ2i 2k k
c. ˆ ˆ(2 2t) i k (l t) d. None of these
Sol (d) Let a = ˆ ˆ ˆ2i j k , ˆ ˆ ˆb i 4 j k and
c = ˆ ˆ ˆi 2 j 2k Then, the equation of the line will be
r = a + t (c – b)
= ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ(2i j k) t (i 2 j 2k i 4 j k)
= ˆ ˆ ˆ ˆ ˆ ˆ(2i j k) t (2i 2 j k)
= ˆ ˆ ˆ(2 2t) i j( 1 2t) k(1 t)
vi. Equation of a plane in vector form
(a) Vector equation of the plane through origin and
parallel to band c is given by r = s b + t c, where
sand t are scalars.
(b) Vector equation of the plane passing through
a and parallel to band c is given by
[r b c] = | a b c] or r = a + sb + tc
where s and t are scalars.
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(c) Vector equation of the plane passing through
a, b and c is r (b × c + c × a + a × b)
= [a bel or r = (1–s –t) a + sb + tc,
where sand t are scalars.
(d) Equation of plane which passes through the
point a and perpendicular to n, is r·n = a.n.
vii. Equation of a plane containing the line of
intersection of two planes
Let two planes be r.n1 = q
1 and r.n
2 =q
2' then
equation (r.na, – q1) + A (r.n
2 – q
2) = 0, where A
be any scalar quantity, is the equation of plane
passing through the intersection line of planes.
viii. Equation of a line of intersection of two planes
Let r-n.=q, and rn, = q2 be two equation of planes,
then the equation of a line of intersection of two
planes, is r = a +t(n,× n2)' where t be any scalar.
ix. Solving of vector equation
Solving a vector equation means determining an .
unknown vector (or a number of vectors satisfying
the given conditions).
Generally, to solve vector equations, we express
the unknown as the linear combination of three
non-coplanar vectors as
r = x a + y b + z (a × b) as a, b and a x bare non-
coplanar and find x, y, z using given conditions.
Sometimes, we can directly solve the given
conditions. It would be more clear from some
examples.
Example 13Solution of the vector equation r × b = a × b,
c = 0 provided that c is not perpendicular to b, is
a. r = a – a.c
b.c
b b. r = b – a.c
b.r
a
c. r = b – b.c
a.c
a d. None of these
Sol (a) We are given; r × b = a × b
(r– a) × b = 0
Hence, (r – a) and b are parallel
(r – a = t b)
and we know r – c = 0,
Taking dot product of Eq. (i) by c we get
r.c – a.c = t (b.c)
0 – a.c. = t (b.c.)
t = – a.c
b.c
From Eqs. (i) and (ii) solution of r is
r = a a.c
b.c
Tetrahedron
Atetrahedron is a three dimensional figure formed
by four triangles.
In figure, ABC tetrahedron
ABC base
OAB, OBC, OCA faces
OA, OB, OC, AB, BC and CA edges
OA, BC, OB, CA, OC and AB pair of opposite
edges.
Properties of Tetrahedron
i. Atetrahedron in which all edges are equal is
called a regular tetrahedron.
ii. If two pairs of opposite edges of a tetrahedron
are perpendicular, then the opposite edges of the
third pair are also perpendicular to each other.
iii. The sum of the squares of two opposite edges
is the same for each pair of opposite edges.
iv. Any two opposite edges in a regular tetrahedron
are perpendicular.
v. Volume of a tetrahedron ABCD is
1
6 [a d,b d,c d]
where a, b, c and d are position vectors.
vi. Volume of a tetrahedren whose three
coterminous edges are in the right handed system
are a, hand C is given by 1
6 [a b c]
viii. centroid of tetrahedron is [a b c d]
4
where a, b, c and d are position vectors
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Algebra of Vectors, Modulus of Vectors,
Collinearity and Coplanarity of Vectors
1. If ABCD is a rhombus whose diagonals cut at the
origin 0, then OA + OB + OC + OD equals to
a. AB+ AC b. O
c. 2 (AB + BC) d. AC + BO
2. Let AD be the angle bisector of A of ABC
such that AD = AB + AC, then
a. = AB
AB AC , = AC
AB AC
b. = AB AC
AB
, =
AB AC
AC
c. = AC
AB AC , = AB
AB AC
d. = AB
AC , = AC
AB
3. If G is the centroid of a triangle ABC, then
GA + GB + GC equals to
a. 0 b. 3GA
c. 3GB d. 3GC
4. If O and O' denote respectively the circumcentre
and orthocentre of ABC, then 0'A + O'B + O'C
is equal to
a. O'O b. OO'
c. 20'0 d. O
5. Consider ABC and A1B
1C
1in such a way that
AB = A1 B
1and M, N, M1, N1 be the mid–points
of AB, BC, A,B1and BP1 respectively, then
a. MM1 = NN
1
b. CC1 = M
1
c. CC1 = NN
1
d. M1= BB
1
6. If the position vector of three points are
a –2b + 3c, 2a + 3b – 4c, – 7b + 10c, then the
three points are
a. collinear b. non–coplanar
c. non–collinear d. None of these
7. The pcsition vectors of the vertices A, B, C of a
ABC are ˆ ˆ ˆi j 3k and ˆ ˆ ˆ5i 2 j 6k respectively. The length of the bisector AD of the
angle LBAC where D is on the line segment BC,
is
a.15
2b.
11
2
c.1
4d. None of these
8. A vector coplanar with vectors ˆ ˆi j and ˆ ˆj k and
parallel to the vector ˆ ˆ ˆ2i 2 j 4k is
a. ˆ ˆi k b. ˆ ˆi j k
c. ˆ ˆi j k d. ˆ ˆ3i 3j 6k
9. If a, b are the position vectors of A, B respectively
and C is a point on AB produced such that
AC = 3 AB, then the position vector of C is
a. 3 a – 2b b. 3b – 2a
c. 3b + 2a d. 2a – 3b
10. Let D, E, F be the middle points of the sides BC,
CA, AB respectively of a triangle ABC.
Then, AD + BE + CF equals to
a. O b. O
c. 2 d. None of these
11. Let ABC be a triangle having its centroid at G. If
S is any point in the plane of the triangle, then
SA + SB + SC is equal to
a. SG b. 2SG
c. 3SG d. 0
12. The figure formed by four points ˆ ˆ ˆi j k ,
ˆ ˆ2i 3j , ˆ ˆ ˆ ˆ ˆ3i 5j 2k,k j
a. parallelogram b. rectangle
C. trapezium d. square
13. Given that the vectors a and b are non–collinear,
the values of x and y for which the vector equality
2u – v = w holds true if
u = xa + 2y b,v = – 2y a + 3xb, w = 4a – 2b are
a. x = 4 6
,y7 7
b. x = 10 4
, y7 7
c. x = 8 2
, y7 7
d. x = 2, y = 3
14. Three points with position vectors a b, c will be
collinear, if there exist scalars x, y, z such that
a. xa + yb = zc b. xa + yb + z c = 0
c. xa + yb + zc = 0 d. xa + yb = c
where x + y + z = 0
15. If the points P (a + 2b + c), Q (2a + 3b),
R (b + tc) are collinear, where a, b, c are three
non–coplanar vectors, the value of t is
a. –2 b. – 1/2
c. 112 d. 2
Exercise - 1
(Topical Problems)
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16. ABCDEF is a regular hexagon with centre at the
origin such that AD + EB + FC = ED. Then, is equal to
a. 2 b. 4
c. 6 d. 3
17. 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
a. A b. B
c. C d. 0
18. In a quadrilateral ABCD, the point P divides DC
in the ratio 1: 2 and Q is the mid-point of AC. If
AB + 2AD + BC – 2 DC = k PQ, then k is equal
to
a. – 6 b. – 4
c. 6 d. 4
19. If m1,m2, m3 and m, are respectively the
magnitudes of the vectors
1 2ˆ ˆ ˆ ˆ ˆ ˆa 2i j k,a 3i 4 j 4k ,
3ˆ ˆ ˆa i j k
4ˆ ˆ ˆa i 3j k , then the correct order of m
1, m
2,
m3 and m
4 is
a. m3 < m
1, < m
4 < m
2
b. m3 < m
1, < m
2 < m
4
c. m3 < m
4, < m
1 < m
2
d. m3 < m
4, < m
2 < m
1
20. If a = (2,1,– 1),b = (1,–1,0), c = (5, – 1,1), then
unit vector parallel to a + b – c but in opposite
direction is
a.1
3 ˆ ˆ ˆ(2i j 2k) b.
1 ˆ ˆ ˆ(2i j 2k)2
c.1 ˆ ˆ ˆ(2i j 2k)3
d. None of these
21. The vectors a = ˆ ˆ ˆi j mk , b = ˆ ˆ ˆi j (m )k
and c = ˆ ˆ ˆi j m k are coplanar, if m is equal to
a. 1
b. 4
c. 3
d. no value ofm for which vectors are coplanar
22. Given, p = ˆ ˆ ˆ2i 2 j 4k , a = ˆ ˆi j , b = ˆ ˆj k
c = ˆ ˆi k and P = xa + y b + zc, then x, y, z are
respectively
a.3 1 5
, ,2 2 2
b.1 3 5
, ,2 2 2
c.5 3 1
, ,2 2 2
d.1 5 3
, ,2 2 2
23. If 2 a + 3b – 5 c = 0, then ratio in which c divides
is
a. 3: 2 internally b. 3: 2 externally
c. 2: 3 internally d. 2: 3 externally
24. If C is the mid–point of AB and P is any point
outside AB, then
a. PA+ PB= PC b. PA + PB + 2 PC = 0
c. PA+ PB – 2PC=0 d. PA + PB+ PC = 0
25. Let a, b, c be three non–zero vectors such that no
two of these are collinear. If the vector a + 2b is
collinear with c, then a + 2b + 6c equals
a. a (0, a scalar)
b. b (0, a scalar)
c. c (0, a scalar)
d. 0
26. If a = ˆ ˆ ˆi j k , b = ˆ ˆ ˆi j k and c =
ˆ ˆxi (x 2) j k and if the vector c lies in the
plane of vectors a and b, then x equals
a. 0 b. 1
c. –2 d. 2
27. A, B, C, D, E, F in that order, are the vertices of
a regular hexagon with centre origin. If the position
vector of the3 vertices A and B are respectively,
ˆ ˆ ˆ4i 2 j k and ˆ ˆ ˆ3i j k , then DE is equal to
a. ˆ ˆ ˆ7i 2 j 2k b. ˆ ˆ ˆ7i 2 j 2k
c. ˆ ˆ ˆ3i j k d. ˆ ˆ ˆ4i 3j 2k 28. If the position vectors of the vertices of ABC
are ˆ ˆ ˆ ˆ ˆ ˆ3i j 2k, i 2 j 7k and ˆ ˆ ˆ2i 3j 5k then the ABC is
a. right angled and isosceles
b. right angled, but not isosceles
c. isosceles but not right angled
d. equilateral
29. Let two non-collinear unit vectors a and b form
an acute angle. A point P moves so that at any
time t the position vector OP (where, 0 is the
origin) is given by it cos t + b sin t. When P is
farthest from origin 0, let M be the length of OP
and u be the unit vector along OP. Then,
a.
ˆa bu
ˆa b
and M = 1/ 2ˆˆ(1 a.b)
b.
ˆa bu
ˆa b
and M = 1/ 2ˆˆ(1 a.b)
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c.
ˆa bu
ˆa b
and M = 1/ 2ˆˆ(1 2a.b)
d.
ˆa bu
ˆa b
and M = 1/ 2ˆˆ(1 2a.b)
30. The non–zero vectors a, band e are related by
a = 8b and e = –7 b. Then, the angle between a
and c is
a. b. 0
c.4
d.
2
31. The position vectors of P and Q are respectively
a and b. If R is a point on PQ such that
PR = 5PQ, then the position vector of R is
a. 5n – 4a b. 5b + 4a
c. 4b – 5a d. 4b + 5a
32. If ABCDEF is a regular hexagon with AB = a and
BC = b, then CE equals
a. b – a b. – b
c. b – 2a d. None
Product of Two Vectors
33. If a and b are two collinear vectors, then which
of the following are incorrect?
a. b = .a, for some scalar A.
b. ˆa b c. The respective components of a and bare
proportional
d. Both the vectors a and b have same direction
but different magnitudes
34. The projection of the vector ˆ ˆ ˆi+ 3j + 7k on the
vector ˆ ˆ ˆ7i j 8k is
a.60
122b.
30
144
c.60
114d.
60
111
35. If (a + b)· (a–b) = 8 and |a| = 8 |b|, then the values
of |a| and |b| are
a.16 2 2 2
,3 7 2 7
b.4 2 2 3
,3 7 3 7
c.12 2 4 2
,5 7 3 7
d. None of these
36. If ˆ ˆ ˆa 2i 2 j 3k , ˆ ˆ ˆb i 2 j k and ˆ ˆc 3i j
such that a + b is perpendicular to e, then the
value of is
a. 2 b. 4
c. 6 d. 8
37. If a· a = 0 and a .b = 0,then what can be conclude
about the vector b?
a. Any vector b. Zero–vector
c. Unit vector d. None of these
38. If the vertices A, B, C of a ABC have position
vectors (1,2, 3), (–1, 0, 0), (0,1, 2) respectively,
then ABC (ABC is the angle between the
vectors BA and BG), is equal to
a.2
b.
4
c. cos–1 10
102
d. cos–1 1
3
39. Let a, band e be three non–coplanar vectors and
let p, q and r be vectors defined by the relations.
p = b c
abc
, q =
c a
abc
and r =
a b
abc
Then, the value of label the expression (a + b)·
p + (b+ c)· q + (e+ a)· r is equal to
a. [x ab]2 b. [xbc]2
c. [x c a]2 d. 0
40. If x.a = x·b = x·c = 0, where x is a non-zero
vector. Then, [a × b b×c c × a] is equal to
a. [x a b]2 b. [x b c]2
c. [x ca]2 d. 0
41. If for a unit vector a,(x – a)· (x + a) = 12, then |x|
is equal to
a. 4 b. 2
c. 13 d. 11
42. For any two non–zero vectors a and b, |a| b + |b| a
and |a| b – |b| a are
a. parallel b. perpendicular
c. non–parallel d. None of these
43. If a, b, c are unit vectors such that a + b + c = 0,
then the value of a .b + b·c + c· a is
a. 0 b. – 1
2
c.3
2d. 2
44. The points A(l, 2, 7), B (2, 6,3) and C(3,10,–1)
are
a. collinear b. coplanar
c. non–collinear d. None of these
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45. The moment about the point M( –2,4, – 6) of the
represented in magnitude and position AB, where
the points A and B have the coordinates (1,2, – 3)
and (3, – 4,2) respectively, is
a. ˆ ˆ ˆ8i 9 j 14k b. ˆ ˆ ˆ2i 6 j 5k
c. ˆ ˆ ˆ3i 2 j 3k d. ˆ ˆ ˆ5i 8j 8k 46. Let a and b be unit vectors inclined at an angle
2 (0 < < n) each other, then | a+ b| < 1, if
a. = 2
b. <
3
c. > 2
3
d.
3
< <
2
3
47. a × [a × (a × b)] is equal to
a. (a × a) (b × a) b. a·(b × a) – b (a × b)
c. [a.a × b)]a d. (a.a) b × a)
48. If a = ˆ ˆ ˆ(i + j + k) = l and a × b = ˆ ˆj k , then b is
a. ˆ ˆ ˆi j k b. ˆ ˆ2 j k
c. i d. ˆ2i
49. If a = ˆ ˆ ˆi j k , b = ˆ ˆi j , c = i and (a × b) × c
= a + a – b, then + is equal to
a. 0 b. 1
c. 2 d. 3
50. [b × c c × a a × b] is equal to
a. [a b c] b. 2 [a b c]
c. [a b c]2 d. ax(b × c)
51.If a = ˆ ˆ ˆi 2 j 3k and
b = ˆ ˆ ˆ ˆ ˆ ˆi (a i) j (a j) k (a k) then length of
b is equal to
a. 12 b. 2 12
c. 3 14 d. 2 14
52. If a, b and c are unit coplanar vectors, then
[2a – b, 2b – c, 2c – a] is equal to
a. 1 b. 0
c. – 3 d. 3
53. [ ˆ ˆ ˆi k j ] + [ ˆ ˆ ˆk ji ] + [ ˆ ˆ ˆjk i ] is equal to
a. 1 b. 3
c. –3 d. –1
54. Let P, Q, Rand S be the points on the plane with
position vectors ˆ ˆ ˆ ˆ2i j,4i,3j and ˆ ˆ3i 2 j
respectively. The quadrilateral PQRS must be a
a. parallelogram, which is neither a rhombus nor
a rectangle
b. square
c. rectangle. but not a square
d. rhombus, but not a square
55. Two adjacent sides of a parallelogram ABCD are
given by AB = ˆ ˆ ˆ2i 10 j 11k and and
AD = ˆ ˆ ˆi 2 j 2k The side AD is rotated by an
acute angle a 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
a.8
9b.
17
9
c.1
9d.
4 5
9
56. If the vectors a = ˆ ˆ ˆi j 2k , b = ˆ ˆ ˆ2i 4 j k and
c = ˆ ˆ ˆi j k are mutually orthogonal, then
(, ) is
a. (–3, 2) b. (2, –3)
c. (–2, 3) d. (3, –2)
57. If p, q and r are perpendicular to q + r, r + p and
p + q respectively and if [p + q] = 6,|q + r| =
4 3 and r p 4 ,then | p+ q + r | is
a. 5 2 b. 10
c. 15 d. 5 e. 25
58. If the scalar product of the vector ˆ ˆ ˆi j 2k with
the unit vector along ˆ ˆ ˆmi 2 j 3k is equal to 2,
then one of the values of m is
a. 3 b. 4
c. S d. 6 e. 7
59. Which one of the following vector is of magnitude
6 and perpendicular to both a = ˆ ˆ ˆ2i 2 j k and
b = ˆ ˆ ˆi 2 j 2k ?
a. 2 ˆ ˆ ˆi j 2k b. 2 ˆ ˆ ˆ2i j 2k
c. ˆ ˆ ˆ3(2i j 2k) d. ˆ ˆ ˆ2(2i j 2k)
e. ˆ ˆ ˆ2(2i j 2k) 60. If |a| = 5, |b| = 6 and a .b = – 25, then Ia x b Iis
equal to
a. 25 b. 6 11
c. 11 5 d. 11 6
61. Vectors a and b are inclined at an angle = 1200.
If | a | = | b | = 2, then [(a + 3b) × (3a + b)]2 is
equal to
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a. 190 b. 275
c. 300 d. 320
e. 192
62. If the projection of the vector a on b is |a × b | and
if 3 b = ˆ ˆ ˆi j k , then the angle between a and b
is
a.3
b.
2
c.4
d.
6
63. If 2 a + 3b + c = 0, then a × b + b × c + c × a is
equal to
a. 6 (b×c) b. 3(b × c)
c. 2(b × c) d. 0
64. (x – y) x (x + y) = ..... where x,yR3
a. 2 (x × y) b. | x |2 – | y|2
c. – (x × y) d. None of these
65. Let a = ˆ ˆ2i k , b = ˆ ˆ ˆi j k and c = ˆ ˆ ˆ4i 3j 7k k, If r is a vector such that r × b = c × b and
r·a = 0, then value of r – b is
a. 7 b. – 7
c. – 5 d. 5
66. If the vectors a = ˆ ˆ ˆ ˆ ˆ2i k,b i j k , and
c = ˆ ˆ ˆ2i 3j k are coplanar, then the value of
is equal to
a. 2 b. 1
c. 3 d. – 1
67. If u, v, w are non–coplanar vectors and p, q are
real numbers, then the equality [3 u p v p w] –
[p v w q u] – [2 w q v q u] = 0 holds for
a. exactly two values of (p, q)
b. more than two but not 0 all values of (p, q)
c. all values of (p, q)
d. exactly one value of (p, q)
68. If r.a = 0, r.b = 0 and r– c = 0 for some non–zero
vector r. Then, the value of [a b c] is
a. 0 b.1
2
c. 1 d. 2
69. If the vectors, ˆ ˆ ˆ ˆ ˆ ˆi 2 j 3k, 2i 3j 4k ,
ˆ ˆ ˆi j 2k are linearly dependent, then the value
of A is equal to
a. 0 b. 1
c. 2 d. 3
70. If a and b are two non–zero, non–collinear vectors,
then ˆ ˆ ˆ ˆ ˆ ˆ2[abi]i 2[abj]j 2[abk]k [aba] is equal
to
a. 2(a × b) b. a × b
c. a + b d. None of these
71. If the volume of the parallelopiped with a, band c
as coterminous edges is 40 cu units, then the
volume of the parallelopiped having b + c, c + a
and a + b as coterminous edges in cubic units is
a. 80 b. 120
c. 160 d. 40
72. The volume of the tetrahedron having the edges
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆi 2 j k, i j k, i j k as coterminous,
is 2
3 cu unit. Then, A equals
a. 1 b. 2
c. 3 d. 4
73. If = x (a × b) + y (b × c) + z (c × a) and [a b c]
= 8
then x + y + z is equal to
a. 8 (a + b + c) b. · (a + b + c)
c. 8 (a + b + c) d. None of these
74. Volume of the parallelopiped having vertices at
O (0, 0, 0), A (2, – 2,1), B (5, – 4,4) and
C (t – 2, 4) is
a. 5 cu units b. 10 cu units
c. 15 cu units d. 20 cu units
75. If ˆ ˆ ˆ ˆi k, i j (1 ) k and ˆ ˆ ˆi j (1 )k
are three coterminal edges of a parallelopiped, then
its volume depends on
a. only b. only c. Both and d. Neither nor
76. The edges of a parallelopiped are of unit length
and are parallel to non–coplanar unit vectors
ˆˆ ˆa,b,c such that it 1ˆ ˆˆ ˆ ˆ ˆa,b b,c c.a2
. Then, the
volume of the parallelopiped is
a.1
2 cu unit b.
1
2 2 cu unit
c.3
2 cu unit d.
1
3 cu unit
77. The vector a = ˆ ˆ ˆi 2 j k in the plane of the
vectors b = ˆ ˆi j and c = ˆ ˆj k and bisects the
angle between b and c. Then, which one of the
following gives possible value of and ?
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a. = 1, = 1 b. = 2, = 2
c. = 1, = 2 d. = 2, = 1
78. If the volume of a parallelopiped with a×b, b × c,
c × a as coterminous edges is 9 cu units, then the
volume of the parallelopiped with
(a × b) × (b × c), (b × c) × (c×a), (c × a) × (a × b)
as coterminous edges is
a. 9 cu units b. 729 cu units
c. 81 cu units d. 27 cu units
e. 243 cu units
79. If a = ˆ ˆ ˆi j k , b = ˆ ˆ ˆi j k , c = ˆ ˆ ˆi j k and
d = ˆ ˆ ˆi j k , then observe the following lists
List I List II
A a·b 1. a – d
B b.c 2. 3
C [a bc ] 3 b·
D b × c 4 ˆ ˆ2i 2k
5 ˆ ˆ2 j 2k6 4
The correct match of List I to List II
Codes
A B C D A B C D
a. 3 1 2 6 b. 3 1 6 5
c. 1 3 2 6 d. 1 2 6 4
80. Three vectors ˆ ˆ ˆ ˆ ˆ ˆ7i 11j k,5i 3j 2k and
ˆ ˆ ˆ12i 8 j k forms
a. an equilateral triangle
b. an isosceles triangle
c. a right angled triangle
d. collinear
81. If the vectors ˆ ˆ ˆ2i 3j 4k and ˆ ˆ ˆi 2 j k and
ˆ ˆ ˆmi j 2k are coplanar, then the value of m is
a.5
8b.
8
5
c. – 7
4d.
2
3
82. If the volume of parallelopiped with coterminous
edges ˆ ˆ ˆ ˆ ˆ4i 5j k, j k and ˆ ˆ ˆ3i 9 j pk is 34
cu units, then p is equal to
a. 4 b. –13
c. 13 d. 6
83. If a· i = 4 then ˆ ˆ ˆ(a j).(2 j k) is equal to
a. 12 b. 2
c. 0 d. –12
84. The value of A, for which the four points
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2i 3j k, i 2 j 3k,3i 4 j 2k, i 6 j k are coplanar, is
a. 2 b. 4
c. 6 d. 8
85. The number of distinct real values of for which
the vectors 2 2ˆ ˆ ˆ ˆ ˆ ˆi j k, i j k and 2ˆ ˆ ˆi j k coplanar, is
a. zero b. one
c. two d. three
86. If a is perpendicular to b and c, |a| = 2, |b| = 3,
|c| = 4 and the angle between band c is 2, then
[a b c] is equal to
a. 4 3 b. 6 3
c. 12 3 d. 18 3
87. If the points whose position vectors are
6 ˆ ˆ ˆ2i j k , ˆ ˆ ˆ6i j 2k and ˆ ˆ ˆ14i 5j pk are
collinear, then the value of p is
a. 2 b. 4
c. 6 d. 8
88. The volume (in cubic unit) of the tetrahedron with
edges ˆ ˆ ˆ ˆ ˆ ˆi j k, i j k and ˆ ˆ ˆi 2 j k is
a. 4 b.2
3
c.1
6d.
1
3
Application of Vectors in Geometry
89. The vector equation of a plane which is at a
distance of 7 units from the origin and normal to
the vector ˆ ˆ ˆ3i 5j 6k is
a.3
70x +
5
70 y –
6
70 z = 7
b. 3x + 5y – 6z = 7
c. 3 70 x + 5 70 y – 6 70 z = 7
d. None of the above
90. Let P(3,2,6) be a point in space and Q be point
on the line r = ˆ ˆ ˆ(i j 2k) + ˆ ˆ ˆ(3i j 5k) .
Then, the value of 11for which the vector PQ is
parallel to the plane x – 4y + 3z = 1 is
a.1
4b. –
1
4
c. 1
8d. –
1
8
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91. Let the vectors, a, b, c and d be such that
(a × b) × (c × d) = 0. Let P1 and P
2 be planes
determined by the pairs of vectors a, b and c, d
respectively. Then, the angle between P1 and P
2
a. 0 b.2
c.3
d.
2
92. Equation of the plane through three points A, B
and C with position vectors
ˆ ˆ ˆ ˆ ˆ ˆ6i 2 j 2k,3i 2 j 4k and ˆ ˆ ˆ5i 7 j 3k is
a. r· ˆ ˆ ˆ(i j 7k) + 23 = 0
b. r· ˆ ˆ ˆ(i j 7k) = 23
c. r· ˆ ˆ ˆ(i j 7k) + 23 = 0
d. r· ˆ ˆ ˆ(i j 7k) = 23
93. For the lines L1 r = a + t (b + c) and L
2 r = b + s
(c + a), then L1 and L
2 intersect at
a. a b. b
c. a + b + c d. a + 2b
94. Let P(3, 2,6) be a point in space and Q be a point
on the line r = ˆ ˆ ˆ(i j 2k) + ˆ ˆ ˆ( 3i j 5k) .
Then, the value of 11for which the vector PQ is
parallel to the plane x – 4y + 3z = 1 is
a.1
4b. –
1
4
c.1
8d. –
1
8
95. A non–zero vector a is parallel to the line of
intersection of the plane determined by the vectors
ˆ ˆ ˆi, i j and the plane determined by the vectors
ˆ ˆ ˆ ˆi j, i k .The angle between a and
ˆ ˆ ˆi 2 j 2k is
a.2
b.
3
c.5
d.
4
96. The two variable vectors ˆ ˆ ˆ2xi yj 3k and
ˆ ˆ ˆxi 4yj 4k are orthogonal to each other, then
the locus of (x, y) is
a. hyperbola b. circle
c. straight line d. ellipse e. parabola
97. The angle between the straight lines
r = (2 – 3t) i + (1+ 2t) j + (2+6t) k and r=(1+ 4s)
+ (2 – 5) j + (8s – 1) k is
a. cos–1 41
34
b. cos –1 1
34
c. cos–1
43
63
d. cos–1 5 23
41
e. cos–1 34
63
98. Find the equation of the perpendicular drawn from
the origin to the plane 2x + 4y – 5z = 10.
a. r = (2k, 5k, 4k), k R
b. r = (2k, 4k, – 5k), k R
c. r = (3k, 4k, Sk), k R
d. None of these
99. If a, b, c are three non–coplanar vectors, then the
vector equation r = (1– p – q) a+pb+ q c represents
a. straight line
b. plane
c. plane passing through the origin
d. sphere
100. The vector equation of the plane passing through
the origin and the line of intersection of the planes
r.a = and r. b = , is
a. r· (a – b) = 0 b. r'(b – 11a) = 0
c. r'(a + b) = 0 d. r'(b + a) = 0
101. The cartesian form of the plane
r = (5 – 2t) i + (3 – t) j + (25 + t) k is
a. 2x – 5y – z – 15 = 0
b. 2x–5y+z–15=0
c. 2x – 5y – z + 15 = 0
d. 2x + 5y – z + 15 = 0
e. 2x + 5y + z + 15 = 0
102. The equation of the plane perpendicular to the
line x 1 y 2 z 1
1 1 2
and passing through the
point (2, 3, 1)
a. r. ˆ ˆ ˆ(i j 2k) = 1
b. r. ˆ ˆ ˆ(i j 2k) = 1
c. r. ˆ ˆ ˆ(i j 2k) = 7
d. r. ˆ ˆ ˆ(i j 2k) = 10
e. None of these
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103. If the planes r. ˆ ˆ ˆ(2i j 3k) = 0 and
r· ˆ ˆ ˆ( i 5j k) = 5 are perpendicular to each
other, then the value of 2 + is
a. 0 b. 2
c. 1 d. 3 e. 4
104. Match the terms of column I with the terms of
column II and choose the correct option from the
codes given below.
Column I Column - II
A 10 kg 1. Scalar
B 2 m North – West 2. Vector
C 40'
D 40 W
E 10–19C
F 20 m/s2
Codes
A B C D E
a. 2 1 1 1 2
b. 1 2 1 1 1
c. 1 2 1 1 2
d. 2 1 1 2 1
105. Match the terms of column I with the terms of
column II and choose the correct option from the
codes given below.
Column I Column -I
A. Times period 1. Scalar
B. Distance 2. Vector
C Force
D Velocity
E Work done
Codes
A B C D E
a. 1 1 2 2 1
b. 1 1 2 1 2
c. 2 1 2 2 1
d. 2 1 2 1 2
106. The vector equation r = ˆ ˆ ˆ ˆ ˆi 2 j k t(6 j k) represents a straight line passing through the points
a. (0,6, –1) and (1,–2, –1)
b. (0, 6, –1) and (–1, – 4, – 2)
C. (1,– 2, –1) and (1,4, – 2)
d. (1,– 2, –1) and (0, – 6,1)
107. In R2, find the unit vector orthogonal to unit vector
x = (cos , sin )
b. (– cos , – sin )
c. (– sin , cos )
d. (cos , sin )
108. Let a = ˆ ˆ ˆi j k , b = ˆ ˆ ˆi j k and c ˆ ˆ ˆi j k be
three vectors. A vector v in the plane of a and b,
whose projection on c is 1
3is given by
a. ˆ ˆ ˆi 2 j 3k
c. ˆ ˆ ˆ3i j 3k
b. ˆ ˆ ˆ3i 3j k
d. ˆ ˆ ˆi 3j 3k 109. Find the distance between the planes
r. ˆ ˆ ˆ ˆ ˆ(2i j 2k) 4and r.(6i 3j 9k 13 0)
a.5
3 14b.
10
3 14
c.25
3 14d. None of these
Scalar Triple Product and its Applications
110. If a, b and c are three non-coplanar vectors and
p, q are r are vectors defmed by p = b c
[abc]
,
q = c a
[abc]
and, then the value of
(a + b).p + (b + e).q + (c + a).r is equal to
a. 0 b. 1
c. 2 d. 3
111. If the volume of the parallelopiped formed by three
non–coplanar vectors a, band e is 4 cu units, then
[a × b b × c c × a] is equal to
a. 64 b. 16
c. 4 d. 8
112. If ˆˆ ˆa,b,c , are unit vectors such that
a × ( b × c ) = 1
2 b then angle between a and c is
a. /6 b. /4
c. /2 d. /3
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Exercise 2
(Miscellaneous Problems)
1. If a and b are unit vectors, then the greatest value
of | a + b | + |a – b| is
a. 2 b. 4
c. 2 2 d. 2
2. If a, band c are non–coplanar vectors and r is a
real number, then the vectors a + 2b + 3c, b + 4
c and (2– 1) c are non–coplanar for
a. no value of b. all except one value of c. all except two values of d. all values of f...
3. Vectors a and b are such that IaI= 1, Ibl = 4 and
a b = 2. If c = 2a × b – 3b, then the angle between
b and c is
a.6
b.
5
6
c.3
d.
2
3
4. The distance of the point ˆ ˆ3i 5k from the line
parallel to ˆ ˆ ˆ6i j 2k and passing through the
point ˆ ˆ ˆ8i 3j k is
a. 1 b. 2
c. 3 d. 4
5. Let a, band c be three non–zero vectors such that
no two of them are collinear and (a × b) × c = 1
3
|b| |c| a. If is the angle between vectors b and c,
then a value of sin is
a.2 2
3b. ˆ ˆi k
c. ˆ ˆi j d. ˆ ˆ ˆi 2 j k
6. A vector of magnitude 2 coplanar with
ˆ ˆ ˆi j k and ˆ ˆ ˆi 2 j k and perpendicular to
ˆ ˆ ˆi j k is
a. ˆ ˆi k b. ˆ ˆi k
c. ˆ ˆi j d. ˆ ˆ ˆi 2 j k
7. Let a, band c be three unit vectors such that a is
perpendicular to the plane of b and c. If the
angle between band c is 3
, then | a × b– x c | is
equal to
a. 1/3 b. 1/2
c. 1 d. 2
8. In a parallelogram ABCD, |AB| = a, |AD| = b and
|AC| = a, then DA·AB is equal to
a.1
2 (a2 + b – c) b.
1
2 (a2 –b2+c2)
c.1
4(a2 + b2 – c2) d. (b2 + c2 – a2)
9. Let 1 2 3 2 3ˆ ˆ ˆ ˆ ˆ ˆa i a j a k,b i b j b k
c = c1
2 3ˆ ˆ ˆi c j c k . If |c| = 1 and (a×b) × c = 0,
then
1 2 3
1 2 3
1 2 3
a a a
b b b
c c c is equal to
a. 0 b. 1
c.2 2a b d. |a × b|2
10. If the sum of two unit vectors is a unit vector,
then the magnitude of their difference is
a. 2 b. 3
c. 5 d. 7
11. The three vectors a, band c with magnitude 3, 4
and 5 respectively and a + b + c = 0, then the
value of a – b+ b· c+ c· a is
a. –23 b. –25
c. 30 d. 26
12. The vectors (a, a + 1,a + 2)(a + 3, a + 4, a + 5),
(a + 6, a + 7, a + 8) are coplanar for
a. a R b. a R
c. a = 3 d. None of these
13. If a = ˆ ˆ ˆi 2 j 3k , ˆ ˆ ˆb i 2 j k and
c = 3 ˆ ˆi j then p such that a + p b is at right
angle to c will be
a. 7 b. 9
c. 3 d. 5
14. Let a = ˆ ˆ ˆi 2 j k , b = ˆ ˆ ˆi j k and c = ˆ ˆ ˆi j k
A vector in the plane a and b whose projection on
c is 3
is
a. ˆ ˆ ˆi j 2k b. ˆ ˆ ˆ3i j 3k
c. ˆ ˆ ˆ4i j 4k d. ˆ ˆ ˆ4i j 4k
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15. If a = ˆ ˆ ˆi 2 j 3k , b = ˆ ˆ ˆ2i j k , c =
ˆ ˆ ˆi 2 j k and [a b c] = 6, then A is equal to
a. – 8 or 3 b. –9 or 3
c. –30r + 9 d. 8 or 5
16. Three points A,B and C with position vectors
a1 =
1ˆ ˆ ˆa 3i 2 j k ,
2ˆ ˆ ˆa i 3j 4k and
3ˆ ˆ ˆa 2i j 2k relative to an origin O. The
distance of A from the plane OBC is (magnitude)
a. 5 b. 3
c. 3 d. 2 3
17. If a and b are two vectors such that a.b < 0 and
|a – b | = | a × b|, then the angle between a and b is
a.3
4
b.
2
3
c.4
d.
3
18. If V is the volume of the parallelopiped having
three coterminous edges, as a, band c, then the
volume of the parallelopiped having three
coterminous edges as
. = (a.a) a + (a.b) b + (a.c) c
= (a.b) a + (b.b) b + (b.c) c
= (a.c) a + (b.c) b + (b.c) c
a. V3 b. 3V
c. V2 d. 2V
19. Let ˆ ˆa 2i j and ˆ ˆb 2i j . If c is a vector such
that a.c = | c|, |c – a| = 2.[2 and the angle between
a × b and c is 30°, then |(a × b) × c| is equal to
a. 2/3 b. 3/2
c. 2 d. 3
20. If the position vectors of P, Q, Rand S are
ˆ ˆ ˆ ˆ2i j, i 3j, and ˆ ˆi j respectively and
PQ || RS, then the value of is
a. –7 b. 7
c. – 6 d. None of these
21. The angle between the vectors
a = ˆ ˆ ˆ2i 2 j k
and b = ˆ ˆ ˆ6i 3j 2k
a. cos–1 3
11b. cos–1
2
11
c. cos–1 4
11d. cos–1
3
22
22. Three vectors a = ˆ ˆ ˆi j k , b = ˆ ˆ ˆi 2 j k and
c = ˆ ˆ ˆi 2 j k , then the unit vector perpendicular
to both a + band b + c is
a.i
3b. ˆ6k
c.k
3d.
ˆ ˆ ˆi j k
3
23. If a, b and c are unit coplanar vectors, then the
scalar triple product [2 a – b, 2b – c, 2 c – a] is
a. 2 b. –3
c. 0 d. None of these
24. The vectors a = ˆ ˆ ˆ2i j 2k , b = ˆ ˆi j. . If c is a
vector such that a.c. | c | and | c–a| = 2 2 ,
angle between a × b and c is 45°, then
I(a × b) × c] is
a.3 2
2b.
3
2
c.3 3
2d. None of these
25. The three vectors a = ˆ ˆ ˆi j k , b = ˆ ˆi j and c
= i and (a × b) × c = + a + b, then the value
of + is
a. 2 b. 3
c. 0 d. None of these
26. If [a × b b × c c × a] = [a b c]2, then A is equal
to
a. 1 b. 2
c. 3 d. 0
27. If the vectors AB = ˆ ˆ3i 4k and AC =
ˆ ˆ ˆ(5i 2 j 4k) are the sides ABC, then the length
of the median through A is
a. 18 b. 72
c. 33 d. 45
28. Let it and 6 be two unit vectors. If the vectors
c = ˆa 2b and d = ˆˆ5a 4b are perpendicular to
each other, then the angle between it and b is
a.6
b.
2
c.3
d.
4
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29. 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
a. r = 3r 3(q.q)
p(p.p)
b. r = – q + (q.p)
p(p.p)
c. r = q – (p.q)
p(p.p)
d. r = –3q + 3(p.q)
p(p.p)
30. If a = 1
10 ˆ ˆ(3i k) and b =
1
7– ˆ ˆ ˆ(2i 3j 6k) ,
then the value of (2 a – b)· [(a × b) × (a + 2b)] is
a. – 3 b. 5
c. 3 d. – 5
31. The vectors a and b are not perpendicular and c
and d are two vectors satisfying b × c = b × d and
a. d = 0. Then, the vector d is equal to
a. c + a.c
ba.b
b. b+ b.c
ca.b
c. c – a.c
ba.b
d. b – b.c
ca.b
32. If the vector ˆ ˆ ˆ ˆ ˆ ˆpi j k, i qj k and
ˆ ˆ ˆi j rk (q q r1) are coplanar, then the
value of pqr – (p + q + r) is
a. – 2 b. 2
c. 0 d. –1
33. Let a, band c be three non–zero vectors which
are pairwise non – collinear. If + 3b is collinear
with c and b + 2c is collinear with a, then
a + 3b + 6 c is
a. a + b b. a
c. c d. 0
34. Let a = ˆ ˆi k and c = ˆ ˆ ˆi j k , then the vector b
satisfying a × b + c = 0 and a b = 3, is
a. ˆ ˆ ˆi j 2k b. ˆ ˆ ˆ2i j 2k
c. ˆ ˆ ˆi j 2k d. ˆ ˆ ˆi j 2k
35. If the vectors a = ˆ ˆ ˆi j 2k , b = ˆ ˆ ˆ2i 4 j k and
c = ˆ ˆ ˆi j k are mutually orthogonal, then
(,) is equal to
a. (–3, 2) b. (2,–3)
c. (–2,3) d. (3,–2)
36. 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
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)
37. The position vector of the point, where the line
r = ˆ ˆ ˆ ˆ ˆ ˆi j k t (i j k) meets the plane
r. ˆ ˆ ˆ(i j k) = 5 is
a. ˆ ˆ ˆ5i j k b. ˆ ˆ ˆ5i 3j 3k
c. ˆ ˆ ˆ2i j 2k d. ˆ ˆ ˆ5i j k
38. The non–zero vectors a, b and c are related by
a = 8b and c = – 7 b. Then, the angle between
a and c is
a. b. 0
c.4
d.
2
39. If u and v are unit vector and is the acute
angle between them, then 2 u × 3 u is a unit
vector for
a. exactly two values of b. more than two values of c. no value of d. exactly one value of
40. Let a = ˆ ˆ ˆi j k , b = ˆ ˆi j 2k and
c = ˆ ˆ ˆxi (x 2) j k . If the vector c lies in the
plane of a and b, then x equals to
a. 0 b. 1
c. – 4 d. – 2
41. If (a × b) × c = a × (b × c), where a, band c are
any three vectors such that a. b = 0, b·c 0, then
a and c are
a. parallel
b. inclined at an angle of 3
between them
c. inclined at an angle of 6
between them
d. perpendicular
42. The value of a, for which the points, A, B and C
with position vectors ˆ ˆ ˆ2i j k ˆ ˆ3j 5k and
ˆ ˆ ˆai 3j k respectively are the vertices of a right
angled triangle with C = 2
are
a. – 2 and – 1 b. – 2 and 1
c. 2 and –1 d. 2 and 1
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43. If C is the mid–point of AB and P is any point
outside AB, then
a. PA + PB + PC = 0 b. PA + P B + 2 PC = 0
c. PA+ PB = PC d. PA + PB = 2PC
44. For any vector a, then the value of
(a × i ) 2 + (a × j ) 2 + (a × k ) 2 is
a. 4 a2 b. 2 a2
c. a2 d. 3a2
45. If ˆ ˆ ˆ ˆa i k,b xi j (1 x) k and
c = ˆ ˆyi xj (1 x y) k Then, [a b c] depends
on
a. Only x b. Only y
c. Both x and y d. Neither x nor y
46. Let a, band c be distinct non–negative numbers.
If the vectors ˆ ˆ ˆai aj ck , ˆ ˆi k and ˆ ˆ ˆci j bk lie
in a plane, then c is
a. the harmonic mean of a and b
b. equal to zero
c. the arithmetic mean of a and b
d. the geometric mean of a and b
47. If a, b, c are non–coplanar vectors and A is a real
number, then [(a + b) c] = [a (b + c) b] for
a. exactly two values of A
b. exactly three values of A
c. no value of A
d. exactly one value of A
48. Let a,b and c be three non–zero vectors such that
no two of these are collinear. If the vector a + 2b
is collinear with c and b + 3 c is collinear with a
( being some non–zero scalar), then a + 2b + 6 c
equals to
a. a b. b
c. c d. 0
49. A particle is acted upon by constant forces
ˆ ˆ ˆ4i j 3k and ˆ ˆ ˆ3i j k which displace it from
a point ˆ ˆ ˆi 2 j 3k to the point 5 I+ 41 + k. The
work done in standard units by the forces is given
by
a. 40 units b. 30 units
c. 25 units d. 15 units
50. If a, band c are non–coplanar vectors and is a
real number, then the vectors a + 2b + 3c,
b + 4 c and (2 –1) c are non–coplanar for
a. all values of A
b. all except one value of A
c. all except two values of A
d. no value of A
51. Let u, v, w be such that
| u | = 1, |v| = 2, |w| = 3
If the projection v along u is equal to that of w
along u and v, ware perpendicular to each other,
then |u – v + w | is equal to
a. 2 b. 7
c. 14 d. 14
52. Let a, band c be non–zero vectors such that
(a × b) × c = 1
3|b| c | a. If is the acute angle
between the vectors band c, then sine equals to
a.1
3b.
2
3
c.2
3d.
2 2
3
53. The value of [(a – b) (b – c) (c – a)] is equal to
a. 2 b. 1
c. 2[a be] d. 2
54. If ˆ ˆ ˆ ˆ ˆi j, j k, i k are the position vectors of
the vertices of a ABC taken in order, then A is
equal to Kerala
a.2
b.
5
c.6
d.
4
e.
3
55. If a, band c are three non–zero vectors such that
each one of them being perpendicular to the sum
of the other two vectors, then the value of
|a + b + c| 2 isa. |a|2 + |b|2 + |c|2 b |a| + |b| + |c|
c. 2|a|2 + |b|2 + |c|2 d.1
2 |a|2 + |b|2 + |c|2
56. Let a = ˆ ˆ ˆi j k , b = ˆ ˆ ˆi j k and c = ˆ ˆ ˆi j k
be three vectors. A vector v in the plane of a and
b, whose projection on c is 1
3 is given by
a. ˆ ˆ ˆ2i j 3k b. ˆ ˆ ˆ3i j 3k
c. ˆ ˆ ˆ3i j 3k d. ˆ ˆ ˆ3i j 3k 57. If r.a = r.b = r.c = 1 where a, b, c are any three
non–coplanar vectors, then r is
a. coplanar with a, b, C
b. parallel to a + b + C
c. parallel to b × c + c × a + a × b
d. parallel to (a × b) × c
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1. If the position vectors of the vertices A, B and C
are ˆ ˆ6i,6 j and k respectively w.r.t. origin O, the
volume of the tetrahedron OABC is
a. 6 b. 3
c.1
6d.
1
3
2. If three vectors ˆ ˆ ˆ2i j k , ˆ ˆ ˆi 2 j 3k and
ˆ ˆ ˆ3i j 5k are coplanar, then the value of A is
a. – 4 b. – 2
c. – 1 d. – 8
3. The vector perpendicular to the vectors
ˆ ˆ ˆ4i j 5k and ˆ ˆ ˆ2i j 2k whose magnitude
is
a. ˆ ˆ ˆ3i 6 j 2k b. ˆ ˆ ˆ3i 6 j 2k
c. ˆ ˆ ˆ3i 6 j 6k d. None of these
4. If in a ABC, 0 and 0' are the incentre and ortho
centre respectively, then (0' A + O' B + O' C) is
equal to
a. 2O'O b. O'O
c. OO' d. 2OO'
5. If a + b + c = 0 and Ia I= 5, Ib I= 3 and Ic I= 7,
then angle between a and b is
a.2
b.
3
c.4
d.
6
6. If u = a – b and v = a + b and | a | = |b| = 2, then
|u×v| is equal to
a. 2 216 (a.b) b. 216 (a.b)
c. 2 4 (a.b) d. 2 24 (a.b)
7. If the vectors a, band c are coplanar, then
a b c
a.a. a.b. a.c.
b.a. b.b b.c is equal to
a. 1 b. 0
c. –1 d. None of the above
8. A vector v is equally inclined to the X–axis,
Y–axis and Z–axis respectively, its direction
cosines are
MHT - CET Corner
a. < 1 1 1
, ,3 3 3
>
b. 216 (a.b)
c. 22 4 (a.b)
d. 22 4 (a.b)
9. If a, b, e are three non–coplanar vectors and
p, q, r are defined by the relations
p = b c
[abc]
, q =
c a
[abc]
and r =
b a
abc
a· p + b· q + c· r is equal to
a. 0 b.
c. 1 d. 3
10. The volume of a parallelopiped whose coterminous
edges are 2a, 2b, 2e, is
a. 2 [a b e] b. 4 [a b c]
c. 4 [a b c] d. 8 [a b c]
11. The position vectors of vertices of a 6 ABC are
ˆ ˆ ˆ ˆ ˆ4i 2 j, i 4 j 3k and ˆ ˆ ˆi 5j k respectively,,
then ABC is equal
a.6
b.
4
c.3
d.
2
12. Given p =
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ3i 2 j 4k,a i j,b j k,c i k
and p = x a + y b + z e, then x, y, z are
respectively
a.3 1 5
, ,2 2 2
b.1 3 5
, ,2 2 2
c.5 3 1
, ,2 2 2
d.1 5 3
, ,5 2 2
13. Volume of the parallelopiped having vertices at
O = (0, 0, 0), A = (2, – 2, 1),B (5, – 4,4) and
C = (1, –2, 4) is
a. 5 cu units b. 10 cu units
c. 15 cu units d. 20 cu units
14. If 2a + 3b – 5c = 0, then ratio in which c divides
AB is
a. 3 : 2 internally b. 3 : 2 externally
c. 2 : 3 internally d. 2 : 3 externally
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15. If the constant forces ˆ ˆ ˆ2i 5j 6k and
ˆ ˆ ˆi 2 j 6k act on a particle due to which it is
displaced from a point A (4, – 3, – 2) to a point
B(6, t – 3), then the work done by the forces is
2008
a. 10 units b. –10 units
c. 15 units d. –9 units
16. If the vectors ˆ ˆ ˆ ˆ ˆi 3j 2k, i 2 j represent the
diagonals of a parallelogram, then its area will be
a. 21 b.21
2
c. 2 21 d.21
4
17. If |a| = 2, |b| = 3 and a, b are mutually
perpendicular, then the area of the triangle whose
vertices are 0, a + b, a – b is
a. 5 b. 1
c. 6 d. 8
18. a × [a × (a × b)] is equal to
a. (a × a)· (b ×a) b. a·(b × a) –b (a × b)
c. [a·(a × b)] a d. (a·a) (b × a)
19. If the vectors a + Ab + 3c, –2a + 3 b – 4e and
a – 3b + 5c are coplanar, then the value of A is
a. 2 b. –1
c. 1 d. –2
20. If the vectors a = 2 2ˆ ˆ ˆ ˆ ˆ ˆi aj a ik,b i bj b k
c = 2ˆ ˆ ˆi cj c k are three non–coplanar vectors
and
2 3
2 3
2 3
a a 1 a
b b a a
c c 1 c
= 0, then the value of abc is
a. 0 b. 1
c. 2 d. –1
21. Let a = ˆ ˆ ˆ ˆ ˆ ˆ2i j k,b i 2 j k and
c = ˆ ˆ ˆi j 2k be three vectors. A vector in the
plane of b and c whose projection on a is of
magnitude 2
3
a. ˆ ˆ ˆ2i 3j 3k b. ˆ ˆ ˆ2i 3j 3k
c. ˆ ˆ ˆ2i 2 j 5k d. ˆ ˆ ˆ2i j 5k
22.a.(b c) b(a b)
b.(c a) (b c)
is equal to
a. 1 b. 2
c. 0 d. 23. If |a | = |b | = 1and | a + b+| = 3 , then the value
of (3a – 4b)· (2a + 5b) is
a. –21 b. – 21
2
c. 21 d.21
2
24. If a is perpendicular to band c, |a| = 2, |b| = 3,
|c| = 4 and the angle between b and c is 27, then
[a b c] is equal to
a. 4 3 b. 6 3
c. 12 3 d. 18 3
25. If a, band e are perpendicular to b + e, e + a and
a + b respectively and if |a + b | = 6, |b + c| = 8
and |c + a| = 10 then |a + b + c| is equal to
a. 5 2 b. 50
c. 10 2 d. 10 2
26. If vectors ˆ ˆ ˆi j k , ˆ ˆ ˆi j k and ˆ ˆ ˆ2i 3j k are coplanar, then is equal to
a. – 2 b. 3
c. 2 d. –3
27. Given a b |a| 1 and if (a + 3b) . (2a – b) = – 10,
then b is equal to
a. 1 b. 3
c. 2 d. 4
28. [a + b b + e e + a] ==[a b c], then
a. [a b c] = 1
b. a, b, c are coplanar
c. [a b c] = –1
d. a, b, c are mutually perpendicular
29. Area of rhombus is , where diagonals are
a = ˆ ˆ ˆ2i 3j 5k and b = ˆ ˆ ˆi j k
a. 21.5 b. 31.5
c. 28.5 d. 38.5
30. Let ABCD be a parallelogram whose diagonals
intersect at P and 0 be the origin, then
OA + OB + OC + OD equals
a. OP b. 2 OP
c. 3 OP d. 4 OP
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Answers
Exercise 1
1. (b) 2. (c) 3. (a) 4. (b) 5. (d) 6. (a) 7. (d) 8. (b) 9. (b) 10. (a)
11. (c) 12. (c) 13. (b) 14. (c) 15. (d) 16. (b) 17. (d) 18. (a) 19. (a) 20. (a)
21. (d) 22. (b) 23. (a) 24. (c) 25. (c) 26. (c) 27. (a) 28. (d) 29. (a) 30. (a)
31. (a) 32. (c) 33. (d) 34. (c) 35. (a) 36. (d) 37. (a) 38. (c) 39. (d) 40. (d)
41. (c) 42. (b) 43. (c) 44. (a) 45. (a) 46. (d) 47. (d) 48. (c) 49. (a) 50. (c)
51. (d) 52. (b) 53. (d) 54. (a) 55. (b) 56. (a) 57. (a) 58. (d) 59. (e) 60. (e)
61. (e) 62. (a) 63. (b) 64. (a) 65. (b) 66. (b) 67. (d) 68. (a) 69. (a) 70. (a)
71. (a) 72. (a) 73. (a) 74. (b) 75. (d) 76. (a) 77. (a) 78. (c) 79. (b) 80. (d)
81. (b) 82. (b) 83. (d) 84. (c) 85. (c) 86. (c) 87. (b) 88. (b) 89. (a) 90. (a)
91. (a) 92. (a) 93. (c) 94. (a) 95. (d) 96. (a) 97. (e) 98. (b) 99. (b) 100. (b)
101. (c) 102. (b) 103. (a) 104. (b) 105. (a) 106. (c) 107. (c) 108. (c) 109. (c) 110. (d)
111. (b) 112. (d)
Exercise 2
1. (c) 2. (c) 3. (b) 4. (c) 5. (a) 6. (a) 7. (c) 8. (a) 9. (d) 10. (b)
11. (b) 12. (a) 13. (d) 14. (c) 15. (a) 16. (c) 17. (a) 18. (a) 19. (b) 20. (c)
21. (c) 22. (b) 23. (c) 24. (c) 25. (c) 26. (a) 27. (c) 28. (c) 29. (b) 30. (d)
31. (c) 32. (a) 33. (d) 34. (a) 35. (a) 36. (d) 37. (b) 38. (a) 39. (d) 40. (d)
41. (d) 42. (d) 43. (d) 44. (b) 45. (d) 46. (d) 47. (c) 48. (d) 49. (a) 50. (c)
51. (c) 52. (d) 53. (a) 54. (e) 55. (a) 56. (b) 57. (c)
MHT–CET Corner
1. (a) 2. (d) 3. (c) 4. (a) 5. (b) 6. (a) 7. (b) 8. (c) 9. (b) 10. (c)
11. (d) 12. (b) 13. (b) 14. (a) 15. (c) 16. (b) 17. (c) 18. (d) 19. (d) 20. (d)
21. (a) 22. (a) 23. (b) 24. (c) 25. (d) 26. (c) 27. (c) 28. (b) 29. (c) 30. (d)
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28
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29
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
30
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
31
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
32
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
33
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
34
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
35
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
36
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
37
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
38
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
39
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
40
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
41
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
42
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
43
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
44
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
45
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
46