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88Vectors

MATHEMATICS – MHT-CET Himalaya Publication Pvt. Ltd.

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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89Vectors


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 î and j and} at

right angle.

i.e.

The vectors x ˆ î 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 î, j,k i.e. if



90Vectors


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



91Vectors


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 î 2 j

that has magnitude 7 units.

a.2 3ˆ î j3 8

b.5 7ˆ î j

85

c.4 2ˆ î j6 8

d.7 14ˆ î j5 5

Sol (d) We have. a = ˆ î 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 î j5 5

• Example 4The position vector of the vertices P, Q and R of

a triangle are ˆ ˆ ˆ ˆ ˆ î 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.



92Vectors


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 î, 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 âi 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 â 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 .



93Vectors


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



94Vectors


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 î.i j. j k.k 1

• ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ î. 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 î i j j k k = 0

• ˆ ˆ ˆ ˆ ˆ ˆ ˆ î 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 â 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 î j k

a a a

b b b



95Vectors


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 î 2 j 2k

17

Sol (a) We have,

OA = ˆ ˆ ˆ2i j k OB = ˆ ˆ î j k

OC = ˆ ˆ2i 3kNow, AB = OB – OA

= ˆ ˆ ˆ ˆ ˆ ˆ(i j 2k) (2i j k) = ˆ ˆ î 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 î j k

1 2 1

0 1 2

= ˆ ˆ ˆ ˆ ˆ î (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 â 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 â 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 â 2i 3j k,b i k and



96Vectors


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 î 3j 2k

and (a + b) × (b + c) =

ˆ ˆ î j k

1 3 2

1 2 0

= ˆ ˆ î(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 â 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)



97Vectors


Example 11

If ˆ ˆ ˆ ˆ ˆ î (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 î (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 î.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 î 4 j k and ˆ ˆ î 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 î 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.



98Vectors


(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



99Vectors


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 î 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 î j and ˆ ˆj k and

parallel to the vector ˆ ˆ ˆ2i 2 j 4k is

a. ˆ î k b. ˆ î j k

c. ˆ î 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 î 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)



100Vectors


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 â 2i j k,a 3i 4 j 4k ,

3ˆ ˆ â i j k

4ˆ ˆ â 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 î j mk , b = ˆ ˆ î j (m )k

and c = ˆ ˆ î 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 î j , b = ˆ ˆj k

c = ˆ î 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 î j k , b = ˆ ˆ î 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.

â bu

â b

and M = 1/ 2ˆˆ(1 a.b)

b.

â bu

â b

and M = 1/ 2ˆˆ(1 a.b)



101Vectors


c.

â bu

â b

and M = 1/ 2ˆˆ(1 2a.b)

d.

â bu

â 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. â 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 î+ 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 â 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



102Vectors


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 î j k b. ˆ ˆ2 j k

c. i d. ˆ2i

49. If a = ˆ ˆ î j k , b = ˆ î 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 î 2 j 3k and

b = ˆ ˆ ˆ ˆ ˆ î (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 î 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 î 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 î j 2k , b = ˆ ˆ ˆ2i 4 j k and

c = ˆ ˆ î 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 î 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 î 2 j 2k ?

a. 2 ˆ ˆ î 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



103Vectors


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 î 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 î 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 î 2 j 3k, 2i 3j 4k ,

ˆ ˆ î 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 î 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 î k, i j (1 ) k and ˆ ˆ î 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 â,b,c such that it 1ˆ ˆˆ ˆ ˆ â,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 î 2 j k in the plane of the

vectors b = ˆ î 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 ?



104Vectors


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



e. 243 cu units

79. If a = ˆ ˆ î j k , b = ˆ ˆ î j k , c = ˆ ˆ î j k and

d = ˆ ˆ î 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 î 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 î j k, i j k and 2ˆ ˆ î 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 î j k, i j k and ˆ ˆ î 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



105Vectors


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 î, i j and the plane determined by the vectors

ˆ ˆ ˆ î j, i k .The angle between a and

ˆ ˆ î 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



106Vectors


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 î 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 î j k , b = ˆ ˆ î j k and c ˆ ˆ î 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 î 2 j 3k

c. ˆ ˆ ˆ3i j 3k

b. ˆ ˆ ˆ3i 3j k

d. ˆ ˆ î 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 â,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



107Vectors


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 î k

c. ˆ î j d. ˆ ˆ î 2 j k

6. A vector of magnitude 2 coplanar with

ˆ ˆ î j k and ˆ ˆ î 2 j k and perpendicular to

ˆ ˆ î j k is

a. ˆ î k b. ˆ î k

c. ˆ î j d. ˆ ˆ î 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 â i a j a k,b i b j b k

c = c1

2 3ˆ ˆ î 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 î 2 j 3k , ˆ ˆ ˆb i 2 j k and

c = 3 ˆ î 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 î 2 j k , b = ˆ ˆ î j k and c = ˆ ˆ î j k

A vector in the plane a and b whose projection on

c is 3

is

a. ˆ ˆ î j 2k b. ˆ ˆ ˆ3i j 3k

c. ˆ ˆ ˆ4i j 4k d. ˆ ˆ ˆ4i j 4k



108Vectors


15. If a = ˆ ˆ î 2 j 3k , b = ˆ ˆ ˆ2i j k , c =

ˆ ˆ î 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 â 3i 2 j k ,

2ˆ ˆ â i 3j 4k and

3ˆ ˆ â 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 â 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 î 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 î j k , b = ˆ ˆ î 2 j k and

c = ˆ ˆ î 2 j k , then the unit vector perpendicular

to both a + band b + c is

a.i

3b. ˆ6k

c.k

3d.

ˆ ˆ î 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


24. The vectors a = ˆ ˆ ˆ2i j 2k , b = ˆ î 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 î j k , b = ˆ î j and c

= i and (a × b) × c = + a + b, then the value

of + is

a. 2 b. 3


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 = â 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



109Vectors


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 î 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 î k and c = ˆ ˆ î j k , then the vector b

satisfying a × b + c = 0 and a b = 3, is

a. ˆ ˆ î j 2k b. ˆ ˆ ˆ2i j 2k

c. ˆ ˆ î j 2k d. ˆ ˆ î j 2k

35. If the vectors a = ˆ ˆ î j 2k , b = ˆ ˆ ˆ2i 4 j k and

c = ˆ ˆ î 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 î 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 î j k , b = ˆ î 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 âi 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



110Vectors


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 â 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 âi aj ck , ˆ î 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 î 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 î 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 î j k , b = ˆ ˆ î j k and c = ˆ ˆ î 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



111Vectors


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



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



112Vectors


15. If the constant forces ˆ ˆ ˆ2i 5j 6k and

ˆ ˆ î 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 î 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 î aj a ik,b i bj b k

c = 2ˆ ˆ î 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 î 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 î j k , ˆ ˆ î 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 î 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



113Vectors


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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