Multiplication of vector...p u vr uZ v B B u() W E upE W B uMB LAMI ’S THEOREM Contact :...

Multiplication of vector

MULTIPLICATION OF VECTORS

Two types of multiplication:

1. Scalar (or dot) product of two vectors.

2. Vector (or cross) product of two vectors.

1. Scalar Product of Two Vectors

(1) Definition : The scalar product (or dot product) of two

vectors is defined as the product of the magnitude of two

vectors with cosine of angle between them.

Thus, if there are two vectors and having angle q

between them, then their scalar product written as is

defined as

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A B.A B

cosAB q.A B

(2) Properties : (i) It is always a scalar which is positive

if angle between the vectors is acute (i.e., < 90°) and

negative if angle between them is obtuse (i.e. 90°< q <

180°).

(ii) It is commutative, i.e.

(iii) It is distributive, i.e.

(iv) As by definitionwww.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

q

A

B

Fig. 0.10

. .A B B A

CABACBA ..)(.

qcos. ABBA

The angle between the vectors


In


2. Geometrical interpretation of scalar product

Let OA & OB represent vectors a & b respectively.

Then,

a = |a| = |OA| = OA

b = |b| = |OB| = OB

Let, M, N be the feet of the

perpendiculars from A, B on OB, OA respectively.

Then, magnitude of projection of a on b

= OM = OA cosq ( cosq = OM/OA in DOMA)

= a cosq


b

A N a O

M B

a.b = ab cosq = a(a cosq) = b. (projection of a on

b)

similarly magnitude of projection of b on a.

= ON = OB cosq ( cosq = ON/OB in DONB)

= b cosq

a.b = ab cosq = a(b cosq) = a. projection of b on a.

Thus a.b can be defined as the product of the

modulus of one vector and the projection of the

other vector upon it.


(3) Example : (i) Work W : In physics for constant

force work is defined as,

…(i)

But by definition of scalar product of two vectors,

…(ii)

So from eqn (i) and (ii) i.e. work is the scalar

product of force with displacement.


cosW Fs q

. cosF s Fs q

.W F s

In


Magnetic flux through an area is given by

…(i)

But by definition of scalar product

...(ii)


cosd Bds q

. cosB d s Bds q

q

sd

B

O

Fig. 0.11

(iv) Potential energy of a dipole U : If an electric

dipole of moment is situated in an electric field

or a magnetic dipole of moment in a field of

induction the potential energy of the dipole is

given by :

and


p

E M

B

.EU p E .BU M B

VECTOR PRODUCT OF TWO VECTORS

(1) Definition : The vector product or cross product

of two vectors is defined as a vector having a

magnitude equal to the product of the magnitudes

of hand screw rule.

Thus, if and are two vectors, then their vector

product written as is a vector defined by


C A B

A B

A B C

ˆsinC A B AB nq

The direction of i.e. is perpendicular to the plane containing

vectors and and in the sense of advance of a right handed screw

rotated from (first vector) to (second vector) through the

smaller angle between them. Thus, if a right handed screw whose

axis is perpendicular to the plane framed by and is rotated

from to through the smaller angle between them, then the

direction of advancement of the screw gives the direction of

i.e.


Fig. 0.12

A B C

A B

A B

BA

BA

A B C

(2) Properties :

(i) Vector product of any two vectors is always a vector

perpendicular to the plane containing these two vectors,

i.e., orthogonal to both the vectors and though the

vectors and may or may not be orthogonal.

(ii) Vector product of two vectors is not commutative,

i.e., [but

Here it is worthy to note that

i.e. in case of vector and magnitudes are equal

but directions are opposite.www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

A B

A B

A B B A ]B A

| | | | sinA B B A AB q

A B B A

(iii) The vector product is distributive when the order

of the vectors is strictly maintained, i.e.

(iv) The vector product of two vectors will be maximum

when i.e.,

i.e. vector product is maximum if the vectors are

orthogonal.

(v) The vector product of two non- zero vectors will be

minimum when minimum = 0, i.e.,

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( )A B C A B A C

sin max 1,q 90oq

maxˆ[ ]A B AB n

| sin |q 0oq

min[ ] 0A B

i.e. if the vector product of two non-zero vectors

vanishes, the vectors are collinear.

(vi) The self cross product, i.e., product of a vector by

itself vanishes, i.e., is null vector

(vii) In case of unit vector so that

(viii) In case of orthogonal unit vectors, in

accordance with right hand screw rule :


ˆsin0 0oA A AA n

ˆ ˆ 0n n ˆ ˆˆ ˆ ˆ ˆ 0i i j j k k

ˆˆ ˆ, ,i j k


i

j

k

i

j

k

Fig. 0.13

(x) In terms of components

(3) Example : Since vector product of two vectors is a

vector, vector physical quantities (particularly

representing rotational effects) like torque, angular

momentum, velocity and force on a moving charge in a

magnetic field and can be expressed as the vector

product of two vectors. It is well – established in physics

that :www.gangwarinstitue.com Contact : 8400-582-582, 8604-582-582

ˆˆ ˆ

x y z

x y z

i j k

A B A A A

B B B

ˆ( )y z z yi A B A B ˆˆ( ) ( )z x x z x y y xj A B A B k A B A B

(i) Torque

(ii) Angular momentum

(iii) Velocity

(iv) Force on a charged particle q moving with

velocity in a magnetic field is given by

(v) Torque on a dipole in a field

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r F

L r p

v r

v B ( )F q v B

E p E

B M B

LAMI’S THEOREM


i.e. for any triangle the ratio of the sine of the angle

containing the side to the length of the side is a constant.

For a triangle whose three sides are in the same order

we establish the Lami's theorem in the following

manner. For the triangle shown

[All three sides are taken in order]

…(i)

…(ii)

Pre-multiplying both sides by


0a b c

a b c

a

If


If

Dividing through out by abc, we have


sin sin sin

a b c

If a vector is perpendicular to the

vector Then the value of is;

(A) –1 (B) 1/2

(C) –1/2 (D) 1.


ˆˆ ˆ2 3 8i j k

ˆˆ ˆ4 4 .j i k

01

A particle acted upon by constant forces

and is displaced from the point

to point . The total work done by the

forces in SI unit is;

(A) 20 (B) 24

(C) 50 (D) 30.


ˆˆ ˆ4 3i j k ˆˆ ˆ3i j k

ˆˆ ˆ5 4i j k

02

ˆˆ ˆ2 3i j k

A body, constrained to move in the Y-direction is

subjected to a force given What

is the work done by this force in moving the body

a distance 10 m along the Y-axis;

(A) 20 J (B) 150 J

(C) 160 J (D) 190 J.


ˆˆ ˆ( 2 15 15 ) .F i j k N

03

Vector which is perpendicular to is

(A) (B)

(C) (D) All of these.


ˆ ˆ( cos sin )a i b jq q

ˆ ˆsin cosb i a jq q1 1ˆ ˆsin cosi ja b

q q

ˆ5k

04

If vector & are

functions of time, then the value to t at which they

are orthogonal to each other is;


ˆ ˆcos sinA ti tj ˆ ˆcos sin2 2

t tB i j

05

A vector points vertically upward and points

towards north. The vector product is

(A) Zero

(B) Along west

(C) Along east

(D) Vertically downward


A B

A B

06

If , then the value of is;


| | 3( . )A B A B | |A B

07

The moment of the force, at (2, 0, –3)

about the point (2, –2, –2) is given;


ˆˆ ˆ4 5 6F i j k

08

A angle between the vectors and is q. The

value of the triple product is

(A) A2B (b) Zero

(C) A2B sinq (d) A2B cosq


A B

.( )A B A

09

What is the unit vector perpendicular to the

following vectors and


ˆˆ ˆ2 2i j k ˆˆ ˆ6 3 2i j k

10

Which

Thank you 9-8

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