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Scalars and vectors

Scalars: It is a physical quantity having only magnitude but no direction.Ex: Distance, speed, mass, time, energy, density, temperature

Vectors: The physical quantities which has both magnitude as well as direction are called vectors.Ex: Displacement, velocity, weight, momentum etc.

Representation of a vector: Vectors are represented by an arrow drawn to a particular scale. Here the length PQ is proportional to the magnitude & the arrow head refers the direction. Avector is represented by a letter with an arrow over it i.e A .

Classification of vectors: 1)

Unit vector ( Aˆ ): A vector having unit magnitude is called unit vector. It is ˆ. represented as A

ˆ A

A A

i.e

unit vector

Vector magnitude of that vector

Note: 1)The unit vectors along x, y, z axis are given by ˆi, ˆj

&

kˆ

respectively. 2) In a unit vector its magnitude is non-zero i.e A 1.

0

Equal vectors: Two vectors are said to be equal if they have same magnitude as well as in the same direction.

2.

Null or Zero vector: A vector with which zero magnitude.

3.

Parallel vectors: Two or more vectors are said to be parallel if they have same direction. 6

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Antiparallel: Two or more vectors are said to be antiparallel if they are opposite in direction

5.

Negative vectors: Two anti-parallel vectors with same magnitude are said to be negative of each other.

6.

Concurrent vectors: Vectors which intersect at a point or vectors which have common initial point. 1)Co –planar: Vectors in a same plane are called co-planar vectors. 2)Collinear vectors: vectors along the same line.

Triangle law of addition: Statement: If two vectors a

&

b are

represented by two sides of a triangle in the same order, then the third side drawn in opposite sense represents the sum of vectors.

Parallelogram law of vector addition: Statement: If two vectors acting at a point are represented both in magnitude & in direction by the adjacent sides of a parallelogram drawn from a point, then the resultant is represented both in magnitude & in direction by the diagonal of the parallelogram drawn from that point. Let p & q be the two vectors at O. The diagonal Oc of the parallelogram OACB is given by the diagonal

R

p

q

Subtraction of vectors: Suppose a vector b is to be subtracted from another vector a , mathematically it can be written as

a

b

a

b

It’s a reverse process of addition. 7

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For more: www.puctime.com Resolution of a vector: The process of splitting up a vector into two or more vectors is known as resolution of a vector. If a vector is resolved into 2 vectors which are mutually perpendicular, then these vectors are called rectangular components of the given vector. Let us suppose the vector p has to be resolved into its rectangular components. To do this, taking the initial point of p has to be resolved along Ox & Oy direction. Draw perpendicular from c to x & y axis to meet at A &B. Let the components be px & p y . From the rectangle OABC

p

px

py

The magnitude of rectangular components are given by: From

le

OCA

cos

px

sin

OA p x OC p pcos py Ac

p OC psin

py Again in

le

CA OA

OCA tan tan

1

(1)

(2)

py Px

py px

This gives the direction of p To find magnitude, square & add equation (1) & (2)

px2

p2y

p2 cos2

p2 sin2

p2 (cos2 sin2 ) p2 px2 p2y p

px2

p2y

Multiplication of a vector by a scalar: On multiplying a vector P by a scalar or a number `n`, a vector R is obtained such that

R

nP

The direction of R is same as that of P . 8

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Scalar or Dot product:

If the product of 2 vectors results in a scalar quantity, it is a scalar product. If A and B are two vectors & can be represented by

A.B

be the angle between them, then the scalar product

ABcos

Hence Dot product can be defined as the product of 2 vectors is equal to the product of the magnitude of one vector & the component of the second in the direction of the first vector. Ex: Work = F × displacement

= FS cosθ In terms of components, dot product can be expressed as follows: Let a

ax ˆi

ay ˆj

az kˆ

b

bx ˆi

by ˆj

b z kˆ

(ax ˆi

ˆ (b ˆi b ˆj b k) ˆ ay ˆj azk). x z y ax bx ˆi.iˆ ax by ˆi.jˆ ax b z ˆi. kˆ ˆ a b ˆj.iˆ a b ˆj.jˆ a b ˆj.k

Then a . b

y x

y y

ˆˆ az bx k.i

y

ˆˆ a zby k.j

z

ˆ ˆ a zb zk.k

Here the dot product of 2 mutually perpendicular vectors are zero. i.e ˆi . ˆj

a.b

ˆj. kˆ

ax bx

ˆi .kˆ

0

ax by

az bz

Cross or vector product: If the product of two vectors results in a vector, then it is called as vector product. If A & B are the 2 vectors & θ be the angle between them, then the vector product is written as

A

B

ABsin nˆ

Where nˆ is the unit vector in the direction of its resultant. Ex: 1) Angular momentum L 2) Torque

r

r

p

F

B A Note : 1) A B 2) Cross product is non-distributive

A

B

c

A

B

A

c 9

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3) ˆi

ˆi 4) ˆi ˆj ˆj ˆi

ˆj ˆj ˆ k; ˆ k;

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kˆ

0

ˆj kˆ ˆi ; kˆ ˆi ; ˆi kˆ ˆj

ˆi kˆ

ˆj ˆj

5) The Cross product is given by

a

b

kˆ

ˆi ax

ˆj ay

az

bx

by

bz

Faraday Michael Faraday's interest in knowledge for its own sake often baffled people of a more practical bent. British Prime Minister William Gladstone, observing Faraday performing a particularly unlikely experiment one day, pointedly asked him how useful such a 'discovery' could possibly be. "Why," Faraday smartly replied, "you will soon be able to tax it!"

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