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SCALARS & VECTORS

SCALAR QUANTITIES

Physical quantities which can completely be specified by a number (magnitude)
having an appropriate unit are known as "SCALAR QUANTITIES".

Scalar quantities do not need direction for their description.
Scalar quantities are comparable only when they have the same physical dimensions.
Two or more than two scalar quantities measured in the same system of units are equal if they have the same magnitude and sign.
Scalar quantities are denoted by letters in ordinary type.
Scalar quantities are added, subtracted, multiplied or divided by the simple rules of algebra.

EXAMPLES

Work, energy, electric flux, volume, refractive index, time, speed, electric potential, potential difference, viscosity, density, power, mass, distance, temperature, electric charge etc.

VECTORS QUANTITIES

Physical quantities having both magnitude and direction
with appropriate unit are known as "VECTOR QUANTITIES".

We can't specify a vector quantity without mention of deirection.
vector quantities are expressed by using bold letters with arrow sign such as:

vector quantities can not be added, subtracted, multiplied or divided by the simple rules of algebra.
vector quantities added, subtracted, multiplied or divided by the rules of trigonometry and geometry.

EXAMPLES

Velocity, electric field intensity, acceleration, force, momentum, torque, displacement, electric current, weight, angular momentum etc.

REPRESENTATION OF VECTORS

On paper vector quantities are represented by a straight line with arrow head pointing the direction of vector or terminal point of vector.

A vector quantity is first transformed into a suitable scale and then a line is drawn with the help of the

scale choosen in the given direction.

Addition of vectors by Head to Tail method (Graphical Method)

Head to Tail method or graphical method is one of the easiest method used to find the resultant vector of two of more than two vectors.

DETAILS OF METHOD

Consider two vectors

and

acting in the directions as shown below:

In order to get their resultant vector by head to tail method we must follow the following steps:

STEP # 1

Choose a suitable scale for the vectors so that they can be plotted on the paper.

STEP # 2

Draw representative line

of vector

Draw representative line

of vector

such that the tail of

coincides with the head of vector

STEP # 3

Join 'O' and 'B'.

represents resultant vector of given vectors

and

i.e.

STEP # 4

Measure the length of line segment

and multiply it with the scale choosen initially to get the magnitude of resultant vector.

STEP # 5

The direction of the resultant vector is directed from the tail of vector

to the head of vector

ADDITION OF VECTORS

PARALLELOGRAM LAW OF VECTOR ADDITION

Acccording to the parallelogram law of vector addition:

"If two vector quantities are represented by two adjacent sides or a parallelogram
then the diagonal of parallelogram will be equal to the resultant of these two vectors."

EXPLANATION

Consider two vectors

. Let the vectors have the following orientation

parallelogram of these vectors is :

According to parallelogram law:

MAGNITUDE OF
RESULTANT VECTOR

Magintude or resultant vector can be determined by using either sine law or cosine law.

UNIT VECTOR-FREE VECTOR-POSITION VECTOR-NULL VECTOR

UNIT VECTOR

"A unit vector is defined as a vector in any specified direction whose magnitude
is unity i.e. 1. A unit vector only specifies the direction of a given vector. "

A unit vector is denoted by any small letter with a symbol of arrow hat (

A unit vector can be determined by dividing the vector by its magnitude.

For example unit vector of a vector A is given by:

In three dimensional coordinate system unit vectors

having the direction of the positive X-axis, Y-axi and Z-axis are used as unit vectors.These unit vectors are mutually perpendicular to each other.

FREE VECTOR

A vector that can be displaced parallel to itself and applied at any point is known as a FREE VECTOR.

A free vector can be specified by giving its magnitude and any two of the angles between the vector and coordinate axes.

POSITION VECTOR

Avector that indicates the position of a point in a coordinate system is referred to as POSITION VECTOR.

Suppose we have a fixed reference point O, then we can specify the position the position of a given point P with respect to point O by means of a vector having magnitude and direction represented by a directed line segment OP .This vector is called POSITION VECTOR.

In a three dimensional coordinate system if O is at origin then,O(0,0,0) and P is any point say P(x,y,z)

in this situation position vector of point P will be:

NULL VECTOR

A null vector is a vector having magnitude equal to zero.It is represented by

. A null vector has no direction or it may have any direction. Generally a null vector is either equal to resultant of two equal vectors acting in opposite directions or multiple vectors in different directions.

MULTIPLICATION & DIVISION OF VECTOR BY A NUMBER (SCALAR)

MULTIPLICATION
OF A VECTOR
BY A SCALAR

When a vector is multiplied by a positive number (for example 2, 3 ,5, 60 unit etc.) or a scalar only its magnitude is changed but its direction remains the same as that of the original vector.
If however a vector is multiplied by a negative number (for example -2, -3 ,-5, -60 unit etc.) or a scalar not only its magnitude is changed but its direction also reversed.

The product of a vector

by a scalar quantity (m) follows the following rules:

(m)

(m) which is called commutative law of multiplication.

m(n

) = (mn)

which is called associative law of multiplication .

(m + n)

= m

+ n

which is called distributive law of multiplication .

DIVISION
OF A VECTOR
BY A SCALAR

The division of a vector

by a scalar number (n) involves the multiplication of the vector by the reciprocal of the number (n) which generates a new vector.

Let n represents a number or scalar and m is its reciprocal then the new vector

is given by :

where m = 1/n

and its magnitude is given by:

The direction of

is same as that of

if (n) is a positive number.

The direction of

is opposite as that of

if (n) is a negative number.

RESOLUTION OF VECTOR

DEFINITION

The process of splitting a vector into various parts or components is called "RESOLUTION OF VECTOR"

These parts of a vector may act in different directions and are called "components of vector".

We can resolve a vector into a number of components .Generally there are three components of vector viz.
Component along X-axis called x-component
Component along Y-axis called Y-component
Component along Z-axis called Z-component

Here we will discuss only two components x-component & Y-component which are perpendicular to each other.These components are called rectangular components of vector.

METHOD OF RESOLVING
A VECTOR INTO
RECTANGULAR COMPONENTS

Consider a vector

acting at a point making an angle q with positive X-axis. Vector is
represented by a line OA.From point A draw a perpendicular AB on X-axis.Suppose OB and BA
represents two vectors.Vector OA is parallel to X-axis and vector BA is parallel to Y-axis.Magnitude
of these vectors are V_x and V_y respectively.By the method of head to tail we notice that the sum of these vectors is equal to vector

.Thus V_x and V_y are the rectangular components of vector

.
V_x = Horizontal component of

.
V_y = Vertical component of

MAGNITUDE OF
HORIZONTAL COMPONENT

Consider right angled triangle DOAB

MAGNITUDE OF
VERTICAL COMPONENT

Consider right angled triangle DOAB

ADDITION OF VECTORS BY RECTANGULAR COMPONENTS METHOD

INTRODUCTION

Rectangular component method of addition of vectors is the most simplest method to add a number of vectors acting in different directions.

DETAILS OF METHOD

Consider two vectors

making angles q₁ and q₂ with +ve x-axis respectively.

STEP #01

Resolve vector

into two rectangular components

and

Magnitude of these components are:

and

STEP #02

Resolve vector

into two rectangular components

and

Magnitude of these components are:

and

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STEP #03

Now move vector

parallel to itself so that its initial point (tail) lies on the terminal point (head) of vector

as shown in the diagram.

Representative lines of

and

are OA and OB respectively.Join O and B which is equal to resultant vector of

and

STEP #04

Resultant vector along X-axis can be determined as:

STEP # 05

Resultant vector along Y-axis can be determined as:

STEP # 06

Now we will determine the magnitude of resultant vector.

In the right angled triangle DBOD:

HYP² = BASE² + PERP²

STEP # 07

Finally the direction of resultant vector will be determined.

Again in the right angled triangle DBOD:

Where q is the angle that the resultant vector makes with the positive X-axis.
In this way we can add a number of vectors in a very easy manner.
This method is known as ADDITION OF VECTORS BY RECTANGULAR COMPONENTS METHOD

	COMMUTATIVE LAW OF VECTOR ADDITION
	Consider two vectors and . Let these two vectors represent two adjacent sides of a parallelogram. We construct a parallelogram
	OACB as shown in the diagram. The diagonal OC represents the resultant vector

	From above figure it is clear that:
	This fact is referred to as the commutative law of vectr addition .
	ASSOCIATIVE LAW OF VECTOR ADDITION
	The law states that the sum of vectors remains same irrespective of their order or grouping in which they are arranged. Consider three vectors , and
	Applying "head to tail rule" to obtain the resultant of (+ ) and (+ )
	Then finally again find the resultant of these three vectors :

	This fact is known as the ASSOCIATIVE LAW OF VECTOR ADDITION.

Multiplication of two vectors

MORE DETAILS ON VECTORS

There are two types of the multiplication of two vectors:

Scalar Product OR Dot Product
Vector Product OR Cross Product

SCALAR PRODUCT
OR
DOT PRODUCT

Type of multiplication of two vectors in which the product is a scalar quantity is referred to as "Scalar Product OR Dot Product"

MATHEMATICAL REPRESENTATION OF SCALAR PRODUCT

Consider two vectors and making an angle q with each other. Their scalar product or dot product is defined as:

Scalar product of two vectors and is a scalar quantity which is equal to the
product of the magnitudes of and and the cosine of the angle between them.

EXPLANATION

EXAMPLE # 01

When we multiply two vector quantities force and displacement we get work which is a scalar quantity. Therefore, we can say that work is the scalar product or dot product of force and displacement.

EXAMPLE # 02

When we multiply two vector quantities electric intensity and normal area we get electric flux which is a scalar quantity. Therefore, we can say that electric flux is the scalar product or dot product of electric intensity and normal area

COMMUTATIVE LAW FOR DOT PRODUCT

COMMUTATIVE LAW
FOR
DOT PRODUCT

This law states that :

"The scalar product of two vectors and is equal to the magnitude
of vector times the projection of onto the direction of vector . "

Consider two vectors

and

,the angle between them is q.

where

represents the projection of vector onto the direction of vector

Similarly,

Where

represents the projection of vector onto the direction of vector

Since

This shows that the dot product of two vectors does not chanfe with the change in the order of the vectors to be multiplied.
This fact is known as the commutative of dot product.

DISTRIBUTIVE LAW FOR DOT PRODUCT

DISTRIBUTIVE LAW FOR
DOT PRODUCT

According to distributive law for dot product:

PROOF

Consider three vectors

and

.Here we will use geometric interpretation of dot product by drawing projection as shown below. First we obtain the sum of vectors

and

by head to tail rule then we draw projection

and

from the terminal point of vector

respectively onto the direction of

.
The dot product

is equal to the projection of vector

onto the direction of

multiplied by the magnitude of

. i.e.

VECTOR PRODUCT

MORE DETAILS ON VECTORS

VECTOR PRODUCT
OR
CROSS PRODUCT

Type of multiplication of two vectors in which the product is also a vector quantity is referred to as "Vector Product OR Cross Product"

MATHEMATICAL REPRESENTATION OF VECTOR PRODUCT

Consider two vectors and making an angle q with each other. Their vector product or cross product is defined as:

vector product of two vectors and is a third vector whose magnitude is equal
to the product of the magnitudes of and and the sine of the angle between them. The direction of the product vector is perpendicular to the plane containing both the vectors.

DIRECTION OF THIRD VECTOR

The vector which represents the cross product or vector product is perpendicular to the plane containing vectors and and points in the direction in such a way as to make A , B and C vectors in order.
Direction of third vector C can be determined by using right hand rule:

EXPLANATION

EXAMPLE # 01

When we multiply two vector quantities force arm and force we get torque which is also a vector quantity. Therefore, we can say that torque is the vector product or cross product of force arm and force.

EXAMPLE # 02

When we multiply two vector quantities velocity and magnetic flux density we get force which is also a vector quantity. Therefore, we can say that force is the vector product or cross product of velocity and magnetic flux density.

AREA OF PARALLELOGRAM


	If two sides of a parallelogram are represented by two vectors A and B, then the magnitude of their cross product will be equal to the area of parallelogram i.e.

	PROOF
	Consider a parallelogram OABC whose two sides are represented by two vectors A and B as shown. The area of parallelogram OABC is equal to :
	D = hA----------------(1)


	Draw perpendicular CD on side OA.
	Consider right angled triangle COD
	sinq = h/OC h= OC sinq h= B sinq
	Putting the value of h in equation (1), we get,
	D = B sinq A D = AB sinq
	OR





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2nd chapter of physics

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