Monday, May 9, 2011

ADDITION OF VECTORS BY RECTANGULAR COMPONENTS METHOD

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.

RESOLUTION OF VECTOR

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 Vx and Vy respectively.By the method of head to tail we notice that the sum of these vectors is equal to vector .Thus Vx and Vy are the rectangular components of vector .
Vx = Horizontal component of .
Vy = Vertical component of .
MAGNITUDE OF
HORIZONTAL COMPONENT

Consider right angled triangle DOAB
MAGNITUDE OF
VERTICAL COMPONENT

Consider right angled triangle DOAB

MULTIPLICATION & DIVISION OF VECTOR BY A NUMBER (SCALAR)

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.

UNIT VECTOR-FREE VECTOR-POSITION VECTOR-NULL VECTOR

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.

PARALLELOGRAM LAW OF VECTOR ADDITION

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.

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

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 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 .
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.