## Computer

Computers have become the part of everybody's life. Engineers and scientists use computers for difficult/tedious mathematical calculations. The commercial artists use computer as a substitute for his/her canvas. The business executives use the same as a decision-support system. It is used as an input/output system in various organizations like railways, airlines, banks, universities, hospitals and most other service providers. It is also used by children for games and other recreational purposes. Experts are trying to develop systems that will recognize the human voice and ultimate interact with the human beings. Expert systems have already been developed to take decisions like human beings. It is fact difficult to visualize an area where computer has not been used or at least is not likely to be used in the near future. Because of the tremendous impact of computers on the society, it is worthwhile to learn a little about the history development of computers. its functions and the history of the developments computer are discussed in-depth

### TYPES OF COMPUTERS

Computers can be classified into three broad categories, namely
() Digital,
(i) Analog, and
(ii) Hybrid, the details of each category are given below

### Digital Computer

A Digital Computer deal with digitals data that can be storage into binary format. Binary data are made of two digits 0 and 1. Such data may be stored in a bit in the computer's memory. The bit may be in either of two states, namely, "on state and off stat", which respectively represent the numbers 1 and 0. Therefore, it is required to convert any type of data (even a graphic image or a musical note ) into binary format before it is stored in computer memories

### Microcomputer

A Microcomputer is one whose CPU is a microprocessor. The progress in the field of microcomputers has been phenomenal. From Intel's 8088 chip in 1980's it has changed to Intel's 80286, 80386, 80486 and Pentium. Microcomputers are used by engineers, business executives and commercial artists. These are also used in homes as personal computers. The word size of a microcomputer is 8 to 32

### Minicomputer

A Minicomputer possesses greater amount of memory and storage capacity as compared to microcomputers. The word length of minicomputer is 32 or more. Minicomputers are usually used as multi-user systems where a large number of people can work together. The PDP and the VAX are popular minicomputers

### Mainframe Computer

Mainframe Computer is a large and powerful system with a vast amount of memory and storage space. Since the power based tremendously and they are much cheaper, the work that was done earlier by mainframes is being done by PC's. Now-a-day' popular mainframe computers s mainframes work as nodes for large computer networks. The word length of mainframe computers is generally 64. The CDC 6600 and the CYBER 170 are popular mainframe computer

### Supercomputer

Supercomputer  consists of a number of processors that can process a variety of jobs in parallel. Today's supercomputers one of instructions per second. These are applied mainly for high tech applications like nuclear research,   military application, space exploration and similar other applications. The PARAM and the ANURAG are supercomputers manufactured in India and exported to different countries

### Analog Computer

Digital computers convert continuous data into digital signals for processing. This process of conversion is known as encoding or digitizing. A digital computer essentially base upon the computer can measure a quantity however an analog computer can be measure a quantity and process it without the need of any digitization. Such computers are used mainly in scientific application’s

### Hybrid Computer

A numerically-controlled machine produces cylindrical pins to very close tolerances. The machine has to keep track of the diameter of pins. This is possible if the diameter of the product is continuously measured. The instant diameter of the pin can be sent to the computer as an analog signal. Suppose the machine has a tool that requires periodic replacement. How can one get a message when the tool requires replacement This requires analysis of the trend of the products produced This is essentially statistical analysis of the past data and best handled as numeric data. One can have a computer to deal with the measurement of the diameter of the pin through an analog device and analyze the trend through statistical analysis of numeric data which is digital in nature. A computer that can deal with the numeric data and analog signals is known as a hybrid computer".

## DALTON'S ATOMIC THEORY

In 1803, John Dalton presented a atomic hypothesis based on the results of experiments and observations, and on the basis of this hypothesis, he composed the composition of matter, the rules of chemical coincidence and the physical and chemical changes. The results of many experiments and observations The authenticity of Dalton's nuclear hypothesis was certified and accepted as a scientific theory. Dalton's atomic theory is called Dalton's

### According to Dalton's nuclear principle,

(1) The matter is composed of very subtle (indivisible) particles called atoms.

(2) Atom is the subtle particle of matter which can not be divided into more subtle particles. The original particle of atomic material is.

(3) Atoms participate in chemical reactions and in chemical reactions the atom is not decomposed or destroyed.

(4) All atoms of an element are identical to all types, that is, their mass, size and properties are similar.

(5) Each element has a fixed atomic weight. The fundamental property of atomic weight is the fundamental property.

(6) The atoms of different elements are different from one another and their atomic masses, shapes, etc. are different.

(7) Nuclear combinations of different elements can be combined into the whole number ratio. | And by combining them, "compound atoms" are formed.
Elements of atomic   ,      compounds atoms
A+B                        =        AB
2A+B                        =        A2B

(8) Nuclear can not be produced and can not be destroyed
Amendments to Dalton's Atomic Principles

#### In light of modern scientific discoveries and explorations, significant modifications have been made in Dalton's original nuclear principle, which are as follows:

(1) Nuclear is divisible. It is true that atoms are not decomposed into physical and chemical changes, but by extraordinary physical methods the atoms can be broken into smaller particles than them. The atom itself can be used in various types of fine particles Are made from

Those are primary particles or subatomic particles. Elecrons are the main constituents of proton and neutron atoms.

(2) The atoms of an element can be of different atomic masses. Those atoms of an element whose atoms are different, are called isotopes.

(3) Atomic masses of atoms of different elements can be similar. Those atoms of different elements. The atoms of which are equal are called isobars.

(4) The basic symptom of the element is its atomic number and not atomic weight. The number of protons present in the atomic nuclei of an element is called the atomic number, z, of that element. Each element is a fixed Atomic numbers are different. Atomic numbers of different elements are different.

(5) Combinations of atoms are not "compound atoms" but molecules are formed.

(6) The ratio of the combination of atoms is not necessary in the smallest quantities. In the proteins, carbohydrate polymers, etc. the connective ratio of atoms is not in the periphery

## Moment of inertia

According to Newton's first law, if any body is in a state of rest, then it remains in a state of rest and if it is. If it is moving in a straight line from the same trick, then it moves in the straight line from the same trick, unless it has to be replaced with some exterior forces and it should not change in the present state. These properties of the body that they oppose their state-change are called inertia (inertia).

Similarly, when a body rotates the back of an axis, then it has its tendency to resist its state-change. This quality of the body due to which it opposes the change in the rotation of an axis: the rotation of the body - the back of the axis: the inertia - the horizontal '. It's often To display. Rotation of any particle of the body - the surface of the axis: Inertia - the mass of the particle that is incomplete and its rotation - equals the product of the square of distance from the axis.

. Assuming M is a strong body of mass. Its vertical axis passing through its point of zero is to find the inertia - the horizontal. For this, the body is made up of several small particles. If the mass of these particles is M1, M2, M3,. . . . And rotate them - away from the axis R1, R2, R3, respectively. . . . . If so, the inertia of the particles - phonera na11, nepr:. $m1r^{2},m2r^{2},m3r^{2}..........$. . . . Will be.

Therefore rotation of the whole body - the balance of the axis: the inertia - horizontal (i) the inertia of all particles - will be equal to the sum of the halves

$I=m1r^{2}+m2r^{2}+m3r^{2}+..........$
$I=\sum&space;MR^{2}$

Or
Here ! (Sigma) means - the sum of all the positions. Thus, the backbone of a given axis is equal to the sum of all the particles of the particles and the sum of the products of the distance groups corresponding to their axis
.
Inertia - In the SI method of acceleration, Molecular $kg-m^{2}$ and C. G. S Method is $gm-cm^{2}$'. Its immense formula is [ML]

From the above, it is clear that any body of the body of an axis depends on the inertia -
(i) on the mass of the body,
(i) the position of the axis relative to the body, and
(iii) on the distribution of mass in the body mass relative to the axis

When the position of the rotating axis of the body is changed, the inertia of the body changes - the horizontal. Therefore, its rotation with the inertia of a body - it is necessary to clarify the position of the axis

### Rotating - Radius of Gyration:

If any mass is considered centered on one point, its rotation - the distance per axis so that the distance of the body is multiplied by the mass of the body, rotation of the object - after the axis: . Strength - When the torque is received, this distance is rotating - the body of the body relative to the axis is called 'rotation - radius'

If a body of the mass of the mass is rotated by a rotating axis then the inertia is i

$I=MK^{2}$

Where the rotation radius relative to the rotation axis of k body is
$K=\sqrt{\frac{I}{M}}$

Therefore, the rotation of a body is called the rotating radius relative to the rotation axis of the body, the square root of the motion of the inertia and the
mass of the mass of the rotating axis.

## Ohm's law

### Ohm's law

In 1826, the relationship between the German scientist, Dr. George Simon Om, and the flow of current and current flowing through the direction of a conductor was expressed by a rule called 'Om Law'.

According to this rule, if there is no change in the physical condition (like heat) of the driver, then the ratio of the probability imposed on its ends and the current flows is fixed.
So if there is a probable variant on the crossing of the bark and the stream flowing in it, then according to the Om rule.

$\frac{V}{I}&space;=COSTANT$

According to the determinant definition, the ratio of 'V' / 'driver' to 'electrical resistance' is R

$\frac{V}{I}&space;=R$

In this way, according to the rule of Oh, unless the conduct of a conductor changes in heat and other physical conditions, the driver's resistance remains fixed even though the extent of the impact on the ends of the driver
.

### note

Ohm's law is only true for metal movements

## Capacitor and Use of Capacitors

### Capacitor

A capacitor is an adjustment in which the amounts of charge are stored on it without changing the size of the driver. Suppose that a driver is given 5 charges, then its valid capacity becomes V. Then the driver's capacitance

$C=&space;\frac{Q}{V}$

However, if we reduce the chance of the driver in any way, then more charges can be given to bring it back to the same extent, thus the capacitance of driver c will increase and increase.

### 16. Use of Capacitors

capacitor use in various purpose describe the given bellow

### (1) To deposit the charge:

The key function of the capacitor is to collect the charge. If there is a transient but strong current in a circuit, its best solution is to connect the ends of the circuit to the charging capacitor. Pulsed electromagnet, through which transient but intense magnetic fields are produced, receives electrons from the charged capacitors only.

### | (2) Energy depositing:

Capacitors are not only for charging but also for energy. Enough energy is stored in the electric field installed between the plates of the charging capacitor. The device that accelerates the electrons is synchocyclodone, a large bank of capacitors in which the energy is stored and the machine keeps taking energy according to its requirement.

### (3) In electrical appliances:

Capacitors have a special place in many electrical appliances. When a stimulus circuit breaks suddenly, instead of breaking often, the spark is often produced. But if a capacitor is felt in the circuit, then the spark does not arise if the circuit induced by the breakdown of the circuit, charging the plateau of the capacitors. The capacitor is applied for the same work in the ignition system of the induction coil and the motor engine. Electric fan also has a valid motor capacitors

### (4) In electronic circuits:

In almost all electronic circuits, capacitors are used. For example, in reducing the voltage in voltage in the power supply, in the transmission of pulsed signal and in the production and transmission of electromagnetic charge of radio frequency (i.e. transmission and transmission of radio and television programs) with capacitors an important role. _

## Coulomb's law and Importance of the rule of Coolam:

### Coulomb's law

We have read that two similar types of fees back one another and the opposite type is mutually intruding. Attracted. This shows that a force works between two charges, which are called 'electric force'. the same . Between the allegations, the power force replication - this attraction between force and opposite charges - is the force. Even if electric charges are located in vacuum, they still have electrical power.

. In 1785, French scientist Kulm gave a rule in relation to the force employed between two charges on the basis of experiments, which is called the 'rule of koolam'. According to this rule, there are two fixed points - the power of attraction or replication between charges, the sequential ratio of the product of the quantity of two orders and the sequential ratio of the curvature of the distance class between them. This force adheres to the line involved in those allegations.

Thus if the two point charge Q1 and Q2 are located at one distance of ' r ', then the force between them

$f=&space;\frac{q1q2}{r^{2}}$

$f=&space;K\frac{q1q2}{r^{2}}$
Where K is a serum proportional element whose value depends on the medium between the charges and the amount of charge, distance and force if the force of the force 'Newton' is the number of distance meters and the amounts of charge and take place in the charge and are located in two charge (Nirvati) So the value of K is 9.0 x$10^{9}$  by experiment
So the point between the points placed in the vacuum,

$f=&space;(9.0\times&space;10^{9})\frac{q1q2}{r^{2}}$

In this equation the Q1 = = Q2 = 1 coolom and absolute R = 1 meter then F = $f=&space;(9.0\times&space;10^{9})$ will be Newton.

Hence; 1 Coulom is a charge that replaces itself with the strength of $(9.0\times&space;10^{9})$ utensils at the distance of 1 meter from its own equally in Nirvana (or air).

If the charge is located in the vacuum, then in the equation for the convenience, the sequential determinant K   $\frac{1}{4\pi&space;\xi&space;0}$ is written

where
$\frac{1}{4\pi&space;\varepsilon&space;0}newton-metter^{2}/coulom^{2}$

constant   $\varepsilon&space;0$  Apesail Zero is called electrification of nirvat.

### Comparison with the force of gravity of the electric force:

We can compare the power force working between two charging objects in which the gravitational force is working between them. These two army work according to the rules, and they are active in both of them. But there are some differences between these two too:

(i) Power Force attraction - force can also be done and replica - force also; While gravity - force always attracts force force. It shows that the charge can be of two types while the mass is only one type.

(ii) Electrical force depends on the medium between two charges. Whereas gravitational force is not dependent on the medium between two people.

(ii) The force of force from the gravitational force is very strong. For example, between two protons, the electric force is 10 times larger than the force of gravity between them. Between two electrons it is 10 times larger than that.

### Importance of the rule of Coolam:

The law of coolam is true from very great distances to very little distance from atomic distance (= $10^{-11}$ meters) and atomic distance (= $10^{-15}$ meters). Therefore, this rule is not only knowledge of forces operating between powers, but also helps to explain those forces that cause the atom's electrons to bind to their nuclei to make two or two atoms, and with more atomic molecules Talk, and many atoms or molecules interact with each other and make concrete and preto. Most of the forces that experience in our daily life, which are not gravitational force, are electric power. | Another, very intense attraction - acts between the particles present in the nucleus of the atom (protons and neutrons), which acts as a force that connects these squads with each other. This is called 'nuclear force'. This force does not depend on the charge or discharge of the particles, nor does it have any relation with the rule of Kulam. But this does not mean that the thermal power replication does not exist between protons (within the nucleus). Power replication force is present (though it is atomic attraction - very small in front of force) and plays an important role within the nucleus. If this force is not there then the heavy nucleus is not radioactive and heavy elements (which are temporary) before uranium are permanent.

## chemistry and branches of chemistry and Importance and expansion of chemicals

What is chemistry
Is a branch of chemistry under which the composition and properties of the substances are studied in their structure and changes in them
In ancient times, the rise of chemicals came in the form of art and crafts; Chemistry developed in the form of chemical arts and crafts till the 17th century was born in the 17th Century. Modern art is not science
Chemistry is an experimental science, which is related to the study of substances, given the extraordinary modern development of chemistry, it has been divided into several branches. The major  chemicals are as follows

1 Inorganic chemicals - All elements and compounds are studied under it (excluding organic compounds)

2 Organic Chemistry - All organic compounds are studied under it

3 Physical Chemistry - Under this, rules and principles of chemical reactions are studied.

4 Analysis Chemistry - The methods for determining the quantity and quantification of the substances are studied.

5 Industrial Chemistry - The rules, reactions, methods etc. are studied in connection with the creation of huge results of the substances under it.

6 Biochemistry - It involves chemical reactions occurring in biosynthesis and chemicals obtained from animals and plants.

7 Agricultural Chemistry - Under this, chemicals related to agriculture are studied.

### Importance and expansion of chemicals

In modern life, chemistry is a very important place. The chemistry of human society in almost all areas. . Designed to make human life prosperous, prosperous, safe and prosperous. Chemical contribution is important in the national economy. The success of the country's development plans depends on the application of some chemicals. All small and large industries require chemical materials. Acid, alkalis and salt metallurgy, metal-purification, petroleum finishing and used in the production of glass, soap, paper, textile, fertilizer, explosive, pigment, drugs etc. Molecular acids, nitric acid, ammonia, cestolic soda and chlorine are pillars of industries. Iron, copper, aluminum, zinc, nickel etc. Many types of alloy - metal, brass and steel - are used in making industries and many things of daily life. Many useful items are made from plastic, teflon, polythene, synthetic rubber and other polymer. The fabrics are made of artificial silk, wool and yarn. Pesticides, pandemic etc. protect crops. Modernities protect varieties and lives. Oils, fats, proteins, carbohydrates, salts and vitamins are an essential part of our diet.
1. Agriculture - fertilizers, insecticides, pesticides etc.

2 Textiles - Artificial silk, wool, artificial thread etc
.
3. Building and road construction: - Cement, steel, wire etc.

4. Health and Life - Medicine, Vitamin Antibiotics etc.

5. Fuel - Gasoline, Diesel, Kerosene, CNG Gaseous Money, LPG etc.

6. Industry cement, glass, textiles, paint, rubber, plastic, leather.
Steel, sugar, fertilizer etc. Almost all industries use different raw materials.

7. War Material: Explosive.

8. Power generation -: Atomic energy, battery, dry cell etc.

9. Metallurgy: Metal - Extraction, Metal - Finishing, Alloy - Metal.

10. Entertainment: Gramophone record, photo film, photography, movies, colorful films, tape, CD etc.

11, Beauty of beauty: oil, perfume, kim, powder, lipstick, nail polish, soap, shampoo etc.

12. Clarifiers: Soaps, insecticides, organic solvents etc.

13 manure - manure preservatives, artificial honey

14 Refrigeration & Air Conditioning - Refrigerators, Fraun

## Cutting a cube

It is a simple fact that if a wood or rod is to be cut in two equal parts, then we cut it one. Similarly, if the wood or rod is cut into three, four or five equal parts, then we cut it once, Or cut the bar n times, it cuts (N-1) times

If the cube with an 8 cm arm has to be cut into small cubes of 2 cm, then each surface will have
N = (  8/2 )  = 4 sections and to divide it into four sections, the cube is divided from three sides to  ( N-1 ) = ( 4 -1) = 3 times, and after partition, $n^{3}$ = $4^{3}$ = 64 small cubes will be received

If a cube is to be cut into 8 small cubes, then the first cube root of 8 is called ∛8 = $\sqrt[3]{2\times&space;2\times&space;2\times&space;2}$  = 2. In this case, we get 2 cubic roots, namely the cube from one on each side (n-1) On cutting, we will get 8 small cubes

### Identifying the cubic or cuboid after cutting it

After cutting a small cubic material into a cub, a cube or cuboid is seen on the various parts of the cube or cuboid

Top cube - Such cube is located at the top of the corner ie every lane of its cube is equal to eight, because any cube has eight top or corners.

Middle cube - Such cube is located right in the middle of each edge

Central cube - Such cube is located at the right center of each surface

Intermediate cube - Such cube is located in the middle of the central cube of each surface, it is not visible from the outside, it is called the Kuclias cube.

The number of total small dances obtained after dividing the larger cube into smaller cubes = $n^{3}$

Where
$N=\frac{One.arm&space;.of&space;.big.cube}{One.&space;arm.of.big.cube}$

Number of top cube                    =             6
Number of middle cube              =            12(n-2)
Number of central cube              =            $6(n-2)^{2}$
Number of intermediate cube    =             $(n-2)^{3}$

## Simple interest

The basic concept of interest is based on the excess amount borrowed by the borrower in relation to the borrowed amount, in addition to the amount and amount of principal and interest paid on the amount paid to the principal in the fixed period of time. The amount made is known as compound. Interest on milling is calculated as simple and compound interest, and for a year, both the interest rate and compound interest are equal when the rate of interest is yearly.

## Simple interest

When calculating interest only for the time being, the principal is called simple interest.

### Finding the Principle on the Rate of Seduction Interest

When the rate of simple interest is different for different years, in such a situation, certain calculations are needed to calculate the principal. Using these sources, saving time in the examination building, along with the Accuracy also It comes

1 -  If the rate of ordinary interest goes from 1% to 2% and in time t is more than m

Principal amount   =
$\frac{M}{r2-r1}\times&space;\frac{100}{t}$

2 -  If the interest rate r1% for time t1 on any money, then rate r2% for time t2

### Finding simple interest on N times of principal

When a wealth gets N times of N times or N times of interest in time, then they experience some difficulty in finding the rate of simple interest because they are N times of themselves and due to the N times of interest, I do not care about it, so its concept is understood as follows

1- If any funding at the rate of simple interest becomes n times in T
rate of interest         =            $\frac{(n-1&space;)\times&space;100}{t}$

2- If any money at the rate of simple interest becomes n times of interest in time

rate of interest        =             $\frac{n\times&space;100}{t}$

## Reflection of light and snail law

A ray of light runs through a straight line, but when the ray of light

goes through a transparent medium to another transparent medium,

then it gets distracted by its path. In the second medium, the beam,

either towards the first medium, either bends towards the

perpendicular or away from the perpendicular

If the light beam goes from one medium to another, it is called

"refraction" from its path, if the refracted ray bends toward

the relation relative to the incident ray, then the second is called

the dense relative to the medium, but if the refracted ray is an

incident ray Removed from the relative angle, then the other

medium is called the relative viral of the first

Light refraction has two rules
1 Incident ray, refracted ray and perpendicular to the point of

view are all in the same plane

2 For any two of the medium and for the light of certain colors

(wavelength), the ratio of sin to the angle of incidence and the

angle of refraction remains constant.

If the angle of incidence is i and the angle of refraction then

$\frac{sin&space;I}{sin&space;R}=costant$

This rule is called the snail rule. This determinant is called the

refractive index of the second medium relative to the first medium

, if we first display the medium 1 and the second medium 2, the

refractive index is displayed from 1N2.

## Analytic method of vector addition

If two vectors display the results from two adjoining side drawn from any point of a parallelogram in the direction of the result, then the result and direction, the parallelogram that is drawn by the diagonal has been drawn from the same point to add the vector Parallel quadratic rules
Assuming vector $\vec{A}$ and $\vec{B}$ are bent at opposite $\theta$ angle, they are displayed in the direction of the parallel quadrilateral OPQS in the direction of OPQS and OP and the OS, then according to the rule of parallelogram, resultant $\vec{R}$ of $\vec{A}$ And $\vec{B}$ will be represented by the diagonal OQ in the result and direction. In order to find the result of the resulting $\vec{R}$, we increase the side OP and pull the vertical QE from the point Q, thus

Thus, in the right angled triangle OEQ

$(OQ)^{2}&space;=&space;(OE)^{2}&space;+&space;(QE)^{2}&space;=&space;(OP+PE)^{2}&space;+&space;(QE)^{2}$

$=&space;(OP)^{2}&space;+&space;(PE)^{2}&space;+&space;2(OP)(PE)&space;+&space;(QE)^{2}$

NOW

$(PE)^{2}&space;+&space;(QE)^{2}&space;=&space;(PQ)^{2}$

SO

$(OQ)^{2}&space;=&space;(OP)^{2}&space;+&space;(PQ)^{2}&space;+&space;2(OP)(PE)$

IN Right-angled triangle

$COS\theta&space;=&space;\frac{PE}{PQ}$
OR

$PE&space;=&space;PQCOS\theta$

So the final equation will be

$(OQ)^{2}&space;=&space;(OP)^{2}&space;+&space;(PQ)^{2}&space;+&space;2(OP)(PQCOS\theta&space;)$   ,  NOW

OP =A , PQ= OS = B , OQ = R ,,    SO

$R^{2}&space;=&space;A^{2}+&space;B^{2}&space;+2AB&space;COS\theta$

$\left&space;[&space;R=\sqrt{A^{2}&space;+B^{2}&space;+2ABCOS\theta&space;}&space;\right&space;]$

To find out the direction of the resulting $\vec{R}$, say that the angle of $\vec{R}$ vector $\vec{A}$ is formed by $\theta$

$Tan&space;\theta&space;=&space;\frac{QE}{OE}&space;,=&space;\frac{QE}{OP+PE}$   NOW  , OP = A , PE = BCOS$\theta$

To find the value of QE, in triangle PEQ

$SIN\theta&space;=&space;\frac{QE}{PQ}$

$\left&space;[&space;Tan\theta&space;=&space;\frac{Bsine\theta&space;}{A+bcos\theta&space;}&space;\right&space;]$

Special cases

(1) when both the vector are in same direction

Again  equation

$R&space;=&space;\sqrt{A^{2}+B^{2}+ABCOS\theta&space;}&space;=&space;\sqrt{A^{2}&space;+B^{2}+2AB}&space;=&space;A+B$   and

$Tan\theta&space;=\frac{b&space;\times&space;0}{a+b}&space;=0&space;,&space;\alpha&space;=0$
Thus the result of result $\vec{A}$ is equal to yoga of both vector $\vec{A}$ and $\vec{B}$ and in $\vec{A}$ and $\vec{B}$ same direction

(2) when the both vector are at right angle to each other

$R=\sqrt{A^{2}+b^{2}+2ABcos\theta&space;}&space;,=\sqrt{A^{2}+B^{2}}$

$Tan\theta&space;=\frac{Bsin90}{A+Bcos90}&space;,&space;=\frac{A}{B}$

(3) When both the vectors are in opposite direction

Again  equation

$R=\sqrt{A^{2}+B^{2}+2ABcos180^{\circ}}&space;=&space;\sqrt{(A-B)^{2}}&space;=A-B,B-A$

$Tan\theta&space;=\frac{Bsin180^{\circ}}{A+Bcos180^{\circ}}&space;=0&space;,nad&space;,\alpha&space;=0,or&space;180^{\circ}$
Thus the result of the resultant vector $\vec{R}$ is equal to the difference of the result of both vectors and in the direction of the large vector
It is clear from the above that the result of the result of both vector is maximized, then both vectors are in the same direction and this is minimal when they are in the direction