Physical quantities having magnitude and direction are called 'Vectors'. The concept of vector has given a precise definite direction to the study of Physics. A thorough understanding of 'Vectors' is like possessing a sharp tool to solve the secrets of nature. Man likes to learn from nature.
All physical quantities do not have direction, distance, mass, temperature, speed, work etc will have only magnitude and they do not need direction. They are called 'Scalars'. You are astonished to note that all fundamental physical quantities are scalars.
Suppose a force F is acting on a body. Then it is quite natural to ask 'What is the direction in which the force is acting?' Which means that 'force' direction has to be mentioned. Thus force is a vector. Similarly the physical quantities acceleration, momentum etc are vectors.
Vector - A sharp arrow!
A vector is a directed line segment in space representing a physical quantity both in direction and magnitude.
Tail 
Head
The length of the straight line AB represents the magnitude and the arrow head gives the direction of the vector. Vectors are also represented as , , , etc.
A vector is a mathematical object with both numerical and geometrical properties. Towards the end of 19th century Josiah Williand Gibbs and Oliver Heariside discovered vector analysis.
Are vectors, tails of monkeys?
Consider three monkeys trying to cross a fast flowing river. As a measure of precaution, the second monkey will hold the tail of the first monkey going a head, and the third monkey will hold the second monkey's tail and all start crossing the river confidently. Suppose after travelling some distance, if the speed of the flowing water increases suddenly, the first monkey will hold the tail of the third and the three monkeys form a closed circuit such that they will come to an equilibrium.
Similarly, if three forces (vectors) 
1, 2, 3 are acting Simultaneously on a body, and if they are represented by the three sides of a triangle taken 'inorder' both in direction and magnitude then that body will be in equilibrium. This is the 'triangle law of forces' (vectors).
Here, the forces 1, 2 and 3 can be imagined to be the monkeys, crossing the stream holding each others tail.
If many forces are acting on a body (if many monkeys are crossing the river) applying the same procedure (attaching the head of the one to the tail of the other) we can arrive at 'Polygon Law of Vectors'.
These laws are called Vector Laws of addition.
Order of addition - No problem!
Vector addition can be done in any order. What you have to do is to attach the 'tail' of one to the head of the other maintaining the magnitude and direction of the Vectors.
1 + 2 + 3 = 1 + 3 + 2 (Commutative Law)
(1 + 2) + 3 = 1 + (2 + 3) (Associative Law)
Negative of a vector and not negative vector
There is nothing like negative vector, but negative of a vector which means another vector of equal magnitude but opposite in direction. It is called 'reversed vector'.
Vector subtraction is addition only
If a vector 
is to be subtracted from vector 
, the vector is reversed (-) and the reversed vector is to be added to the first.
- = + (- )
The 'Relative velocity' is the vector difference between the velocities.
Flying bird gives Parallelogram Law of forces
Suppose two forces are acting simultaneously on a body in different directions. We can learn the resultant of these two forces from a 'flying bird', in the form of 'Parallelogram Law of forces'.
Let a flying bird pushes air in the directions AO and BO with its wings. These are actions.
Then the reaction forces 
and 
act on the wings. The resultant of these forces OC = R, makes the bird to fly up in the air.
* The Laws of Vector addition illustrates how 'Man likes to learn from Nature' (Monkeys and birds)!!
Why two vector products?
When a is multiplied by b, we get one product a × b = ab only.
But when a vector is multiplied by another vector , we obtain some times a scalar and in certain cases a vector. In order to accommodate this paradox in vector analysis, two types of products i) Dot (scalar) product and ii) Cross product are introduced.
i) . = scalar ii) × = vector
All this is due to 'Work' in physics
In physics, force only does 'work' and not an agency. 'Work' (W) is said to be done by a force
() only when it displaces a body in its direction and is measured by the product of the force and displacement 
.
Since the direction of the force is already mentioned, the direction for work need not be mentioned separately and hence we don't mention direction for force and thus it becomes a scalar. Thus the product of work which is a scalar is mentioned as dot product (W = .). So work is a scalar not because it is a dot product. Dot product is defined to justify the concept of 'work'.
i) . = ab cos θ (dot product)
e.g. : Work, power, energy, electric potential.
ii) × sin θ (cross product)
e.g. : Torque, Angular momentum, Angular velocity.
Unit Vector: 'A vector of unit length in a given direction'. Unit vectors along X, Y and Z axes are represented as 
, 
, 
.
A vector 
can be written in terms of unit vectors as = Ax + Ay + Az where Ax, Ay and Az are scalar components of vector . Magnitude of
