Dear Shubham,
The formula for the length of a 2D vector is the Pythagorean Formula. Say that the vector is represented by   (x, y)T.   Put the vector with its tail at the origin. Now make a triangle by drawing the two sides:
side_1   =  (x, 0)T

side_2   = (0, y)T.

The length of side_1 is x, and the length of side_2 is y, so:




length (x, y)T   =    ( x2 + y2 )




In this formula,    means the positive square root. We don''t (of course) want the length to be negative.
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Dear Shubham,
 vector quantity has both magnitude and direction. Acceleration, velocity, force and displacement are all examples of vector quantities. A scalar quantity is has only magnitude (so the direction is not important). Examples include speed, time and distance.
Unit Vectors
A unit vector is a vector which has a magnitude of 1. There are three important unit vectors which are commonly used and these are the vectors in the direction of the x, y and z-axes. The unit vector in the direction of the x-axis is i, the unit vector in the direction of the y-axis is j and the unit vector in the direction of the z-axis is k.
Writing vectors in this form can make working with vectors easier.
The Magnitude of a Vector
The magnitude of a vector can be found using Pythagoras''s theorem.


The magnitude of ai + bj = √(a2 + b2)


We denote the magnitude of the vector a by | a |
Position Vectors
Position vectors are vectors giving the position of a point, relative to a fixed point (the origin).
For example, the points A, B and C are the vertices of a triangle, with position vectors a, b and c respectively:

You can draw in the origin wherever you want. 
Notice that  = - a + b = b - a because you can get from A to B by going from A to O and then going from O to B.
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Vector Formulas
A vector can also be defined as an element of a vector space.  Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector  is often called a two-vector, an n-dimensional vector is often called an n-vector, and so on.
The important formulas of vectors are given below:
 
1. The position vector of any point p(x,y) is
 or OP = ( x,y ).
 
2.The magnitude of position vector
 and direction 
 
3. The unit vector =  where the magnitude of unit vector is 1
Or,the unit vector = 
 
4.The two vectors  and  are parallel if  and  where k and m are the scalars.
 
5.If  then  is the result vector which is the triangle law of vector addition.
 
6. The scalar or dot product of any two vectors .
 
7. The angle between two vectors is 
 
8.  and  , then :
 where 
 
9. If the position vector of A is  , position vector of point B is  and position vector of mid-point M is m then 
 
10. If the point P divides Ab internally in the ratio m:n then position vector of P is given by  which is a section formula.
 
11.If P divides AB externally in the ratio m:n then 

PRODUCT OF TWO VECTORS
 
1.Scalar Product ( dot product )
 
Let  then dot product of  &  is devoted by  read as  dot  and defined by 
 
Note:
if
 
OR
 
The scalar product of  &  is devoted by  ,
 where  being  angle between  & 
 
Note:1

Note:2
 &  are perpendicular if  = 
i.e  or 
 
2.Properties of Scalar Product
i. .
ii. .
iii. 
iv. 
v. If then 
 
3.Vector (cross) Product of two vectors.
Let  be two vectors then the cross product of is devoted by and defined by
 
 = 
 
We can define in terms of determinants as follows
 
 = 
 
Note:1                being angle between
 & 
Note:2               
Note:3               If  , the  and  &  are parallel if .
 
4. Properties of cross product
 
i. 
ii. 
iii. 
iv. 
v.  is perpendicular to both  and 
vi.  is a Area of paralelogram with sides  and 
vii.  = area of triangle having  ,  ,  as position vectors of vertices of a triangle.