A Vector is a quantity which has both magnitude and direction.
Example: Force, velocity, momentum etc.
A vector is represented by a directed line segment. If the vector has the initial point P and final point Q then the vector is denoted by 
or PQ.
The length of the vector is called its magnitude or length or modulus. The length of a vector PQ is denoted by |PQ|.
Since a vector is not just a number they cannot be added just like numbers. There are some laws which can be used with the vector addition.
Triangle law of addition:
Let a and b be any two vectors. Let O be any point. Let us assume that OA and OB are two line segments such that OA = a and AB = b. Join O and B. Then OB is defined as the sum of the vectors a and b.
i.e. a+b = OA+OB = OB.
This is known as the triangle law of addition.
Parallelogram law of addition:
Let OA=a and OB = b. Complete the parallelogram OABC with OA and OB as sides. Since OC is parallel to AB, we get OC=AB. Thus we have OC=AB=b.
So, OC= a+b = OA+AB = OA+OC.
i.e. The sum of two co-initial vectors OA and OC is given by OB where OB is the diagonal of the parallelogram OABC having OA and OC as the adjacent sides.
This is known as parallelogram law of addition.
Polygon law of vectors:
Let OA = a, AB = b, BC= c and CD = d.
a+b+c+d = OA+OB+OC+OD
=OB +BC + CD
= OC + CD
=OD
Thus the addition of vectors a,b,c and d is defined by OD.
Properties of vector addition:
For any 3 vectors, a,b and c,
1) a+b = b+a
2) a+(b+c) = (a+b) +c
3) a+0 = a
4) a+ (-a)=0
5) k(a+b)= ka+kb.
6) a-b = a+ (-b)
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