Let  be represented by  and  the two adjacent sides of parallelogram OPSQ. In the figure  is the diagonal of parallelogram.

Since  is equal and parallel to 
∴             
     Also  is equal and parallel to 
∴           
         Now applying triangle law of vector addition to  we get
                             ...(1)
  Similarly applying triangle law of vector addition to  we get
                              ...(2)
From (1) and (2),
               
Hence the vector addition follows commutative law.

To subtract vector  from vector  first change the vector  into  and then add it to vector 


Then, we have

                          

This way, the vectors can be subtracted. 

Let two vectors  of magnitude A and B be acting at angle 

We will find the magnitude of  

The vector  can be written as .
Thus,  is equal to resultant of 

 As the angle between  is , therefore angle between  is 

Thus,