Properties of Vector Addition

Property:-  For any two vectors `vec a`and `vec b`,
`vec a + vec b` = `vec b + vec a`         (Commutative property)
Proof: Consider the parallelogram ABCD

Let `vec (AB) = vec a` and `vec (BC) = vec b`, then using the triangle law, from triangle ABC , we have `vec (AC) = vec a + vec b`
Now, since the opposite sides of a parallelogram are equal and parallel, from abov fig.
we have , `vec (AD) = vec (BC) = vec b` and `vec (DC) = vec (AB) = vec a`
 Again using triangle law, from triangle ADC, we have  `vec (AC) = vec (AD) + vec (DC) = vec b + vec a`
Hence `vec a + vec b = vec b + vec a`

Property:-  For any three vectors `vec a , vec b and vec c`   
`(vec a + vec b) + vec c = vec a + (vec b + vec c)`                
(Associative property)
Proof: Let the vectors `vec a , vec b and vec c` be represented by `vec (PQ) , vec (QR)` and `vec (RS)`, respectively , as shown in following fig.

Then `vec a + vec b = vec (PQ) + vec (QR) = vec (PR)`

and  `vec b + vec c = vec (QR) + vec (RS) = vec (QS)`

So, `(vec a + vec b) + vec c = vec (PR) + vec (RS) = vec (PS)`

and `vec a + (vec b+vec c) = vec (PQ) + vec (QS) = vec (PS)`

Hence `(vec a + vec b) + vec c = vec a + (vec b+vec c)`

Remark: Any vector `vec a`, we have 
`vec a + vec 0 =vec 0 + vec a = vec a` 
Here, the zero vector `vec 0`  is called the additive identity for the vector addition.