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**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.