Adding vectors is commutative because adding coordinates is commutative and vector addition is merely two applications of that same law vectors $A$, $B$
$$A + B = (\overrightarrow{a_1,b_1}) + (\overrightarrow{a_2, b_2}) = (a_1 + b_1, a_2 + b_2) + (c_1 + d_1, c_2 + d_2)$$
which implies Commutativity be default due to the fact that we are merely adding real numbers.
Edit: Mistook commutativity for associativity!
Best Answer
Your proof is confusing because $c_1, c_2, d_1$, and $d_2$ appear out of nowhere. But I think your general idea is correct. The proof is as follows.
The commutative property of two-dimensional real vectors is: For all two-dimensional real vectors $a = (a_1, a_2), b = (b_1, b_2)$, we must have $a + b = b + a$. This is true since
$$ a + b = (a_1 + b_1, a_2 + b_2) = (b_1 + a_1, b_2 + a_2) = b + a $$
where the second equality follows from the commutative property of real numbers and the other equalities follow from the definition of vector addition.