Consider the vector $\vec{A}$ and $\vec{B}$. According to Parallelogram of vector addition we get the figure.

Here, $\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{OP}}=\overrightarrow{\mathrm{RQ}} ; \overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{OR}}=\overrightarrow{\mathrm{PQ}}$

Draw a Parallelogram,

From $\Delta \mathrm{OPQ} \quad \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{OP}}+\overrightarrow{\mathrm{PQ}}$

$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{OQ}}$

From $\Delta$ $ORQ$ $\vec{B}+\vec{A}=\overrightarrow{O R}+\overrightarrow{R Q}$

$=\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{OP}}[\because \overrightarrow{\mathrm{OR}}=\overrightarrow{\mathrm{PQ}} \text { and } \overrightarrow{\mathrm{RQ}}=\overrightarrow{\mathrm{OP}}]$

$\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{OQ}} \ldots \text { (ii) }$

From $(i)$ and $(ii)$ we get,

$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{A}}$