Prove the associative law of vector addition.

Consider the vectors of shown in the figure. To find the associative law of $\vec{A}, \vec{B}$ and $\vec{C}$ Draw $\vec{A}=\overrightarrow{O P}, \vec{B}=\overrightarrow{P Q}$ and $\vec{C}=\overrightarrow{Q R}$

By triangle law of vector we get, Figure $(b)$

From the figure $\Delta \mathrm{OPQ}$

$\rightarrow \quad \rightarrow \quad \rightarrow \quad \rightarrow$

$\mathrm{A}+\mathrm{B}=\mathrm{OP}+\mathrm{PQ}$

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

By adding $\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{QR}}$ in both the sides

$(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}})+\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{OQ}}+\overrightarrow{\mathrm{QR}}$

We get $\quad(\vec{A}+\vec{B})+\vec{C}=\overrightarrow{O R} \ldots \ldots$ $(i)$