Prove the associative law of vector addition.

(N/A) To prove the associative law of vector addition,$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$,consider three vectors $\vec{A}, \vec{B}$,and $\vec{C}$ represented by the sides of a polygon.
Let $\vec{A} = \overrightarrow{OP}$,$\vec{B} = \overrightarrow{PQ}$,and $\vec{C} = \overrightarrow{QR}$.
Using the triangle law of vector addition in $\Delta OPQ$:
$\vec{A} + \vec{B} = \overrightarrow{OP} + \overrightarrow{PQ} = \overrightarrow{OQ}$.
Now,adding $\vec{C} = \overrightarrow{QR}$ to both sides:
$(\vec{A} + \vec{B}) + \vec{C} = \overrightarrow{OQ} + \overrightarrow{QR} = \overrightarrow{OR} \quad \dots (i)$.
Next,consider $\Delta PQR$:
$\vec{B} + \vec{C} = \overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PR}$.
Now,adding $\vec{A} = \overrightarrow{OP}$ to both sides:
$\vec{A} + (\vec{B} + \vec{C}) = \overrightarrow{OP} + \overrightarrow{PR} = \overrightarrow{OR} \quad \dots (ii)$.
From equations $(i)$ and $(ii)$,we get:
$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$.
This proves the associative law of vector addition.