In an octagon $ABCDEFGH$ of equal side, what is the sum of $\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}+\overrightarrow{AG}+\overrightarrow{AH},$ if, $\overrightarrow{\mathrm{AO}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-4\hat{\mathrm{k}}$

We know,

$\because \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{OE}}+\overrightarrow{\mathrm{OF}}+\overrightarrow{\mathrm{OG}}+\overrightarrow{\mathrm{OH}}=\overrightarrow{0}$

By triangle law of vector addition, we can write

$\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OB}};\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OC}}$

$\overrightarrow{\mathrm{AD}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OD}};\overrightarrow{\mathrm{AE}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OE}}$

$\overrightarrow{\mathrm{AF}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OF}} ;\overrightarrow{\mathrm{AG}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OG}}$

$\overrightarrow{\mathrm{AH}}=\overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OH}}$

Now

$\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{AE}}+\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{AG}}+\overrightarrow{\mathrm{AH}}$

$=\left(7\overrightarrow{\mathrm{AO}}\right)+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{OE}}+\overrightarrow{\mathrm{OF}}+\overrightarrow{\mathrm{OG}}+\overrightarrow{\mathrm{OH}}$

$=\left(7\overrightarrow{\mathrm{AO}}\right)+\overrightarrow{0}-\overrightarrow{\mathrm{OA}}$

$=\left(7\overrightarrow{\mathrm{AO}}\right)+\overrightarrow{\mathrm{AO}}$

$=8\overrightarrow{\mathrm{AO}}=8\left(2\hat{i}+3\hat{j}-4\hat{k}\right)$

$=16\hat{i}+24\hat{j}-32\hat{k}$