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  • In an octagon ABCDEFGH of equal side, what is the sum of <img alt="\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}+\overrightarrow{AG}+\overrightarrow{AH}" src="https://entrancecorner.codecogs.co

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{AO} = 2\widehat{i}+3\widehat{j}-4\widehat{k}
Option: 1 16\widehat{i }+24\widehat{j}-32\widehat{k}
Option: 2 16\widehat{i }+24\widehat{j}+32\widehat{k}
Option: 3 -16\widehat{i }-24\widehat{j}+32\widehat{k}
Option: 4 16\widehat{i }-24\widehat{j}+32\widehat{k}

Answers (1)

best_answer

We know

\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}......(1)

By triangle law of vector addition, we can write

\begin{array}{l} \overrightarrow{A O}+\overrightarrow{O B}=\overrightarrow{A B} \\ \overrightarrow{A O}+\overrightarrow{O C}=\overrightarrow{A C} \\ \overrightarrow{A O}+\overrightarrow{O D}=\overrightarrow{A D} \\ \overrightarrow{A O}+\overrightarrow{O E}=\overrightarrow{A E} \\ \overrightarrow{A O}+\overrightarrow{O F}=\overrightarrow{A F} \\ \overrightarrow{A O}+\overrightarrow{O G}=\overrightarrow{A G} \\ \overrightarrow{A O}+\overrightarrow{O H}=\overrightarrow{A H} \end{array}

After adding them and using equation (1)

\begin{aligned} 8 \overrightarrow{A O} &=\overrightarrow{A B}+\overrightarrow{A C}+\overrightarrow{A D}+\overrightarrow{A E}+\overrightarrow{A F}+\overrightarrow{A G}+\overrightarrow{A H} \\ &=8(2 \hat{i}+3 \hat{j}-4 \hat{k}) \\ &=16 \hat{i}+24 \hat{j}-32 \hat{k} \end{aligned}

 

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avinash.dongre

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