In a regular octagon ABCDEFGH with center O,w | vedclass

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(B) For a regular polygon with center $O$,the sum of vectors from the center to the vertices is zero:$\overrightarrow{ OA }+\overrightarrow{ OB }+\ove

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

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(B) For a regular polygon with center $O$,the sum of vectors from the center to the vertices is zero:$\overrightarrow{ OA }+\overrightarrow{ OB }+\overrightarrow{ OC }+\overrightarrow{ OD }+\overrightarrow{ OE }+\overrightarrow{ OF }+\overrightarrow{ OG }+\overrightarrow{ OH }=\overrightarrow{0}$Using the triangle law of vector addition,we can express each vector $\overrightarrow{ AX }$ as $\overrightarrow{ AO }+\overrightarrow{ OX }$:$\overrightarrow{ AB }=\overrightarrow{ AO }+\overrightarrow{ OB }$$\overrightarrow{ AC }=\overrightarrow{ AO }+\overrightarrow{ OC }$$\overrightarrow{ AD }=\overrightarrow{ AO }+\overrightarrow{ OD }$$\overrightarrow{ AE }=\overrightarrow{ AO }+\overrightarrow{ OE }$$\overrightarrow{ AF }=\overrightarrow{ AO }+\overrightarrow{ OF }$$\overrightarrow{ AG }=\overrightarrow{ AO }+\overrightarrow{ OG }$$\overrightarrow{ AH }=\overrightarrow{ AO }+\overrightarrow{ OH }$Summing these seven vectors:$\sum = 7 \overrightarrow{ AO } + (\overrightarrow{ OB }+\overrightarrow{ OC }+\overrightarrow{ OD }+\overrightarrow{ OE }+\overrightarrow{ OF }+\overrightarrow{ OG }+\overrightarrow{ OH })$From the initial property,the sum in the parenthesis is equal to $-\overrightarrow{ OA }$,which is $\overrightarrow{ AO }$:$\sum = 7 \overrightarrow{ AO } + \overrightarrow{ AO } = 8 \overrightarrow{ AO }$Given $\overrightarrow{ AO }=2 \hat{ i }+3 \hat{ j }-4 \hat{ k }$,we have:$\sum = 8(2 \hat{ i }+3 \hat{ j }-4 \hat{ k }) = 16 \hat{i}+24 \hat{j}-32 \hat{k}$

In a regular octagon $ABCDEFGH$ with center $O$,what is the sum of $\overrightarrow{ AB }+\overrightarrow{ AC }+\overrightarrow{ AD }+\overrightarrow{ AE }+\overrightarrow{ AF }+\overrightarrow{ AG }+\overrightarrow{ AH }$ if $\overrightarrow{ AO }=2 \hat{ i }+3 \hat{ j }-4 \hat{ k }$?

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