In an octagon ABCDEFGH of equal side, what is the sum of AB→+AC→+AD→+AE→+AF→+AG→+AH→, if AO→=2i^+3j^−4k^?

∵OA→+OB→+OC→+OD→+OE→+OF→+OG→+OH→=0→By triangle law of vector addition, we can writeAB→=AO→+OB→;AC→=AO→+OC→AD→=AO→+OD→;AE→=AO→+OE→AF→=AO→+OF→;AG→=AO→+OG→AH→=AO→+OH→NowAB→+AC→+AD→+AE→+AF→+AG→+AH→=7AO→+OB→+OC→+OD→+OE→+OF→+OG→+OH→=7AO→+0→−OA→=7AO→+AO→=8(2ı^+3ȷ^−4k^)=16ı^+24ȷ^−32k^

In an octagon ABCDEFGH of equal side, what is the sum of AB→+AC→+AD→+AE→+AF→+AG→+AH→, if AO→=2i^+3j^−4k^?

∵OA→+OB→+OC→+OD→+OE→+OF→+OG→+OH→=0→By triangle law of vector addition, we can writeAB→=AO→+OB→;AC→=AO→+OC→AD→=AO→+OD→;AE→=AO→+OE→AF→=AO→+OF→;AG→=AO→+OG→AH→=AO→+OH→NowAB→+AC→+AD→+AE→+AF→+AG→+AH→=7AO→+OB→+OC→+OD→+OE→+OF→+OG→+OH→=7AO→+0→−OA→=7AO→+AO→=8(2ı^+3ȷ^−4k^)=16ı^+24ȷ^−32k^

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In an octagon ABCDEFGH of equal side, what is the sum of $\vec{A B} + \vec{A C} + \vec{A D} + \vec{A E} + \vec{A F} + \vec{A G} + \vec{A H}$ , if $\vec{A O} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$ ?

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

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

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

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

answer is B.

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Detailed Solution

$∵ \vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} + \vec{O E} + \vec{O F} + \vec{O G} + \vec{O H} = \vec{0}$

By triangle law of vector addition, we can write

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

Now

$\begin{array}{l} \vec{A B} + \vec{A C} + \vec{A D} + \vec{A E} + \vec{A F} + \vec{A G} + \vec{A H} \\ = (7 \vec{A O}) + \vec{O B} + \vec{O C} + \vec{O D} + \vec{O E} + \vec{O F} + \vec{O G} + \vec{O H} \\ = (7 \vec{A O}) + \vec{0} - \vec{O A} \\ = (7 \vec{A O}) + \vec{A O} \end{array}$