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

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