In an octagon ABCDEFGH of equal side, what is the sum of → AB + →...

We know,

$∵ \overset{\to}{OA} + \overset{\to}{OB} + \overset{\to}{OC} + \overset{\to}{OD} + \overset{\to}{OE} + \overset{\to}{OF} + \overset{\to}{OG} + \overset{\to}{OH} = \overset{\to}{0}$

By triangle law of vector addition, we can write

$\overset{\to}{AB} = \overset{\to}{AO} + \overset{\to}{OB}; \overset{\to}{AC} = \overset{\to}{AO} + \overset{\to}{OC}$

$\overset{\to}{AD} = \overset{\to}{AO} + \overset{\to}{OD}; \overset{\to}{AE} = \overset{\to}{AO} + \overset{\to}{OE}$

$\overset{\to}{AF} = \overset{\to}{AO} + \overset{\to}{OF}; \overset{\to}{AG} = \overset{\to}{AO} + \overset{\to}{OG}$

$\overset{\to}{AH} = \overset{\to}{AO} + \overset{\to}{OH}$

Now

$\overset{\to}{AB} + \overset{\to}{AC} + \overset{\to}{AD} + \overset{\to}{AE} + \overset{\to}{AF} + \overset{\to}{AG} + \overset{\to}{AH}$

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

$= \left(7\overset{\to}{AO}\right) + \overset{\to}{0} - \overset{\to}{OA}$

$= \left(7\overset{\to}{AO}\right) + \overset{\to}{AO}$

= 8\overset{\to}{AO} = 8\left(2\overset{^}{i} + 3\overset{^}{j} - 4\overset{^}{k}\right)

= 16\overset{^}{i} + 24\overset{^}{j} - 32\overset{^}{k}