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Question
In an octagon ABCDEFGH of equal sides, what is the sum of −−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH if −−→AO=2^i+3^j−4^k ?
In an octagon ABCDEFGH of equal sides, what is the sum of −−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH if −−→AO=2^i+3^j−4^k ?
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Solution
The correct option is A 16^i+24^j−32^k
From figure,
−−→AO+−−→OB=−−→AB
−−→AO+−−→OC=−−→AC
−−→AO+−−→OD=−−→AD
−−→AO+−−→OE=−−→AE
−−→AO+−−→OF=−−→AF
−−→AO+−−→OG=−−→AG
−−→AO+−−→OH=−−→AH
On adding all the above equations, we get,
8−−→AO=−−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH
∵−−→OB is cancelled by −−→OF and similarly −−→OC, −−→OD. Also, −−→AO=−−→OE
⇒−−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH=8(2^i+3^j−4^k)
⇒−−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH=16^i+24^j−32^k
From figure,−−→AO+−−→OB=−−→AB
−−→AO+−−→OC=−−→AC
−−→AO+−−→OD=−−→AD
−−→AO+−−→OE=−−→AE
−−→AO+−−→OF=−−→AF
−−→AO+−−→OG=−−→AG
−−→AO+−−→OH=−−→AH
On adding all the above equations, we get,
8−−→AO=−−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH
∵−−→OB is cancelled by −−→OF and similarly −−→OC, −−→OD. Also, −−→AO=−−→OE
⇒−−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH=8(2^i+3^j−4^k)
⇒−−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH=16^i+24^j−32^k
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