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if overset(to)(a), overset(to)(b) " and ...

if `overset(to)(a), overset(to)(b) " and " overset(to)(c )` are unit vectors satisfying
`|overset(to)(a)-overset(to)(b)|^(2)+|overset(to)(b)-overset(to)(c)|^(2)+|overset(to)(c)-overset(to)(a)|^(2)=9`
`|2overset(to)(a) +5overset(to)(b)+5overset(to)(c)|` is equal to

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The correct Answer is:
`(3)`
PLAN if a b, c are any three vectors
Then `|vec(a)+vec(b)+vec(c )|^(2) ge 0`
`rArr |vec(a)|^(2) + |vec( c)|^(2) + 2(vec(a) ". " vec(b) + vec(b)"."+vec(c )"." vec(a)) ge 0`
` :. , vec(a)". " vec(b)+ vec(b) "." vec(c ) + vec(c ) ". " vec(a) ge (-1)/(2) (|vec(a)|^(2)+|bec(b)|^(2) + |vec(b)|^(2))`
Given `|vec(a) -vec(b)|^(2)+ |vec(b)-vec(c )|^(2) + |vec( c)-vec(a)|^(2) =9`
`rArr |vec(a) |^(2) + |vec(b)|^(2)-2vec(a) "." vec(b) + |vec(b) + |vec(b)|^(2) + |vec(c )|^(2) =2 vec(b)". " vec( c )+ |vec(c )|^(2)+ |vec(a)|^(2)`
`-2 vec( c) "." vec(a)=9`
`rArr 6-2 (vec(a) ". " vec(b) + vec(b)". " vec(c ) +vec(c ).". " vec(a)) =9 [:' |vec(a)|= |vec(b)|=|vec(c )|=1]`
`rArr vec(a)". " bec(b) + vec(b) "." vec(c ) + vec( c) ". " vec(a) = - (3)/(2)`
Also `vec(a) ". " vec(b) + vec(b) "." vec(c ) + vec( c) ". " vec(a) ge (-1)/(2) (|vec(a)|^(2)+|vec(b)|^(2)+|vec(c )|^(2))`
` ge-(3)/(2) `
From Eqs. (i) and (ii) `|vec(a)+vec(b) +vec(c ) |=0`
as`vec(a) "." vec(b) + vec(b) "." vec(c ) + vec(c ) "." vec(a)` is minimum when `|vec(a) + vec(b) + vec( c)|=0`
`rArr vec(a) + vec(b) +vec( c) =0`
`:. |2 a+5 b + 5c|=|2a +5 (b+c ) |=|2a -5a|=3`
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