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If `overset(to)(a) " and " overset(to)(b)` are vectors in space given by `overset(to)(a) = (hat(i) -2hat(j))/(sqrt(5)) " and " overset(to)(b) = (2hat(i) + hat(j) +3hat(k))/(sqrt(14)) ` then the value of
`(2overset(to)(a) + overset(to)(b)).[(overset(to)(a) xx overset(to)(b)) xx (overset(to)(a) -2overset(to)(b))]` is .........

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Verified by Experts

The correct Answer is:
5
From the given information ist is clear that
`vec(a)= (hat(i) -2hat(j))/(sqrt(5)) rArr |vec(a)| =1, |vec(b)| =1, vec(a) ". " vec(b)=0`
Now `(2vec(a) +vec(b)) .[(vec(a) xx vec(b)) xx (vec(a) -2vec(b))]`
` =(2vec(a) +vec(b)) . [a^(2) vec(b)- (vec(a) ". " vec(b)) "." vec(a) +2 b^(2) "." vec(a) -2 (vec(b) "." vec(a)) "." vec(b)]`
` =[2 vec(a) + vec(b))][vec(b)+2vec(a)] =4vec(a)^(2)+vec(b)^(2)`
`=4.1+1 =5 [as a.b =0]`
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