Let `hata` and `hatb` be two unit vectors such that `hata.hatb=1/3` and `veca xx vecb=vecc`, Also `vecF=alphahata+betahatb+lambdahatc`,
where, `alpha,beta,lambda` are scalars. If `alpha=k_(1)(hatF.hata)-k_(2)(hatF.hatb)` then the value of `2(k_(1)+k_(2))` is

Options

A. `2sqrt(3)`
B. `sqrt(3)`
C. 3
D. 1

Answer

C

Explanation

`vecF = alphahata+betahatb+lambdahatc`
`therefore vecF.(hatb xx hatc) = alpha[hatahatbhatc]`
`therefore alpha=(hatF.(hatb xx hatc))/[hatahatbhatc]`
Now, `hatb xx hatc = hatb xx (hata xx hatb) = (hatb.hatb)hata-(hatb.hata)hatb`
`=hata-1/3hatb`
`[hatahatbhatc]=[hata xx hatb)^(2)=|a|^(2)|hatb|^(2)-(hata.hatb)^(2)`
`=1-1/9=8/9`
`therefore alpha=9/8{hatF.hata-1/3hatF.hatb}`
`therefore k_(1)=9/8, k_(2)=3/8`
`therefore k_(1)+k_(2)=12/8=3/2`