Let `overset(to)(a) =2hat(i) + hat(j) -2hat(k) " and " overset(to)(b) = hat(i) + hat(j) . " If " overset(to)(c ) ` is a vectors such that `|overset(to)(a)"." overset(to)(c ) = |overset(to)( c)| , |overset(to)(c )- overset(to)(a)|= 2sqrt(2)` and the angle between `(overset(to)(a) xx overset(to)(b)) " and " overset(to)( c ) " is " 30^(@), " then "|(overset(to)(a) xx overset(to)(b)) xx overset(to)( c )|` is equal to

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `2`
D. `3`

Answer

B

Explanation

Note in this question vectors `vec(c ) ` is not given therfore we cannot apply the formulae`vec(a) xx vec(b) xx vec(c )` (vector triple product )
Now `|(vec(a) xx vec(b)) xx vec(c ) | = | vec(a) xx vec(b)||vec(c )| sin 30^(@)`
Again `|vec(a) xxvec(b) | = |{:(hat(i) ,,hat(j) ,,hat(k)),(2,,1,,-2),(1,,1,,1):}|=2hat(i) -2hat(j) + hat(k)`
`rArr |vec(a) xx vec(b) | = sqrt(2^(2) + (-2)^(2)+1) = sqrt(4+4+1) = sqrt(9) =3`
Since `|vec(c )- vec(a)| = 2 sqrt(2)`
` rArr |vec(c )- vec(a) |^(2)=8`
`rArr (vec(c )- vec(a)) ". " (vec(c ) - vec(a)) =8`
`rArr vec(c ) ". " vec(c ) - vec( c) vec(a) - vec(a) "." vec(c ) + vec(a) ". " vec(a) =8`
` rArr |vec(c)|^(2)+ |vec(a)|^(2)- 2 vec(a) ". " vec(c ) =8`
` rArr |vec(c )|^(2) -2|vec(c)|+ 1=0`
`rArr (|vec(c )|-1)^(2) =0 rArr |vec(c )|=1`
From Eq. (i) , `|(vec(a) xx vec(b)) xx vec(C )| = (3) (1) . ((1)/(2)) = (3)/(2)`