Let `overset(to)(a) , overset(to)(b) " and " overset(to)( c)` be three non-zero vectors such that no two of them are collinear and `(overset(to)(a) xx overset(to)(b)) xx overset(to)( c) = (1)/(3) |overset(to)(b)||overset(to)(c )|overset(to)(a).`
If 0 is the angle between vectors `overset(to)(b) " and " overset(to)(c )` then a value of sin 0 is

Options

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

Answer

A

Explanation

Given `(vec(a) xx vec(b)) xx vec(c ) = (1)/(3) |b||vec(c )|vec(a)`
`rArr -vec(c ) xx (vec(a) xx vec(b)) = (1)/(3) |vec(b)||vec(c )|vec(a)`
`rArr -(vec(c )". " vec(b)) "." vec(a) + (vec(c ) "." vec(b)) vec(b) = (1)/(3) |vec(b)||vec(c )|vec(a)`
`rArr [ (1)/(3) |vec(b)||vec(c )|+ (vec(c ) ". " vec(b))] a = (vec(c ) ". " vec(a)) vec(b)`
Since a and b are not collinear .
`:. vec(c ) ". " vec(b) + (1)/(3) |vec(b)||vec(c )| =0 " and " vec(c ) ". " vec(a) =0`
` rArr |vec(b) ||vec(c )| cos 0 + (1)/(3) |vec(b) ||vec(c)| =0 rArr |vec(b)||vec(c )| (cos 0+ (1)/(3)) =0`
` rArr cos 0 + (1)/(3) =0 [ :' |b| ne 0 , |c | ne 0]`
`rArr cos 0= (1)/(3) rArr sin 0 = (sqrt(8))/(3) = (2sqrt(2))/(3)`