Let `veca,vecb` and `vecc` be non-zero vectors such that no two are collinear and
`(vecaxxvecb)xxvecc=1/3|vecb||vecc|veca`
If `theta` is the acute angle between the vectors `vecb` and `vecc` then `sin theta` equals

Options

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

Answer

A

Explanation

We have
`(vecaxxvecb)xxvecc=1/3|vecb||vecc|veca`
`implies(veca.vecc)vecb-(vecb.vecc)veca=1/3|vecb||vecc|veca`
`implies(veca.vecc)vecb-{(vecb.vecc)+1/3|vecb||vecc|}veca=vec0`
`impliesveca.vecc=0` and `vecb.vecc+1/3|vecb||vecc|=-` [`veca, vecb` are non collinear ]
`implies|vecb||vecc|cos theta+1/3|vecb||vecc|=0`
`implies cos theta=1/3impliessin theta=sqrt(8/9)=(2sqrt(2))/3`