If ` vec a , vec b , and vec c` be non-zero vectors such that no two are collinear or `( vec axx vec b)xx vec c=1/3| vec b|| vec c| vec adot` If `theta` is the acute angle between vectors ` vec b and vecc` , then find the value of `sin thetadot`

Explanation

we have `(veca xx vecb ) xx vecc = 1/3 |vecb||vecc|veca`
`or (veca .vecc) vecb - (vecb .vecc)veca = 1/3 |vecb||vecc|veca`
`or (veca. Vecc) vecb- {(vecb.vecc) + 1/3 |vecb|vecc|} veca =vec0`
`Rightarrow veca.vecc =0 and vecb.vecc + 1/3 |vecb|vecc|=0`
(`veca and vecb` are non-collinear)
or `|vecb||vecc|cos theta+ 1/3 |vecb |vecc| =0`
`or cos theta = -1//3`
`Rightarrow sin theta = sqrt(8/9) = ( 2sqrt2)/3`