If ` vec a , vec b ,a n d vec c` be non-zero vectors such that no tow 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 ba n d vecc` , then find the value of `s inthetadot`

Options

A. `1//3`
B. `sqrt2//3`
C. `2//3`
D. `2sqrt2//3`

Answer

D

Explanation

Since, `1/3|b||c|a=(axxb)xxc`
We know that,
`(axxb)xxc=(a*c)b-(b*c)a`
`implies1/3|a||c|a=(a*c)b-(b*c)a`
On comparing, we get
`1/3|b||c|=-|b||c|cos theta`
`impliescos theta=-1/3`
`impliescos ^(2)theta=1/9`
`implies1-sin ^(2)theta=1/9`
`impliessin^(2)theta=8/9`
` impliessin theta=(2sqrt2)/(3)`