If `veca, vecb, vecc` be three vectors of magnitude `sqrt(3),1,2` such that `vecaxx(vecaxxvecc)+3vecb=vec0` if `theta` angle between`veca` and `vecc` then `cos^(2)theta` is equal to

Options

A. `3/4`
B. `1/2`
C. `1/4`
D. none of these

Answer

A

Explanation

We have
`vecaxx(vecaxxvecc)+3vecb=vec0`
`implies(veca.vecc)veca-(veca.veca)vecc+3vecb=vec0`
`implies(2sqrt(3)cos theta)veca-3vecc+3vecb=vec0`
`implies(2 cos theta )veca-sqrt(3)vecc+sqrt(3)vecb=vec0`
`implies|(2cos theta)veca-sqrt(3)vecc|=|-sqrt(3)vecb|`
`implies|2cos theta veca-sqrt(3)vecc|^(2)=|-sqrt(3)vecb|^(2)`
`implies4cos^(2)theta|veca|^(2)+3|vecc|^(2)-4sqrt(3)cos theta(veca.vecc)=3|vecb|^(2)`
`implies12cos^(2)theta+12-4sqrt(3)cos theta xx sqrt(3)xx2cos theta=3`
`implies12cos^(2)+9-24cos^(2)theta=0`
`implies12cos^(2)theta=9impliescos^(2)theta=9/12=3/4`