If `vec(alpha), vec(beta),vec(gamma)` are three non-collinear unit vectors such that `vec(alpha)+2vec(beta)+3vec(gamma)` is collinear with `vec(beta)+vec(y)` and `vec(alpha)+2vec(beta)` is collinear with `vec(beta)-vec(gamma)`, then `2vec(alpha).vec(beta)+6vec(alpha).vec(gamma) +3vec(beta).vec(gamma)` equal to

Options

A. `-(11)/(2)`
B. `-5`
C. `-7`
D. `-9`

Answer

C

Explanation

`vec(alpha)+2vec(beta)+3vec(gamma)=m(vec(beta)+vec(gamma))`
`implies vec(alpha) + vec(beta) (2-m)+vec(gamma)(3-m)=vec(0)`
Also, `vec(alpha)+vec(beta) (2-n)+nvec(gamma) = vec(0)`
`vec(beta) (n-m)=vec(gamma)(n+m-3)`
implies n=m =(3)/(2) implies |2vec(alpha) + vec(beta) + 3vec(gamma)| = 0`
` implies 2vec(alpha).vec(beta)+ 6vec(alpha).vec(gamma)+3vec(beta).vec(gamma)=-7`