Let `vec(alpha)=ahati+bhatj+chatk, vec(beta)=bhati+chatj+ahatk` and `vec(gamma)=chati+ahatj+bhatk` be three coplnar vectors with `a!=b`, and `vecv=hati+hatj+hatk`. Then `vecv` is perpendicular to

Options

A. `vec(alpha)`
B. `vec(beta)`
C. `vec(gamma)`
D. all of these

Answer

D

Explanation

It is given `vec(alpha),vec(beta)` and `vec(gamma)` are coplanar vectors.
`:.[(vec(alpha), vec(beta),vec(gamma))]=0`
`=|(a, b, c),(b, c, a),(c,a, b)|=0`
`implies 3abc-a^(3)-b^(3)-c^(3)=0`
`impliesa^(3)+b^(3)+c^(3)-3abc=0`
`implies(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)=0`
`impliesa+b+c=0 [ :' a^(2)+b^(2)+c^(2)-ab-bc-ca!=0]`
`impliesveca.vec(alpha)=vecv.vec(beta)=vec(alpha).vec(gamma)=0`
`impliesvecc` is perpendicular to `vec(alpha),vec(beta)` and `vec(gamma)`.