Points `veca , vecb vecc and vecd` are coplanar and `(sin alpha)veca + (2sin 2beta) vecb + (3sin 3gamma) vecc - vecd= vec0` . Then the least value of `sin^(2) alpha + sin^(2) 2beta + sin^(2) 3gamma` is

Options

A. `1//14`
B. 14
C. 6
D. `1//sqrt6`

Answer

a

Explanation

Points `veca , vecb , vecc and vecd` are coplanar, therefore,
`sin alpha + 2 sin 2 beta + 3 sin 3 gamma =1 `
` now |sin alpha + 2sin 2 beta + 2sin 3 gamma|`
`le sqrt(1+4+9). sqrt(sin^(2)alpha +sin^(2)2beta + sin^(2) 3gamma)`
`or sin ^(2) alpha + sin^(2) 2beta + sin ^(2) 3 gamma le 1/14`