Let `veca, vecb, and vecc` be three non- coplanar vectors and `vecd` be a non -zero , which is perpendicular to `(veca + vecb + vecc). Now vecd = (veca xx vecb) sin x + (vecb xx vecc) cos y + 2 (vecc xx veca) `. Then

Options

A. `(vecd. (veca + vecc))/([veca vecb vecc])=2`
B. `(vecd. (veca + vecc))/([veca vecb vecc])=-2`
C. minimum value of `x^(2) + y^(2) is pi^(2)//4`
D. minimum value of `x^(2) + y^(2) is 5 pi^(2)//4`

Answer

b,d

Explanation

`vecd. veca = [veca vecb vecc] cos y=-vecd. (vecb+vecc)`
` or cos y = (vecd. (vecb +vecc))/([veca.vecb.vecc]) `
similarly, ` sin x = (vecd.(veca+vecb))/([veca vecb vecc]) and (vecd.(veca+vecc))/([veca vecb vecc])=-2`
`sin x + cos y + 2 = 0`
` or sin x + cos y =-2 `
` or sin x =-1 , cos y = -1 `
since we want the minimum value of `x^(2) + y^(2) x = -pi//2 , y = pi`
,Therefore, the minimum value of `x^(2) + y^(2) is 5 pi^(2)//4`