Let `alpha epsilon R` and the three vectors `veca=alpha hati+hatj+3hatk, vecb=2hati+hatj-alpha hatk` and `vecc=alpha hati-2hatj+3hatk`. Then the set `S={alpha: veca, vecb` and `vecc` are coplanar}:

Options

A. contains exactly two positive numbers
B. contains exactly two numbers only one of which is positive
C. is empty
D. is singleton

Answer

C

Explanation

`|(alpha, 1,3),(2,1,-alpha),(alpha,-2,3)|=0`
`alpha(3-2alpha)-1(6+alpha^(2))+3(-4- alpha)=0`
`3alpha -2 alpha^(2)-6alpha^(2)-12-3alpha=0`
`3alpha^(2)=-18`
`alpha^(2)=-6`