Let `vec a, vec b, and vec c` be three non coplanar unit vectors such that the angle between every pair of them is `pi/3`. If `vec a xx vec b+ vecb xx vec c=p vec a + q vec b + r vec c` where p,q,r are scalars then the value of `(p^2+2q^2+r^2)/(q^2)` is

Answer

4

Explanation

`|veca|=|vecb|=|vecc|=1`
`veca. Vecb=vecb.vecc= vecc.veca= 1//2`
Also , `veca xx (vecb xx vecc) =p + q (veca.vecb) + r(veca.vecc)`
`p + q/2 + r/2 = [veca vecb vecc] `
Similarly, taking dot product with vector `vecb` we get
` p/2 + q+ r/2 =0`
And , taking dot product with vector `vecc` , we get
`p/2 + q/2 = r= [veca vecb vecc]`
solving (1) (2) and (3), we get
p=r=-q
` Rightarrow (P^(2) +2q^(2)+r^(2))/(q^(2)) = 4`