Prove that if `cos alpha ne 1, cos beta ne1 and cos gamma ne 1`, then the vectors `a=hati cos alpha+hatj+hatk,b=hati+hatj cos beta+hatk` and `c=hati+hatj+hatk cos gamma` can never be coplanar.

Explanation

Suppose that, a,b and c are coplanar.
`implies|(cosalpha,1,1),(1,cosbeta,1),(1,1cos gamma)|=0`
On applying `R_(2) to R_(2)-R_(1) and R_(3) to R_(3)-R_(1)`
`implies|(cos alpha,1,1),(1-cosalpha,cosbeta-1,0),(1-cosalpha,0,cosgamma-1)|=0`
`implies cosalpha(cosbeta-1)(cosgamma-1)-(1-cosalpha)(cosgamma-1)-(1-cosalpha)(cosbeta-1)=0`
On dividing throughout by `(1-cosalpha)(1-cosbeta)(1-cosgamma),` we get
`(cosalpha)/(1-cosalpha)+(1)/(1-cosbeta)+(1)/(1-cosgamma)=0`
`implies(-(1-cosalpha)+1)/(1-cosalpha)+(1)/(1-cosbeta)+(1)/(1-cosgamma)=0`
`implies-1+(1)/((1-cosalpha))+(1)/((1-cosbeta))+(1)/((1-cosgamma))=0`
`(1)/(1-cosalpha)+(1)/(1-cosbeta)+(1)/(1-cosgamma)=1`
`implies"cosec"^(2)(alpha)/(2)+"cosec"^(2)(beta)/(2)+"cosec"^(2)(gamma)/(2)=2,` which is not possible
As, `"cosec"^(2)(alpha)/(2)ge1,"cosec"^(2)(beta)/(2)ge1`
and `"cosec"^(2)(gamma)/(2)ge1`
`because`They cannot be coplanar.