If `|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-a)^2|=0` and vectors ` vec A , vec B ,a n d vec C , w h e r e vec A=a^2 hat i+a hat j+ hat k` , etc, are non-coplanar, then prove that vectors ` vec X , vec Ya n d vec Z ,w h e r e vec X=x^2 hat i+x hat j+ hat k` , etc. may be coplanar.

Explanation

` D = D_(1) D_(2)` (see determinants)
`=2|{:(a^(2),a,1),(b^(2),b,1),(c^(2),c,1):}||{:(1,x,x^(2)),(1,y,y^(2)),(1,z,z^(2)):}| = 0`
since `vecA, vecB and vecC` are non- coplanar, `D_(1) ne 0`,
`D_(2)= 0 or |{:(x^(2),x,1),(y^(2),y,1),(z^(2),z,1):}|=0`
or `vecX , vecY and vecZ` are coplanar.