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 `vecA, vecB and vecC` , where `vecA=a^(2)hati=ahatj+hatk` etc. are non-coplanar, then prove that vectors `vecX, vecYand vecZ " where " vecX =x^(2)hati+xhatj+hatk` . 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.