Consider the vectors u=a i +b j +c k , v=a^2 i +b^2 j +c^2 k and w=a^3 i +b^3 j +c^3 k . These vectors are coplanar if and only if

Step-by-step solution

Here aligned u & =a i +b J +c k v & =a^2 i +b^2 j +c^2 k w & =a^3 i +b^3 j +c^3 k aligned u , v , w are coplanar. aligned & | array lll a & b & c a^2 & b^2 & c^2 a^3 & b^3 & c^3 array |=0 & a b c | array ccc 1 & 1 & 1 a & b & c a^2 & b^2 & c^2 array |=0 & a b c | array ccc 1 & 0 & 0 a & b-c & c-a a^2 & b^2-c^2 & c^2-a^2 array |=0 [using & a b c(a-b)(b-c)(c-a)=0 aligned Either one of a, b, c is zero or any two of a, b, c are equal.