Suppose vector(a, b, c) are coplanar
If vector(a = 0), then vector(a*b = 0) and hence
Let vector{a ≠ 0}. If a, b are collinear (parallel) then vector{a*b = 0} and hence
Let vector{a,b} be non-collinear. Then vector c lies in the plane generated by vector{a,b} and hence vector c is perpendicular to vector{a*b}.
Thus 
Conversely suppose that 
If vector{a = 0 or b = 0 or c = 0, then a, b, c} are coplanar.
Let vector{a ≠ 0, b ≠ 0, c ≠ 0}.
If vector{a,b} are collinear (parallel), then vector{b = αa} for some scalar α. Since any two vectors are coplanar, vector{a,c} are coplanar
=> vector{αa, c} are coplanar
=> vector{b, c} are coplanar
=> vector{a, b, c} are coplanar.
Let vector{a, b} be non-collinear. Then vector{a*b ≠ 0}.
=> vector{a*b} is perpendicular to vector c
Since vector[a*b} is a vector perpendicular to the plane generated by vector{a, b} , it follows that vector c lies in the plane generated by vector{a, b} . Thus vector{a, b, c} are coplanar.
