Question

Prove that vector a, vector b, vector c are non coplanar if and only if vector a*vector b,vector b+vector c,vector c+vector a are non coplanar

Answer 1

Yes, in fact a⃗ +b⃗ a→+b→ is coplanar with both a⃗ a→ and b⃗ b→, b⃗ +c⃗ b→+c→ is coplanar with both b⃗ b→ and c⃗ c→ and c⃗ +a⃗ c→+a→ is coplanar with both c⃗ c→ and a⃗ a→

So, as there does not exist a common plane for all of the vectors, there cannot exist neither a common plane for vectors that are coplanar to each.