Does a negative vector imply scalar multiplication?

Question

Let's say I have a vector (or something that behaves like a vector) called $u$. Every vector in a vector space has corresponding negative vector $-u$. In this case does $-u = -1u$? I ask, because if we redefine scalar multiplication by some scalar $k$ such that $k(x, y, z) = (k^2x, k^2y, k^2z)$ then perhaps $-1u=u$ because $(-1)^2=1$. If this is the case then one of the axioms of vector spaces may have been violated, namely that there must exist some inverse of $u$ such that $u+(-u)=0$. Although I don't know if $-u=u$ implies that there exists no inverse of $u$ for which the prior condition is true just that this vector would not be denoted by $-u$.