The Zero Vector, Negatives and Differences in a Vector Space

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $K$, with $+$, $\cdot$, $0$ and $1$, be a field, let $V$, with its addition and scalar multiplication, be a vector space over $K$, and let $u,v\in V$.

(i) Zero vector. The zero vector $0_{V}$ of $V$ is the unique $z\in V$ such that $x+z=x$ for every $x\in V$, given by Uniqueness of the Zero Vector and of Negatives (i).

(ii) Negative. The negative $-v$ of $v$ is the unique $w\in V$ with $v+w=0_{V}$, given by Uniqueness of the Zero Vector and of Negatives (ii).

(iii) Difference. The difference of $u$ and $v$ is $u-v=u+(-v)$.