Euclidean Space $\mathbb{R}^n$ is a Real Vector Space

Let $n$ be a natural number. Let $\mathbb{R}$ denote the real numbers, which form in particular a field, with additive identity $0$, multiplicative identity $1$, and additive inverse $-t$ of an element $t$; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$.

Then Euclidean space $\mathbb{R}^n$, equipped with the sum of points as its addition and the scalar multiple as its scalar multiplication, is a real vector space.

Moreover, let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points of $\mathbb{R}^n$. Then the following hold.

1. (Zero vector) The origin $0_{\mathbb{R}^n}$ is a zero vector of this vector space, and it is the only one.

2. (Additive inverse) The point $(-x_1,\dots,-x_n)$ is the unique point $w$ of $\mathbb{R}^n$ satisfying $x+w=0_{\mathbb{R}^n}$, and it equals the scalar multiple $(-1)x$. It is denoted $-x$.

3. (Agreement with the difference) The difference $x-y$ of points of $\mathbb{R}^n$ satisfies

$$x-y=x+(-y).$$