What's the definition of line segments in the context of Euclidean spaces?

Question

Let $\mathbb R^3$ be the set ${\{(r_1,r_2,r_3):r_1,r_2,r_3 \in \mathbb R}\}$.

Let $d:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a function defined by $d(x,y)=\sqrt {(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}$ where $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$.

The structure $(\mathbb R^3,d)$ is then called the Euclidean space.

How are line segments defined in it?

Answer 1

The segment $\overline {xy}$ is defined as the set of points $\left \{s \in \Bbb R^3:s=tx+(1-t)y, t\in \Bbb R, 1\ge t \ge0 \right \}$. This works in $\Bbb R^n$ $\forall n \in \Bbb N$.

Why is it defined like that? We are just bounding from $x$ to $y$ the only straight line for $x$ and $y$. In fact if $t=1$ then we have $x$, for $t=0$ we have $y$, otherwise we have a point on the straight line between $x$ and $y$.