N-Dimensional Lines

Definition

Given two distinct points $x$ and $y$ in $\R ^n$, the line through $x$ and $y$ is the set of points expressable as the sum of $x$ and $\alpha (y-x)$ where $\alpha \in \R $.

In other words, the line through $x$ and $y$ is the set

\[ \Set{z \in \R ^n}{\exists \alpha \in \R , z = x + \alpha (y - x)} = \Set{x + \alpha (y-x)}{\alpha \in \R } \]

The second expression is notation for the first, and is called the set-generator notation. Notice that if $z = x + \alpha (y-x)$, then

\[ z = (1 - \alpha )x + \alpha y, \]

where $\alpha \in \R $ and $x, y \in \R ^n$. This highlights the obvious (and obviously desirable) property that the definition is symmetric in $x$ and $y$.

Visualization in the plane

Notation

We denote the line through $x$ and $y$ by $L(x,y)$.

Properties

There are a few nice properties. If $x \neq y$, then $L(x, y) = L(y, x)$. Also, if $v$ and $w$ are in $L(x, y)$ and $v \neq w$, then the line through $v$ and $w$ is the same as the line through $x$ and $y$. In symbols

\[ L(v, w) = L(x, y) \]