We have seen in the definition of a straight line that in the $n$-dimensional number space, a straight line can be defined given two points $A$ and $B,$ $A\neq B$ with the Cartesian coordinates $$\begin{array}{rcl}A&=&(x_1,\ldots,x_n)\\B&=&(y_1,\ldots,y_n)\end{array}$$

as a map $AB:\mathbb R\to \mathbb R^n$ with the formula $$AB(t):=(x_1+t(y_1-x_1),x_2+t(y_2-x_2)\ldots,x_1+t(y_n-x_n)).$$ We want to explain this seemingly complicated formula. First of all, note that the above formula is very similar to what we have seen in our discussion about points vs. vectors in a number space. Using the argumentation given there, the above formula can be written much more concisely as follows:

Definition: Describing a Straight Line Using Two Vectors

Let $A,B$ be points of an $n$-dimensional number space $\mathbb R^n$ with $A\neq B$ and the origin $O\in\mathbb R^n.$ The subset $L\subset \mathbb R^n$ of points defined

$$L:=\{P\in\mathbb R^n:\;\overrightarrow{OP}=\overrightarrow{OA}+t\cdot \overrightarrow{AB}\},$$ where $\overrightarrow{OA}$ and $\overrightarrow{AB}$ are vectors and $t\cdot \overrightarrow{AB}$ is a scalar multiplication with $t\in\mathbb R$ is called a straight line in $\mathbb R^n.$


You can study the behavior of this formula in the following figure for $n=2$ (i.e. in the number plane). Of course, the situation is the same for all dimensions $n.$

In the above figure, you can move the points $A$ and $B$ as well as the point $AB(t)$ depending on the parameter $t$ (try it!). The figure demonstrates that as the parameter $t$ changes, taking all possible real values, the point $AB(t)$ will move along the points of a straight line. In this sense, the set of all points $P$ whose vectors $\overrightarrow{OP}$ are given by $\overrightarrow{OA}+t\cdot \overrightarrow{AB}$ for all $t\in\mathbb R$ is a subset of the number plane which actually is the straight line $AB.$

Lemmas: 1
Proofs: 2 3 4
Propositions: 5
Theorems: 6

Thank you to the contributors under CC BY-SA 4.0!