Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors

Let $A,B,C,D$ be points in of an $n$-dimensional number space $\mathbb R^n$ with $A\neq B$ and $C\neq D,$ and the origin $O\in\mathbb R^n,$ and let the straight lines. $$\begin{array}{rcl}L_1&:=&\{P\in \mathbb R^n:\; \overrightarrow{OP}:=\overrightarrow{OA}+s\cdot \overrightarrow{AB}\}\\L_2&:=&\{Q\in \mathbb R^n:\; \overrightarrow{OQ}:=\overrightarrow{OC}+t\cdot \overrightarrow{CD}\}\end{array}$$

be given. Then the second straight line $L_2$ is, in fact, equal the first straight line $L_1,$ if and only if $C\in L_1$ and $\overrightarrow{CD}=c\cdot \overrightarrow{AB}$ with $c\in\mathbb R,$ $c\neq 0.$

Proofs: 1

Proofs: 1 2

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