Prove that two parallel lines are either equal or have empty intersection

by RoRi

April 29, 2016

Let $L(P;A)$ and $L(Q;A)$ be two parallel lines in $\mathbb{R}^n$ . Prove that $L(P; A) \cap L(Q;A) = \varnothing$ or $L(P;A) = L(Q;A)$ .

Proof. Assume $L(P;A) \neq L(Q;A)$ . Suppose that the intersection is not empty, so that there exists a point $X \in L(P;A)$ and $X \in L(Q;A)$ . Then, we know there exist $s,t \in \mathbb{R}$ such that

$X = P + tA, \qquad \text{and} \qquad X = Q + sA.$

But then

$\begin{align*} P+tA = Q+sA && \implies && P = Q + (s-t)A \\ && \implies && P \in L(Q;A). \end{align*}$

But, by Theorem 13.2, this means $L(P;A) = L(Q;A)$ , contradicting our assumption. Hence, either $L(P;A) = L(Q;A)$ or the intersection is empty $. \qquad \blacksquare$

Stumbling Robot

Prove that two parallel lines are either equal or have empty intersection

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