Parallel Lines

Topics: Geometry - Line - 3D Space

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Two lines $l_1,l_2$ in ${\mathbb{R}}^3$:

$$l_1=\{P+tv\mid t\in \mathbb{R}\}$$

$$l_2=\{Q+su\mid s\in \mathbb{R}\}$$

…are parallel if and only if $u$ and $v$ are parallel. That is, if and only if $u$ and $v$ are multiples of each other ($\exists \lambda \in \mathbb{R}$ such that $u=\lambda v$).

Example 1

Let:

$$l_1=\{t(1,1,1)\mid t\in \mathbb{R}\}$$

$$l_2=\{(1,1,1)+t(1,1,1)\mid t\in \mathbb{R}\}$$

$l_1$ and $l_2$ are parallel because the vectors in their definition are the same one:

$$u=v$$

Example 2

Let:

$$l_1:\frac{x-3}{2}=\frac{y-3}{5}=\frac{z-7}{4}$$

$$l_2:\frac{x-4}{8}=\frac{y+9}{20}=\frac{z+11}{16}$$

For $l_1$, we got $u=(2,5,4)$, For $l_2$, we got $v=(8,20,16)$.

Since $v=4u$, both vectors are parallel and thus, $l_1$ and $l_2$ are also parallel.