I'm trying to solve this innocent problem.
Let $X,Y\subset \mathbb{R}^3$ be two 1-dimensional manifolds. Show that there exists $v\in \mathbb{R}^3$ such that $X$ and $Y+v$ are disjoint.
I know how to show that for any two manifolds $X,Y\subset \mathbb{R}^n$, $X$ and $Y+a$ intersect transversely for $a\in \mathbb{R}^n$ generic, which is too weak in the above context.
As $X$ and $Y$ are one-dimensional, $\operatorname{dim}(T_pX + T_pY) \leq 2 < 3 = \operatorname{dim}(T_p\mathbb{R}^3)$. Therefore, if $p \in X\cap Y$, $X$ and $Y$ are not transverse at $p$. So if $X$ and $Y$ are transverse, they cannot intersect; that is, they must be disjoint.
Your genericity result shows that there is $a$ such that $X$ and $Y + a$ intersect transversely; in particular, they are disjoint.