Identifying a line with an equivalence class

Question

I'm really struggling to come up with an equivalence relation that I feel should be simple: If we describe a line $L$ by a direction vector $\textbf{v}$ and a point $\textbf{x}$, can we introduce an equivalence relation on pairs $(\textbf{x}, \textbf{v})$ so that a given line $L$ can be identified to an equivalence class. (i.e I'm looking to find an equivalence relation $(\textbf{x}, \textbf{v})$ ~ $(\textbf{y}, \textbf{u})$ only if they represent exactly the same line.

Answer 1

How about $(\mathbf{x},\mathbf{v}) \sim (\mathbf{y}, \mathbf{u)}$ iff $\mathbf{v} = \alpha \mathbf{u}$ for $\alpha \ne 0$ and $\mathbf{y} = \mathbf{x} + \beta \mathbf{v}$ for some $\beta$? (Edit: the original answer said erroneously said $\beta \ne 0$) I believe this satisfies your needs.