My geometry textbook states that the vectors $(a, b, c)^T$ and $k(a, b, c)^T$ represent the same line for any non-zero $k$; in other words, two such vectors related by an overall scaling are considered equivalent. It then goes on to state that an equivalence class of vectors under this equivalence relationship is known as a homogeneous vector.
The Wikipedia page for homogeneous coordinates states that for non-zero elements of $\mathbb{R}^3$, $(x_1, y_1, z_1) \sim (x_2, y_2, z_3)$ is defined to mean there is a non-zero $\lambda$ so that $(x_1, y_1, z_1) = (\lambda x_2, \lambda y_2, \lambda z_2)$. Then $\sim$ is an equivalence relation.
I'm confused as to how this could be an equivalence relationship, since equivalence relations are binary relations, whereas this is a triple relation?
If this is an equivalence relation, then what is the relation in set theoretic notation? I have the following: If $X = \{(a, b, c)^T \mid a, b, c \in \mathbb{R} \}$, then the relation on $X$ is $R = \{ (x, y, z)^T \mid (x, y, z)^T = k(a, b, c)^T \ \text{for some $k \in (\mathbb{R} \setminus \{ 0 \}) $} \}$. Is this correct? If not, then what's wrong with it?
I would greatly appreciate it if people could please take the time to clarify this.