Let $X = \mathbb{R}^{n}\setminus \{0\}$, where $0$ is the origin in $\mathbb{R}^n$.
For $x, y \in X $ we define $x \sim y$ as follows:
$x \sim y$ if there exists a nonzero real number $λ$ such that $x = λy$ ($λ$ may depend on the particular $x$ and $y$, it is not a fixed constant).
(a) Show that $\sim$ is an equivalence relation on $X$.
(b) Show that any two equivalence classes $[x]$ and $[\tilde{x}]$, $x, \tilde{x} \in X $, are either disjoint or they coincide.
(c) Describe the set $X/ ∼$ geometrically, i.e., what is the equivalence class $[x]$ if $x ∈ X$?
You have to proof the axioms of a equivalence relation: Axioms
The following are just hints
(a) Reflexivity $\lambda = 1$, Symmetry use $\frac{1}{\lambda}$, Transitivity use the product of the factors which appear
(b) is just a consequence of the fact that $\sim$ is a equivalence relation. Transitivity is the key.
(c) $1$-dimensional subspace that contains $x$ (a line)