This question is from the book Abstract Algebra: Theory and Applications by Thomas W. Judson.
Chapter 1 – Question 29
Projective Real Line. Define a relation on $\mathbb{R}^2 \setminus \{(0, 0)\}$ by letting $(x_1, y_1) \sim (x_2, y_2)$ if there exists a nonzero real number $\lambda$ such that $(x_1, y_1) = (\lambda x_2, \lambda y_2)$. Prove that $\sim$ defines an equivalence relation on $\mathbb{R}^2 \setminus \{(0, 0)\}$. What are the corresponding equivalence classes? This equivalence relation defines the projective line, denoted by $P(\mathbb{R})$, which is very important in geometry.