Examine a given action of the additive real numbers on the Cartesian plane

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.3
Show that the additive group $\mathbb{R}$ acts on the $xy$-plane $\mathbb{R} \times \mathbb{R}$ by $r \cdot (x,y) = (x+ry,y)$.

Solution: Let $(x,y) \in \mathbb{R} \times \mathbb{R}$. We have $$0 \cdot (x,y) = (x + 0y,y) = (x,y).$$ Now let $r_1,r_2 \in \mathbb{R}$. Then \begin{align*}r_1 \cdot (r_2 \cdot (x,y)) = &\ r_1 \cdot (x + r_2 y, y)\\ =& \ (x + r_2 y + r_1 y, y) \\= & \ (x + (r_1 + r_2)y, y) \\=&\  (r_1 + r_2) \cdot (x,y).\end{align*}