It is not hard to see that $(\mathbb R^2,+)$ with this product
$ {r\cdot(x,y)=(rx,ry) } $
is vector space over field $\mathbb R$.
I'm looking for another product that $(\mathbb R^2,+)$ is vector space over $\mathbb R$. I know
$n*(x,y)=\underbrace{(x,y)\oplus (x,y) \oplus \cdots \oplus (x,y)}_n=(nx,ny)$ but I have no idea for arbitrary element of $\mathbb R$. Any Suggestion. Thanks