Making $\mathbb R^n$ a vector space over $\mathbb R$ of dimension $1$ Is it possible to endow $V_{\mathbb R}=\mathbb R^n$ with $+$ and $\cdot$ operations in such a way that with those operations, the vector space $(V_{\mathbb R}, + , \cdot)$ over $\mathbb R$ is having dimension $1$? If yes what are examples of those operations?

Of course this is possible. Choose a bijection $b: \mathbb R^n \to \mathbb R$ and define $\lambda \cdot v = b^{-1}(\lambda b(v))$ and $v+w=b^{-1}(b(v)+b(w))$.