Scalar Multiple of a Point of $\mathbb{R}^n$

Let $n$ be a natural number, let $\lambda$ be a real number, and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$.

The scalar multiple $\lambda x$ is the point of $\mathbb{R}^n$ defined by

$$\lambda x=(\lambda x_1,\dots,\lambda x_n),$$

where in each coordinate the product is the multiplication of the field of real numbers.