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**Ackbeet** Well, addition is defined using the usual vector addition in $\displaystyle \mathbb{R}^{3}.$ That is, given two vectors

$\displaystyle \vec{r}_{1}=\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}$ and $\displaystyle \vec{r}_{2}=\begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix},$ their sum is

$\displaystyle \vec{r}_{1}+\vec{r}_{2}=\begin{bmatrix}x_{1}+y_{1} \\ x_{2}+y_{2}\\ x_{3}+y_{3}\end{bmatrix}.$

I would try adding two vectors in S and seeing if the result looks like a vector in S. Where does that lead you?