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**scorpion007** For the first one: $\displaystyle S=\lbrace (a,2a,3a) \mid a \in R \rbrace \subset R^3$.

Closure under vector addition:

Let $\displaystyle \vec{x}=(a_1,2a_1,3a_1), ~\vec{y}= (a_2,2a_2,3a_2)\in S$.

$\displaystyle \vec{x}+\vec{y}=(a_1+a_2, 2a_1+2a_2, 3a_1+3a_2)$

Grouping coefficients,

$\displaystyle = (a_1+a_2,2(a_1+a_2), 3(a_1+a_2)) \in S$.

Thus S is closed under vector addition.

Scalar multiplication can be showed similarly. (Hint, define an $\displaystyle \alpha \in R$ and show that $\displaystyle \forall \vec{x} \in S,~ \alpha \vec{x} \in S$).