Is there a subspace of $R^3$ of dimension $1$ that contains the vectors $v=(1,1,2)$ and $w=(1,-1,2)$? 
Is there a subspace of $R^3$ of dimension $1$ that contains the vectors $v=(1,1,2)$ and $w=(1,-1,2)$?

I see that $v$ and $w$ linearly independent so I think that there isn't a subspace of dimension $1$ that contains both vectors. But I think that there is a subspace of dimension 2 that contains $v$ and $w$.
My question is whether my reasoning is correct and how to justify it.

It suffices to note that since $\vec v$ and $\vec w$ are linearly independent they span, as a basis, a subspace with dimension $2$. Therefore there isn't a subspace of dimension $1$ that contains both vectors. 
Indeed if such subspace would exist with basis $\{\vec u\}$ we had


*

*$\vec v=a \vec u \quad \vec w=b \vec u\implies \vec v=c\vec w$


which is not true.
