Suppose 2 vectors v and w having, $$ \textbf{v}= \left[ {\begin{array}{c} 0\\ 1\\ 1\\ \end{array} } \right] $$
$$ \textbf{w}= \left[ {\begin{array}{c} 1\\ 1\\ 0\\ \end{array} } \right] $$
How do the combination, $$ av + bw$$ of these vectors occupy a 2-D plane in 3-D
Because $v$ and $w$ are linearly independent they will span a two dimensional subspace. This is a consequence of the fact that $av+bw=0$ if and only if $a=b=0$. If you want the equation of this plane, take the cross product of the two vectors to get a vector normal to this plane.