Write a linear equation defining the subspace of R_3 spanned

Every element of your linear subspace can be written as a linear combination of ${\displaystyle v_1}$ and ${\displaystyle v_2}$. If ${\displaystyle (x_1,y_1,z_1)}$ and ${\displaystyle (x_2,y_2,z_3)}$ both satisfy
${\displaystyle ax+by+cz=0}$
then so does any linear combination of the two. Since ${\displaystyle v_1,v_2}$ span a 2 dimensional subspace of a 3 dimensional space, we only need one equation to describe it, and both ${\displaystyle v_1}$ and ${\displaystyle v_2}$ must solve it. So we need
${\displaystyle a\star 0+b\star 0+c\star 2=0}$
and
${\displaystyle a\star (-3)+b\star \left(1\right)+c\star (-1)=0}$
Solving these equations gives (a,b,c) up to some factor that we can choose, since
${\displaystyle ax+by+cz=0}$
and
${\displaystyle kax+kby+kcz=0}$
have the same solution set if ${\displaystyle k\ne 0}$