Find an orthogonal basis for $\mathbb R^4$ Find orthogonal basis for $\mathbb R^4$ that contains the vectors: $v_1=$
$\begin{bmatrix}
       2            \\[0.3em]
       1            \\[0.3em]
       0            \\[0.3em]
       -1
     \end{bmatrix}$ and $v_2=$
$\begin{bmatrix}
       1            \\[0.3em]
       0            \\[0.3em]
       3            \\[0.3em]
       2
     \end{bmatrix}$
Right now I'm trying to find any basis for $\mathbb R^4$ that contains $v_1$ and $v_2$. Multiplying both equations by $\begin{bmatrix}
       x            \\[0.3em]
       y            \\[0.3em]
       z            \\[0.3em]
       w
     \end{bmatrix}$ I got $2x + y -z = 0 | x + 3y + 2z = 0$. What to do from here? How can I find these vectors so I can proceed to applying the Gram-Schmidt Process?

Take the set of vectors (say there are $k$ of them), augment it with the standard basis, apply the gram schmidt process by  first going through the vectors in the set you started with then continue with the standard basis. Then, the original set of vectors and the non-zero vectors you get after the first $k$ form a basis for the whole space. It is orthogonal by the gram schmidt process since the first $k$ are orthogonal and the remainder are also orthogonal by gram schmidt. If you get a vector which is $0$ during the gram schmidt process, drop it from the process and continue until you get 4 vectors (which is the # of elements in a basis)
That is, you start with $\{v_1, v_2, e_1, e_2, e_3, e_4\}$. Applying Gram Schmidt to this will give you $\{v_1,v_2, a,b\}$ 
