Find a simple vector equation for $B$ and describe the set geometrically.
So we let corresponding constants be $c_1, c_2, c_3 \in \mathbb{R}$ for the vectors.
$$\overrightarrow{x} = c_1\overrightarrow{v_1} + c_2\overrightarrow{v_2} + c_3\overrightarrow{v_3} $$
So I get
$$\overrightarrow{x} = \begin{bmatrix} c_1 + c_2 + 3c_3 \\ c_3 \\ c_1 + 2c_2 \\ c_1 + c_2 \end{bmatrix}$$
But what does this represent geometrically?
Since the vectors are linear independent, their span is a plane $(3-$flat) in $R^4$ through the zero vector.
Hint
You need to solve for $s,t,u, v$ in and get an equation involving these. \begin{align*} c_1 + c_2 + 3c_3 & = s\\ c_3 & = t\\ c_1 + 2c_2 & = u\\ c_1 + c_2 & = v \end{align*} Which amounts to $$ \begin{bmatrix} 1 & 1 & 3 & | s\\ 0 & 0 & 1 & | t\\ 1 & 2 & 0 & | u\\ 1 & 1 & 0 & | v\\ \end{bmatrix} \longrightarrow \begin{bmatrix} 1 & 1 & 3 & | &s\\ 0 & 1 & -3 & | &u-s\\ 0 & 0 & 1 & | &t\\ 0 & 0 & 0 & | &v-s+3t \end{bmatrix} $$ So for consistency you need...