The vector \(\vec{b}\) is in the span of the other vectors.

RREF of vectors augmented by \(\vec{b}\):

\vec b = 
\begin{bmatrix} 
 20  \\ 
 -40  \\ 
 -20
\end{bmatrix}

Vectors:

\begin{bmatrix} 
 2  \\ 
 -2  \\ 
 -3
\end{bmatrix} \quad 
\begin{bmatrix} 
 2  \\ 
 -6  \\ 
 -1
\end{bmatrix} \quad 
\begin{bmatrix} 
 -12  \\ 
 20  \\ 
 14
\end{bmatrix} \quad 
        

\begin{solution}
The vector \vec{b}  is  in the span of the other vectors.
\[
\begin{bmatrix} 
1 & 0 & -4 & 5   \\ 
0 & 1 & -2 & 5   \\ 
0 & 0 & 0 & 0
\end{bmatrix}
\]
\end{solution}