Answer
,$z=(0,-2,-3).$
Work Step by Step
Since we have $u=(1,2,3)$, $v=(2,2,-1)$, $w=(4,0,-41)$ then
\begin{align*}
2u+v-w-3z=0 &\Longrightarrow 3z=2u+v-w\\
&\Longrightarrow z=\frac{1}{3}(2u+v-w)\\
&\Longrightarrow z=\frac{1}{3}(2(1,2,3)+(2,2,-1)-(4,0,-41))\\
&\Longrightarrow z=\frac{1}{3}(0,-6,-9)\\
&\Longrightarrow z=(0,-2,-3).
\end{align*}
Hence, $z=(0,-2,-3).$