Performing the indicated operations (Definition CVA, Definition CVSM), we obtain the vector equations
\begin{align*}
\colvector{3\alpha \\ 2\alpha\\ - 2\alpha} + \colvector{6\\1\\2}
&= \colvector{3\alpha + 6 \\ 2\alpha + 1\\ -2\alpha + 2}
= \colvector{0\\-3\\6}
\end{align*}
Thus, if a solution $\alpha$ exists, by Definition CVE then $\alpha$ must satisfy the three equations:
\begin{align*}
3\alpha + 6 &= 0\\
2\alpha + 1 &= -3\\
-2 \alpha + 2 &= 6
\end{align*}
which leads to $3\alpha = -6$, $2\alpha = -4$ and $-2\alpha = 4$.
And thus, the solution to the given vector equation is $\alpha = -2$.