Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} -1\\4\\ \frac{3}{2} \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=(3,-3,0)= a(1,2,3)+b(1,2,0)+c(0,-6,2).$$
We get the system
\begin{align*}
a+b&=3\\
2a+2b-6c&=-3\\
3a+2c&=0.
\end{align*}
By solving the above system we have the soluiton
$$a=-1, \quad b=4, \quad c=\frac{3}{2}.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} -1\\4\\ \frac{3}{2} \end {array} \right].$$
