Answer
$$[x]_{B'}=\left[ \begin {array}{ccc} 0\\ -1\\2 \end {array} \right].$$
Work Step by Step
Writing $x$ as a linear combination of the basis $B'$ as follows
$$x=\left(11,18,-7\right)= a\left(4,3,3\right)+b\left(-11,0,11\right)+c\left(0,9,2\right).$$
We get the system
\begin{align*}
4a-11b&=11\\
3a+9c&=18\\
3a+11b+2c&=-7.
\end{align*}
By solving the above system we have the soluiton
$$a=0, \quad b=-1, \quad c=2.$$
Thus, the coordinate matrix of $x$ in $R^3$ relative to the
basis $B'$ is
$$[x]_{B'}=\left[ \begin {array}{ccc} 0\\ -1\\2 \end {array} \right].$$
