Center the data set:
$$\begin{align*}\vec x^{(1)}&= (4, -2, 2)^T \\\vec x^{(2)}&= (2, 3, 0)^T \\\vec x^{(3)}&= (3, -1, 1)^T
\end{align*}$$
Solution
First, compute the mean:
\[\bar{\vec x} = \frac{1}{3}\left[(4, -2, 2)^T + (2, 3, 0)^T + (3, -1, 1)^T\right]= \frac{1}{3}(9, 0, 3)^T = (3, 0, 1)^T
\]
Then subtract the mean from each data point:
$$\begin{align*}\tilde{\vec x}^{(1)}&= \vec x^{(1)} - \bar{\vec x} = (4, -2, 2)^T - (3, 0, 1)^T = (1, -2, 1)^T \\\tilde{\vec x}^{(2)}&= \vec x^{(2)} - \bar{\vec x} = (2, 3, 0)^T - (3, 0, 1)^T = (-1, 3, -1)^T \\\tilde{\vec x}^{(3)}&= \vec x^{(3)} - \bar{\vec x} = (3, -1, 1)^T - (3, 0, 1)^T = (0, -1, 0)^T
\end{align*}$$