= Solution
Let $s_i=w_{\alpha_i}$. The <root reflection> formula is $s_i(\gamma)=\gamma-\langle\gamma,\alpha_i^\vee\rangle\alpha_i$. Applying the <Cartan integers> from part (b), the images of the chosen <simple roots> are
$$
\boxed{\begin{array}{c|ccc}
&\alpha_1&\alpha_2&\alpha_3\\\hline
s_1&-\alpha_1&\alpha_1+\alpha_2&\alpha_1+\alpha_3\\
s_2&\alpha_1+\alpha_2&-\alpha_2&\alpha_3\\
s_3&\alpha_1+\alpha_3&\alpha_2&-\alpha_3
\end{array}.}
$$
In particular, the two end-node <Weyl reflections> fix the opposite end node.
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