= Solution
The <additive identity> for the shifted operation must satisfy $\mathbf x+\mathbf0_{\oplus}-\mathbf a=\mathbf x$, giving $\mathbf0_{\oplus}=\mathbf a$. The <additive inverse> must satisfy $\mathbf x+\mathbf x'-\mathbf a=\mathbf a$, giving
$$
\boxed{\mathbf0_{\oplus}=\mathbf a,\qquad\mathbf x'=2\mathbf a-\mathbf x.}
$$
The notation distinguishes the new <additive identity> from the ordinary zero <vector>. This is a translated <abelian group> structure on the same underlying set.
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