I'm studying isometries right now, and I'm kind of confused as to how a translation changes coordinates. My book uses the notation where $t_p(\vec{x})$ denotes a translation on $\vec{x}$ by $\vec{p}$; this all makes sense so far. However, I don't understand how the translation works in practice; does the translation move the origin to the point $(6,4)$, for example, so that $(6,4)$ is the new origin, and if so, how does that even work with the rule for change of coordinates $t_p(\vec{x}') = \vec{x}$, where $\vec{x}'$ is the new coordinate vector of our original vector $\vec{x}$? Thanks in advance!
EDIT: Per the book, let $x$ and $y$ be vectors such that $y = m(x)$ for an isometry $m$. Our change in coordinates will be given by some isometry, let's denote it by $\eta$. Let the new coordinate vectors of $x$ and $y$ be $x'$ and $y'$. The new formula $m'$ is the one such that $m'(x') = y'$. We also have the formula $\eta(x') = x$ analogous to the change of basis formula $PX' = X$. To clarify, the book is Artin's Algebra.