a translation
A translation is the simplest motion of all: a pure slide. Every single point of the plane moves the same distance in the same direction — as if the whole sheet were shoved sideways without any turning or flipping. Picture pushing a book straight across a desk: it ends up in a new place but pointing exactly the same way, perfectly parallel to where it began.
We capture 'same distance, same direction' with a single arrow called the translation vector, written v = (a, b). The translation then does one thing to every point: it adds the vector. In coordinates, the point (x, y) goes to (x + a, y + b). For example, translating by v = (3, -2) sends (1, 5) to (4, 3) and sends (0, 0) to (3, -2) — the same shift, (+3, -2), applied to all. The size of the slide is the length |v| = sqrt(a^2 + b^2), and its direction is the direction of the arrow. A translation has no fixed points at all (unless v is the zero vector, which leaves everything where it is), and it preserves orientation: a clockwise-labelled triangle stays clockwise.
Translations are a direct (orientation-preserving) isometry, and composing two translations just adds their vectors — slide by v, then by w, and you have slid by v + w in one go. They show up everywhere a pattern repeats by simple shifting: the bricks in a wall, the repeating motif of wallpaper, a frieze running along a wall. In fact translational symmetry is the defining feature that separates the infinite repeating wallpaper and frieze patterns from the finite symmetry of a single rosette.
Translate triangle with vertices P(2, 1), Q(5, 1), R(5, 4) by the vector v = (-4, 3). Add (-4, 3) to each: P' = (-2, 4), Q' = (1, 4), R' = (1, 7). Side |PQ| = 3 before and |P'Q'| = 3 after; the triangle has simply slid up-and-to-the-left, unturned and unflipped.
One vector, added to every point — a pure slide with no turning, no flipping, no fixed point.
The vector, not a single moved point, defines the translation: knowing only that one point went from A to A' fixes the whole slide, since the same arrow A-to-A' moves every other point identically.