Translation

Slide a book across a table. Nudge a chess piece three squares along a row. Push a sofa straight across the room. In each case the object moves but never turns, never flips, and never changes size — every part of it travels the same distance in the same direction. The book that arrives is the very same book, just somewhere new.

That sliding motion is a translation. It is the gentlest of all the ways to move a shape: no spinning, no mirror, no stretching. Just a slide.

Describing a slide with a vector

A translation slides a shape across the plane without turning it or flipping it. Every point moves the same distance in the same direction. That direction-and-distance is captured by a column vector \begin{pmatrix} a \\ b \end{pmatrix}, read as "a across, b up".

The top number tells you the sideways move (right if positive, left if negative); the bottom number tells you the up-and-down move (up if positive, down if negative). In coordinates, the vector is just added to each point:

(x, y) \;\longrightarrow\; (x + a,\; y + b)

Because every point moves by exactly the same amount, the shape's size, angles and orientation are all unchanged — the image is an identical copy, just in a new place.

Seeing it on a grid

Take a triangle and translate it by \begin{pmatrix} 4 \\ 2 \end{pmatrix} — 4 across, 2 up. Step through the figure: the arrow shows the vector, and every vertex lands 4 right and 2 up from where it started. The image is the same triangle, just slid.

Worked example 1 — sliding a point

Translate the point (2, 5) by the vector \begin{pmatrix} 3 \\ -4 \end{pmatrix}.

Notice the negative bottom number moved the point down. To translate a whole shape you do this to every corner in turn — same vector each time — and join up the new corners.

Worked example 2 — finding the vector

A shape's corner sits at (-3, 0) and its matching image corner is at (3, 1). What vector slid it there? Just subtract the start from the finish:

Because a translation moves every point the same way, you can pick any pair of matching corners and you will always get the same vector. Step through the figure to see the arrow between the two triangles.

Worked example 3 — one slide after another

Slide a shape by \begin{pmatrix} 3 \\ 2 \end{pmatrix}, then slide the result by \begin{pmatrix} 1 \\ -5 \end{pmatrix}. What single translation does the same job? You simply add the vectors, top to top and bottom to bottom:

\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ -5 \end{pmatrix} = \begin{pmatrix} 3 + 1 \\ 2 + (-5) \end{pmatrix} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}

So two slides in a row are the same as one slide of \begin{pmatrix} 4 \\ -3 \end{pmatrix} — 4 across and 3 down. This is exactly how you add vectors: nose-to-tail, and the totals combine.

Two traps snare almost everyone with column vectors:

Try it: slide a shape 3 right then 2 up, or 2 up then 3 right — you land in exactly the same spot. Translations don't care what order you do them in, because adding vectors gives the same total either way: \begin{pmatrix} 3 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 2 \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \end{pmatrix} + \begin{pmatrix} 3 \\ 0 \end{pmatrix}. Rotations are fussier — spin then slide usually lands somewhere different from slide then spin — so this "order doesn't matter" freedom is a special treat of translations.

A translation is the plainest kind of rigid motion: it preserves everything — size, shape and orientation — which is exactly why a single vector is enough to describe it. No centre, no angle, no mirror line; just two numbers.

That little arrow is the school-level seed of the vectors used everywhere in physics (forces, velocities) and computer graphics. When a game slides your character across the screen, it is adding the same vector to every point of the character's shape — a translation, thousands of times a second. The book you slid across the table and the hero dashing through a game are doing the very same maths.

See it explained