Vector Translations

The starting shape is the object and the translated shape is its image. An image may be labelled using a prime: for example, $A$ may become $A'$.

Recording movement with a column vector

A translation in two dimensions has a horizontal part and a vertical part. It is described by a column vector

where the top component, $a$, gives the horizontal movement and the bottom component, $b$, gives the vertical movement.

The signs determine the direction:

• $a>0$: move right;
• $a<0$: move left;
• $b>0$: move up;
• $b<0$: move down.

A zero component means no movement in that direction. For example,

means no horizontal movement and a movement of 4 units down. The entries are written inside curved brackets with no dividing line: a column vector is not a fraction.

In the diagram, $\begin{pmatrix}3\\-2\end{pmatrix}$ means 3 units right and 2 units down. The same movement carries every vertex of the object to its corresponding vertex on the image.

Translating coordinates

A point $(x,y)$ translated by $\begin{pmatrix}a\\b\end{pmatrix}$ moves to

For example, under the vector $\begin{pmatrix}-3\\2\end{pmatrix}$, the point $(6,-1)$ moves to

The negative top component causes movement to the left; the positive bottom component causes movement upwards. To translate a whole polygon, apply the same change to every vertex and then join the new vertices in the same order.

Describing a translation from a diagram

Choose one clear point on the object and identify the corresponding point on the image. Count the horizontal movement first and the vertical movement second. Equivalently, subtract the original coordinates from the image coordinates:

If a vertex moves from $(-4,-1)$ to $(1,2)$, the horizontal change is $1-(-4)=5$ and the vertical change is $2-(-1)=3$. The translation is therefore

Checking a second pair of corresponding vertices should give the same vector. If it does not, the points have been mismatched or the movement has been counted incorrectly.

Direction matters

A vector describes movement in a particular direction. If the translation from shape $P$ to shape $Q$ is

then the translation from $Q$ back to $P$ is

The distance in each direction is unchanged, but both signs reverse.