Translation in the Cartesian Plane

What is Translation?

In geometry, translation is a type of transformation that slides a shape or point from one position to another without rotating, resizing, or flipping it. It involves moving every point of a shape or object the same distance in a given direction.

In the Cartesian plane, translation is described using a vector, which tells us how far and in which direction to move the points.

Translation Vector

• A translation vector is written as $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ is the horizontal movement and $b$ is the vertical movement.
• For example, the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ means move 3 units right and 2 units down.

How to Translate a Point

1. Identify the original coordinates: Start with the coordinates of the point you want to translate, say $(x, y)$.
2. Apply the translation vector: Add the vector $\begin{pmatrix} a \\ b \end{pmatrix}$ to the original coordinates. The new coordinates will be $(x + a, y + b)$.

Examples

Example 1: Translating a Point

Translate the point $(2, 3)$ using the vector $\begin{pmatrix} 4 \\ -1 \end{pmatrix}$.

Example 2: Translating a Shape

Translate the triangle with vertices $(1, 2), (3, 4), (5, 6)$ using the vector $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$.