Vector Translations

Understanding 2D Column Vectors

To mathematically describe this movement, we use a column vector. This is a vertical stack of two numbers inside brackets, written as $\binom{x}{y}$.

• Horizontal component ($x$): The top number dictates left/right movement. A positive value means movement to the right, while a negative value means movement to the left.
• Vertical component ($y$): The bottom number dictates up/down movement. A positive value means movement up, while a negative value means movement down.

Step-by-Step: Describing a Translation from a Grid

In AQA exams, you will often need to identify the vector that maps a shape onto its new position.

1. First, select one specific corner (vertex) on the original object.
2. Then, identify the exact corresponding vertex on the new image.
3. Next, count the horizontal steps from the object vertex to the image vertex to find the horizontal component.
4. Finally, count the vertical steps between the same two vertices to find the vertical component, and write your final answer in the $\binom{x}{y}$ format.

Worked Example: Identifying a Vector

Triangle $T$ has a vertex $A$ at $(2, 3)$. It is translated so that the corresponding vertex on the image, $A'$, is at $(-1, 7)$. Find the column vector for this translation.

Step 1: Calculate the horizontal shift ($x$).

• Moving from $x = 2$ to $x = -1$ is $3$ units to the left.
• $x = -3$

Step 2: Calculate the vertical shift ($y$).

• Moving from $y = 3$ to $y = 7$ is $4$ units up.
• $y = 4$

Step 3: State the final vector.

• The translation vector is $\binom{-3}{4}$.

Calculating Vectors Using Coordinates

You can also abstract the counting process into a simple calculation. To find the translation vector between an object point $(x_1, y_1)$ and an image point $(x_2, y_2)$, subtract the object coordinates from the image coordinates:

$$\text{Translation Vector} = \binom{x_2 - x_1}{y_2 - y_1}$$

If you need to perform a reverse translation (mapping the image back to the original object), you simply negate the signs of both components. For example, the reverse of the translation $\binom{3}{-2}$ is $\binom{-3}{2}$.

Successive Translations (Resultant Vectors)

If an object undergoes two successive translations (first moving by vector $\mathbf{a}$ and then moving by vector $\mathbf{b}$), you can describe the total movement as a single transformation. The single equivalent translation is found by adding the two vectors together to find the resultant vector: $\mathbf{a} + \mathbf{b}$. When handwriting vectors in your working out, you should underline them (e.g., $\underline{a}$) to distinguish them from standard mathematical variables.