To understand how to find the vector in a translation transformation, we first need to know what translations are.

Translations are when we move a shape to a different location, but we don’t change its size or how it looks.

In Year 10 Mathematics, students learn that we can use a vector to show these translations.

Steps to Identify the Vector

1. Look at the Original Position:

   • First, find the coordinates (the points that tell us where the shape is) of the original shape.
   • For example, imagine a triangle with points at $A(1, 2)$, $B(3, 4)$, and $C(5, 1)$.

2. Find the New Position:

   • Now, check where the shape has moved to after translation.
   • If the triangle's new points are $A'(4, 5)$, $B'(6, 7)$, and $C'(8, 4)$, these are the new coordinates.

3. Calculate the Change in Coordinates:

   • To find the translation vector, we look at how much each coordinate has changed.
   • For point $A$, we calculate:
     • Change in $x$: $4 - 1 = 3$
     • Change in $y$: $5 - 2 = 3$
   • So, the translation vector from $A$ to $A'$ is $(3, 3)$.

4. Write the Vector in Notation:

   • The translation vector can be written as $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}$.
   • This means we move 3 units to the right and 3 units up.

General Formula for Translation

The general translation vector looks like this: $\begin{pmatrix} a \\ b \end{pmatrix}$ where $a$ represents the horizontal shift and $b$ represents the vertical shift.

Conclusion

Finding the vector in translation transformations is really important. It shows how far and in which direction a shape moves.

Understanding this helps students grasp the basics of geometric transformations in coordinate geometry. The vector gives us specific movements, showing that shapes keep their properties during these translations. This keeps math nice and tidy!