Describing Translations as 2D Vectors Revision Notes | GCSE AQA Higher Maths

1. What Is a Translation — and Why Does the Vector Form Matter?

Picture sliding a book across a table. It does not spin, flip, or change size — it moves. That is exactly what a translation does to a shape on a coordinate grid. Every single point shifts by the same amount, in the same direction. In geometric terms, this is not just about how the shape changes position — it's about understanding how we can mathematically represent that movement with precision using vectors, specifically 2D column vectors. A key GCSE skill is describing translations accurately using formal vector notation, ensuring clarity and correctness in your answers. Every aspect of this lesson builds toward that goal.

The challenge is describing that movement precisely. Saying "the shape moved right and up a bit" is not enough for an exam. Use a 2D column vector — a compact, unambiguous way to record both the horizontal and vertical components of the shift in one notation.

Think of the column vector like a set of instructions: "go $a$ steps across, then $b$ steps up or down." The sign of each number tells you the direction, and the magnitude tells you how far.

• $a > 0$: move right; $a < 0$: move left
• $b > 0$: move up; $b < 0$: move down
• $a = 0$: no horizontal movement (purely vertical shift)
• $b = 0$: no vertical movement (purely horizontal shift)

Mastering the sign convention is the single most important skill here — examiners regularly award or withhold marks based on whether students get the signs right.

2. Reading and Writing Translation Vectors

Now that you know what the vector means, here is how to read one from a diagram and how to write one from a description. The process always has two steps: count the horizontal shift, then count the vertical shift.

To find the vector, pick any matching pair of vertices — say $A(1, 1)$ and $A'(4, -1)$. Count how far right (or left) you travel, then how far up (or down). Here, you move $3$ units right and $2$ units down, giving $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$.

Worked example: Finding the Translation Vector from a Diagram

Triangle T has vertex $A$ at $(1, 1)$. After a translation, the image $T'$ has the corresponding vertex $A'$ at $(4, -1)$. Write the column vector for this translation.

1. $a = x_{\text{image}} - x_{\text{object}} = 4 - 1 = 3$ — Subtract the original $x$-coordinate from the image $x$-coordinate. Positive result means movement to the right. [M1]
2. $b = y_{\text{image}} - y_{\text{object}} = -1 - 1 = -2$ — Subtract the original $y$-coordinate from the image $y$-coordinate. Negative result means movement downwards.
3. $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ — Top number = horizontal component; bottom number = vertical component. [A1]
4. $B(3,1) \to B'(6,-1)$: horizontal $= 6 - 3 = 3$ ✓, vertical $= -1 - 1 = -2$ ✓ — Checking a second pair of corresponding vertices confirms the vector is consistent across the whole shape — a useful sanity check.

Answer: $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$

3. Applying a Translation Vector to Find the Image

Reading a vector from a diagram is one skill; applying a given vector to find where a shape ends up is another. This is the direction you will often work in exam questions: you are given the object and the vector, and you must find the image coordinates.

The method is straightforward — add the vector components to each vertex coordinate. Add the top number to every $x$-coordinate, and add the bottom number to every $y$-coordinate.

Worked example: Applying a Translation Vector to a Shape

Shape $P$ has vertices at $(2, 5)$, $(5, 5)$, and $(5, 2)$. Translate shape $P$ by the vector $\begin{pmatrix} -3 \\ -4 \end{pmatrix}$. Write down the coordinates of the image $P'$.

1. $a = -3$ (move 3 left), $\quad b = -4$ (move 4 down) — Both components are negative, so the shape moves left and down. Recognising this before calculating helps catch sign errors early.
2. $(2, 5) \to (2 + (-3),\; 5 + (-4)) = (-1, 1)$ — Add $a$ to the $x$-coordinate and $b$ to the $y$-coordinate. [M1]
3. $(5, 5) \to (5 + (-3),\; 5 + (-4)) = (2, 1)$ — The same vector is applied to every vertex — this is what makes it a translation.
4. $(5, 2) \to (5 + (-3),\; 2 + (-4)) = (2, -2)$ — All three vertices shift by exactly the same amount. [A1 for all three correct image coordinates]
5. $P'$ has vertices at $(-1, 1)$, $(2, 1)$, and $(2, -2)$ — The shape is congruent to the original — same size, same orientation, just in a new position.

Answer: $P'$ has vertices $(-1, 1)$, $(2, 1)$, $(2, -2)$

4. The Inverse Translation

There is a neat consequence of how vectors work: if a vector maps shape $A$ onto shape $B$, then the vector that maps $B$ back onto $A$ is found by negating both components. This is called the inverse translation.

For example, if the translation from $A$ to $B$ is $\begin{pmatrix} 5 \\ -2 \end{pmatrix}$, then the translation from $B$ back to $A$ is $\begin{pmatrix} -5 \\ 2 \end{pmatrix}$. Each component flips sign; the magnitude stays the same.

6. Exam Technique and Common Mistakes

You now have the full method. This final section brings together the mistakes that cost students marks most often, along with the language and habits that win them.

Describing translations as 2D column vectors is the core higher-tier specification requirement for this topic: every translation question at GCSE expects you to produce or interpret the $\begin{pmatrix} a \\ b \end{pmatrix}$ form, with correct signs and correct orientation.