GCSE · Maths · AQA · Spec 8300

Translations as 2D vectors

Slide a shape across a grid without turning or resizing it, and every corner makes exactly the same journey. So just two numbers can describe the whole move.

Translations on a grid

Follow corner A

View: Right and up. Showing 3 layers: Grid, Object ABC, Image 5 right, 2 up

View

try all four views — watch the signs

Corner A goes 5 right, then 2 up. Now look at B and C: their dashed arrows are exactly the same length and point exactly the same way. One journey, three corners. We write it as a column vector, across on top and up/down underneath. Typed on one line, that is (5 over 2).

Grey is the object: where the shape starts. Gold is the image: where it ends up. The object never moves; only the image changes between views.

Which one would you write?

Four answers, one translation

On a squared grid, triangle P is 2 squares wide. Triangle Q is an exact copy of P. It sits to the right of P with 3 empty squares between them, and it is 2 squares higher.

Which column vector describes the translation from P to Q?

(3 over 2). There are 3 empty squares between the triangles, and Q is 2 higher.(5 over 2). Each corner of P goes 5 right and 2 up to its copy on Q.(2 over 5). Q is 2 squares higher and 5 squares along.(−5 over −2). From Q back to P is 5 left and 2 down.

How sure are you?

I'm sureI think soI'm guessing

(3 over 2). There are 3 empty squares between the triangles, and Q is 2 higher.

This is a common misconception

The 3 empty squares are the gap between the shapes, not the distance any corner travels. Put your finger on P's left-hand corner and slide it to Q's left-hand corner: it crosses P's own 2 squares of width and then the 3-square gap. That is 5 across, so the translation is (5 over 2).

(5 over 2). Each corner of P goes 5 right and 2 up to its copy on Q.

This matches the evidence

Spot on. You followed one corner to the same corner on the image: across the 2-square width of P plus the 3-square gap makes 5 right, then 2 up. Right and up are both positive, and across goes on top.

(2 over 5). Q is 2 squares higher and 5 squares along.

This is a common misconception

Your distances are right, but they are stacked the wrong way up. The top number is always the movement across and the bottom number is up or down. So 5 right and 2 up is written (5 over 2). The order matters: (2 over 5) means 2 right and 5 up, which is a different move.

(−5 over −2). From Q back to P is 5 left and 2 down.

This is a common misconception

That describes the journey from Q back to P: the reverse move. "The translation from P to Q" starts at the object, P, and ends at the image, Q. From P, the corners go right and up, so both numbers are positive: (5 over 2).

Pick one corner of the object, find the same corner on the image, and count that corner's journey: across first (right +, left −), then up or down (up +, down −).

Your turn to supply the steps

There and back again

Triangle A has corners (−2, 3), (1, 3) and (−2, 5). Triangle B has corners (3, −1), (6, −1) and (3, 1). Describe the translation from A to B as a column vector, then the translation from B back to A.

Pick one corner of A and its match on B: (−2, 3) moves to (3, −1).Any matching pair works, because every corner makes the same journey.
missing step

Which line is step 2?

Across: 3 − (−2) = 5, so 5 right. Top number: 5.Across: 3 − 2 = 1, so 1 right. Top number: 1.Across: −2 − 3 = −5, so 5 left. Top number: −5.

Now read a vector

Which corner went the wrong way?

Translate triangle T, which has corners (1, 1), (4, 1) and (1, 3), by the vector (−2 over 5).

A student's answer — which line goes wrong?

The vector (−2 over 5) means 2 left and 5 up.
So every x-coordinate goes down by 2 and every y-coordinate goes up by 5.
(1, 1) moves to (1 − 2, 1 + 5) = (−1, 6).
(4, 1) moves to (4 + 2, 1 + 5) = (6, 6).
(1, 3) moves to (1 − 2, 3 + 5) = (−1, 8).
The image has corners (−1, 6), (6, 6) and (−1, 8).

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

In a translation every point slides by the same amount. So two signed numbers, across on top and up or down underneath, tell you everything.

What you need to know

A translation slides a shape to a new position. The image is the same size, the same shape and the same way round as the object.
Every point of the shape moves the same distance in the same direction, so one corner's movement describes the whole translation.
We describe a translation with a column vector. The top number is the movement across; the bottom number is the movement up or down.
Signs give the direction: right and up are positive, left and down are negative.
Always measure from the object to the image. The translation back from the image to the object has the same numbers with both signs flipped.

The big picture

A translation slides a shape to a new position without turning, flipping or resizing it. Every point moves the same distance in the same direction, so one column vector describes the whole move. The top number is the movement across (right is positive, left is negative). The bottom number is the movement up or down (up is positive, down is negative). To describe a translation, follow one corner from the object to the matching corner of the image.

Key points

1Translation = slide. No turning, no flipping, no change of size.

2Column vector: across on top, up or down underneath.

3Right +, left −. Up +, down −.

4Count from a corner of the object to the SAME corner of the image, never across the gap between the shapes.

5With coordinates: across = image x − object x, and up/down = image y − object y.

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Worked example

Problem

Under a translation, the corner (3, −2) of shape S moves to (−1, 4). Where does the corner (5, 1) of S move to?

⚠ Watch out

Describing the move backwards, from the image to the object. "The translation from P to Q" starts at P. If P's corner is at (1, 1) and Q's is at (4, 3), the answer is (3 over 2), not (−3 over −2), because you count from P to Q.

🧠

Memory hook

Along the corridor, then up or down the stairs: across goes on top, up or down goes underneath. Right and up are plus; left and down are minus.

✓

Check yourself

Shape F's corner (4, −1) has its image at (−2, 3). Which vector takes F to its image, and which takes the image back? ((−6 over 4); (6 over −4).)

Flashcards

(6)

What does a translation do to a shape?

Slides it to a new position. Same size, same shape, same way round: nothing turns or flips.

What do the two numbers in a translation's column vector mean?

Top: movement across (right +, left −). Bottom: movement up or down (up +, down −).

Why does one corner's movement describe the whole translation?

Every point of the shape moves the same distance in the same direction.

How do you find a translation vector from coordinates?

Take one corner and its image. Across = image x − object x. Up/down = image y − object y.

The translation from P to Q is (a over b). What takes Q back to P?

(−a over −b): the same distances with both directions reversed.

Why is counting the empty squares between two shapes wrong?

The gap leaves out the width of the shape itself. Count from a corner to the SAME corner on the image.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Four answers, one translation

There and back again

Which corner went the wrong way?

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