Translations as vectors | Year 11

01 / LEARN

Understand the idea.

Add the same displacement

A translation adds the same displacement vector to every position: $p \mapsto p + t$ . The first component of $t$ is the horizontal change and the second is the vertical change, independent of the starting point.

Translation preserves the difference between any two position vectors: $(q + t) - (p + t) = q - p$ . Thus lengths, angles and orientation stay unchanged. To find a translation vector from a pair of points, subtract the starting position from the final position.

Linear transformations fix the origin

An ordinary $2 \times 2$ matrix acting on position vectors always fixes the origin, because $A 0 = 0$ . A non-zero translation moves the origin to $t$ , so it requires vector addition rather than a matrix acting alone.

An affine rule $q = A p + t$ applies the matrix first and then translates. If $A$ is invertible, undo the operations in reverse order: subtract $t$ from the output, then multiply by $A^{- 1}$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Translate a triangle

Translate $P = (1, 2)$ , $Q = (3, 2)$ , $R = (2, 5)$ by $t = (- 2, 1)$ .

Worked solution

1
Write the common coordinate rule.
$(x, y) \mapsto (x - 2, y + 1)$
Every vertex receives the same displacement $(- 2, 1)$ .
2
Translate the first vertex.
$P^{'} = (1 - 2, 2 + 1) = (- 1, 3)$
Subtract 2 horizontally and add 1 vertically.
3
Translate the other two vertices.
$\begin{aligned} Q^{'} & = (3 - 2, 2 + 1) = (1, 3) \\ R^{'} & = (2 - 2, 5 + 1) = (0, 6) \end{aligned}$
The displacement is independent of the original vertex.
4
Check the displacement of each vertex.
$P^{'} - P = Q^{'} - Q = R^{'} - R = (- 2, 1)$
Equal displacement vectors confirm that this is a translation of the whole triangle.

Answer

$P^{'} = (- 1, 3)$ , $Q^{'} = (1, 3)$ , $R^{'} = (0, 6)$ .

EXAMPLE 02

Undo an affine rule

The rule sends $(x, y)$ to $(2 x + 3, 2 y - 1)$ . Find the input whose image is $(11, 5)$ .

Worked solution

1
Separate the scale and translation.
$q = 2 p + (\begin{array}{cccccccccc} 3 \\ - 1 \end{array}), q = (\begin{array}{cccccccccc} 11 \\ 5 \end{array})$
The coordinate rule doubles both coordinates before adding the fixed vector.
2
Remove the translation from the image.
$2 p = (\begin{array}{cccccccccc} 11 - 3 \\ 5 - (- 1) \end{array}) = (\begin{array}{cccccccccc} 8 \\ 6 \end{array})$
Subtract the translation before undoing the scale.
3
Halve both coordinates.
$p = \frac{1}{2} (\begin{array}{cccccccccc} 8 \\ 6 \end{array}) = (\begin{array}{cccccccccc} 4 \\ 3 \end{array})$
The inverse of dilation by 2 is dilation by $\frac{1}{2}$ .
4
Check the recovered input.
$(2 (4) + 3, 2 (3) - 1) = (11, 5)$
The original rule sends $(4, 3)$ to the required image.

Answer

Input $(4, 3)$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

What translation sends $(2, - 1)$ to $(- 3, 4)$ ?

Show answer & reasoning

$(- 5, 5)$ .

The displacement is $(- 3, 4) - (2, - 1) = (- 5, 5)$ . Adding this vector to the starting point gives $(2 - 5, - 1 + 5) = (- 3, 4)$ , as required.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Translate $(4, - 2)$ by $(3, 5)$ .
2 marks
Find the displacement from $(- 1, 6)$ to $(3, 2)$ .
2 marks
Translate by $(- 4, 2)$ , then by $(1, - 5)$ . State the single equivalent translation.
2 marks
Explain why no $2 \times 2$ matrix alone sends every point two units right.
2 marks
Find $p$ if $3 p + (2, - 1) = (8, 5)$ .
2 marks
Under a translation, $P = (1, 4)$ becomes $(6, 2)$ . Find the image of $Q = (- 2, 0)$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Translate $(4, - 2)$ by $(3, 5)$ .
2 marks
Add the corresponding components.
$(7, 3)$ .
Find the displacement from $(- 1, 6)$ to $(3, 2)$ .
2 marks
Subtract start from finish.
$(4, - 4)$ .
Translate by $(- 4, 2)$ , then by $(1, - 5)$ . State the single equivalent translation.
2 marks
Add the displacement vectors.
$(- 3, - 3)$ .
Explain why no $2 \times 2$ matrix alone sends every point two units right.
2 marks
The origin would need to move to $(2, 0)$ , contradicting $A 0 = 0$ .
Every matrix fixes the origin.
Find $p$ if $3 p + (2, - 1) = (8, 5)$ .
2 marks
Subtract $(2, - 1)$ to obtain $(6, 6)$ .
Divide by 3.
$p = (2, 2)$ .
Under a translation, $P = (1, 4)$ becomes $(6, 2)$ . Find the image of $Q = (- 2, 0)$ .
2 marks
The translation vector is $(5, - 2)$ .
Add this to $Q$ .
$(3, - 2)$ .

Checking saved progress…

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

In this lesson

Understand the idea.

Add the same displacement

Linear transformations fix the origin

See the method in action.

Translate a triangle

Write the common coordinate rule.

Translate the first vertex.

Translate the other two vertices.

Check the displacement of each vertex.

Undo an affine rule

Separate the scale and translation.

Remove the translation from the image.

Halve both coordinates.

Check the recovered input.

Try it yourself.

Check your understanding

Keep practising