Translations as vectors | Year 11

01 / LEARN

The idea, step by step

Add the same displacement

A translation moves every point through the same displacement. If its vector is t, a position vector p becomes p+t. Distances, angles and orientation are preserved because differences between position vectors do not change.

Read the components as horizontal and vertical changes. The displacement does not depend on the starting point.

Linear transformations fix the origin

Every ordinary matrix product sends the zero vector to zero: $A 0 = 0$ . A non-zero translation moves the origin, so it cannot be represented by a 2 by 2 matrix acting alone on two-dimensional position vectors.

An affine rule $p \mapsto A p + t$ combines a matrix transformation with a translation. Its inverse first subtracts t, then applies the inverse matrix, when it exists.

A common trap

Using a diagonal matrix to represent a translation.

Instead: A diagonal matrix scales coordinates; a translation adds a fixed vector.

02 / WORKED EXAMPLES

See the thinking

Example 1

Translate a triangle

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

Add -2 to each x-coordinate and 1 to each y-coordinate.
For example P maps to (1-2,2+1)=(-1,3).

P^{'} = (- 1, 3)

Q^{'} = (1, 3)

R^{'} = (0, 6)

Example 2

Undo an affine rule

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

Subtract the translation (3,-1), giving (8,6).
Divide both coordinates by 2.

Input

(4, 3)

Check your understanding

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

Show answer

$(- 5, 5)$ .

Subtract start from finish.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 by 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 by 2 matrix alone sends every point two units right.
2 marks
The origin would need to move to (2,0), contradicting A0=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.
$(3, - 2)$ .

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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 ↗

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