Multiplication as rotation/dilation | Year 12

01 / LEARN

The idea, step by step

The geometry of a fixed multiplier

For $w = a z$ with $a = ρ cis ϕ \neq 0$ , every nonzero point has its distance from the origin multiplied by $ρ$ and its direction increased by $ϕ$ . This is a dilation of scale factor $ρ$ and an anticlockwise rotation through $ϕ$ about the origin.

The multiplier $i$ rotates by $π / 2$ , $- 1$ rotates by $π$ , and a positive real multiplier only changes scale. The origin remains fixed. Multiplication by zero collapses the whole plane to zero and is not an invertible rotation-dilation.

Translation and inverse transformations

A map $w = a z + b$ first multiplies by $a$ and then translates by $b$ . Its inverse is $z = (w - b) / a$ when $a \neq 0$ . Keep the order: translating first would generally give a different result.

If a known nonzero point $z$ maps to $w$ under a pure multiplication, then $a = w / z$ . For two successive multipliers, multiply them: scales multiply and angles add.

A common trap

Multiplication by $- 2$ is a reflection in the real axis.

Instead: It is a scale factor 2 followed by a half-turn. Reflection in the real axis is conjugation, $z \mapsto \bar{z}$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A quarter-turn

Under $w = i z$ , find the image of $z = 2 + i$ and describe the transformation.

$i (2 + i) = 2 i + i^{2} = - 1 + 2 i$ .
$∣ i ∣ = 1$ , so no change of scale occurs.

w = - 1 + 2 i

; anticlockwise rotation through

π / 2

about the origin.

Example 2

Recover the multiplier

A pure multiplication maps $1 - i$ to $4$ . Find the multiplier and describe it.

$a = 4 / (1 - i) = 4 (1 + i) / 2 = 2 + 2 i$ .
Its modulus is $\sqrt{8} = 2 \sqrt{2}$ and its argument is $π / 4$ .

a = 2 + 2 i = 2 \sqrt{2} cis (π / 4)

; scale by

2 \sqrt{2}

, rotate anticlockwise by

π / 4

Check your understanding

Describe multiplication by $2 cis (- π / 3)$ .

Show answer

Scale by 2 and rotate clockwise by $π / 3$ about the origin.

A negative change in argument is clockwise.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the image of $3 - 2 i$ under multiplication by $i$ .
2 marks
Describe the map $w = - 3 z$ .
2 marks
A robot step is represented by $z = 2 + i$ . A turn-and-scale instruction multiplies it by $1 + i$ . Find the new step.
3 marks
Find the multiplier taking $2$ to $- 2 + 2 \sqrt{3} i$ .
2 marks
Find the inverse of $w = 2 i z + 3$ .
3 marks
Two instructions multiply successively by $2 i$ and $- i / 2$ . What is the combined transformation?
2 marks

Read the worked solutions

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

Find the image of $3 - 2 i$ under multiplication by $i$ .
2 marks
$i (3 - 2 i) = 3 i - 2 i^{2} = 2 + 3 i$ .
$2 + 3 i$ .
Describe the map $w = - 3 z$ .
2 marks
$- 3 = 3 cis π$ .
Dilation of scale factor 3 and rotation through $π$ about the origin.
A robot step is represented by $z = 2 + i$ . A turn-and-scale instruction multiplies it by $1 + i$ . Find the new step.
3 marks
$(1 + i) (2 + i) = 2 + 3 i + i^{2} = 1 + 3 i$ . The instruction scales by $\sqrt{2}$ and rotates by $π / 4$ .
$1 + 3 i$ .
Find the multiplier taking $2$ to $- 2 + 2 \sqrt{3} i$ .
2 marks
Divide image by original: $a = (- 2 + 2 \sqrt{3} i) / 2$ . Its modulus is 2 and quadrant-II angle is $2 π / 3$ .
$- 1 + \sqrt{3} i = 2 cis (2 π / 3)$ .
Find the inverse of $w = 2 i z + 3$ .
3 marks
Subtract 3 first, then divide by $2 i$ .
$z = (w - 3) / (2 i) = - \frac{i}{2} (w - 3)$ .
Two instructions multiply successively by $2 i$ and $- i / 2$ . What is the combined transformation?
2 marks
The combined multiplier is $(2 i) (- i / 2) = - i^{2} = 1$ .
The identity transformation.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics complex numbers: Cartesian and polar representations, de Moivre, roots and roots of unity, Argand geometry and loci, complex factorisation and conjugate roots of real polynomials. Real-polynomial techniques are taught in the separate Real Polynomials chapter.

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