2020HSCShort answer

Q14 (a) (i)

2 marks

Question

Let $z_1$ be a complex number and let
$z_2=e^{i\pi/3}z_1.$
The diagram shows points $A$ and $B$ , which represent $z_1$ and $z_2$ , respectively, in the Argand plane.

(i) Explain why triangle $OAB$ is an equilateral triangle.

Uncropped source page 10 for question 14.a.i; requires human review and cropping — Reviewed visual · source page 10

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Solution

Criteria	Marks
Provides correction solution	2
Observes either that $	z_2

Sample answer:

Multiplying $z_1$ by $e^{i\pi/3}$ rotates it anticlockwise through an angle $\pi/3$ , giving $z_2$ .

Also $|z_1|=|z_2|$ .

So $\triangle OAB$ is isosceles, giving all angles equal to $\pi/3$ , and hence $\triangle OAB$ is equilateral.

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Syllabus alignment

Working MathematicallyComplex Numbers

Geometrical implications of complex numbersInterpret complex-number operations geometrically and solve problems involving roots and loci in the complex plane.

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