2022TrialShort answer

Q14 (c) (i)

2 marks

Question

On the Argand diagram, $z$ and $zw$ are represented by $A$ and $B$ . Given $z=re^{i\theta}$ and $w=e^{i\pi/3}$ , where $r>0$ , explain why $OAB$ is an equilateral triangle.

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Solution

Multiplying by $e^{i\theta}$ by $e^{i\pi/3}$ rotates the vector represented by $z$ anticlockwise through an angle of $\pi/3$ .

The length of $z$ is unchanged as $|w|=1$ .

Thus $|zw|=|z|=r$ and $\angle AOB=\pi/3$ .

Any isosceles triangle with the angle between the equal sides being $\pi/3$ is an equilateral triangle.

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Complex 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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