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Worked solution

Rotations on an Argand diagram

The point AA represents a fixed complex number aa where π/4<arg⁡(a)<π/2\pi/4<\arg(a)<\pi/2. The complex numbers −ia-ia and −a-a are represented by the points BB and CC respectively.

On a single clearly labelled Argand diagram, mark the points AA, BB and CC.

Describe the geometrical relationship between

(a) AA and BB, and

(b) AA and CC.

[2023/TJC_NYJC_VJC/FM/I/6(modified)]

CJC · Discussion · Question 9

Answer

B: 90° clockwise rotation of A about O. C: 180° rotation of A about O. All three have equal modulus.

Argand diagram

Put A in the first quadrant with argument strictly between π/4\pi/4 and π/2\pi/2. B lies in the fourth quadrant at the same distance from O, obtained by multiplying a by −i. C is opposite A in the third quadrant, also at the same distance from O. A, O and C are collinear, and OA is perpendicular to OB.

How did this part go?

(a)

B is obtained by rotating A 90° clockwise about the origin O.

How did this part go?

(b)

C is obtained by rotating A 180° about the origin O.

How did this part go?

Compare your sketch

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A represents a in the first quadrant with π/4 < arg(a) < π/2. B=−ia is in the fourth quadrant and C=−a is opposite A in the third quadrant. OA=OB=OC, A/O/C are collinear and OA is perpendicular to OB. B is a 90° clockwise rotation of A about O; C is a 180° rotation. The illustrated direction is one permitted example, not an extra numerical condition.
A represents a in the first quadrant with π/4 < arg(a) < π/2. B=−ia is in the fourth quadrant and C=−a is opposite A in the third quadrant. OA=OB=OC, A/O/C are collinear and OA is perpendicular to OB. B is a 90° clockwise rotation of A about O; C is a 180° rotation. The illustrated direction is one permitted example, not an extra numerical condition.
A represents a in the first quadrant with π/4 < arg(a) < π/2. B=−ia is in the fourth quadrant and C=−a is opposite A in the third quadrant. OA=OB=OC, A/O/C are collinear and OA is perpendicular to OB. B is a 90° clockwise rotation of A about O; C is a 180° rotation. The illustrated direction is one permitted example, not an extra numerical condition.

Common mistakes

The inequalities for arg(a) are strict; neither boundary ray is permitted.

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