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Worked solution

Modulus and principal argument in the third quadrant

Let z=−1−3iz=-1-\sqrt3i.

(i) State the value of ∣−1−3i∣|-1-\sqrt3i| and arg⁡(−1−3i)\arg(-1-\sqrt3i).

(ii) Give the geometrical meanings of the statements “∣−1−3i∣|-1-\sqrt3i|” and “arg⁡(−1−3i)\arg(-1-\sqrt3i)”.

CJC · Discussion · Question 10

Answer

∣z∣=2|z|=2, arg⁡z=−2π/3\arg z=-2\pi/3. They describe distance from O and directed angle from the positive real axis.

(i)

The modulus is 12+(3)2=2\sqrt{1^2+(\sqrt3)^2}=2. The point lies in the third quadrant, so its principal argument is −(π−tan⁡−13)=−2π/3-(\pi-\tan^{-1}\sqrt3)=-2\pi/3.

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(ii)

The modulus is the distance from (−1,−3)(-1,-\sqrt3) to the origin. The argument is the directed angle locating the position vector relative to the positive real axis. Here the principal angle is −2π/3, a clockwise turn of 2π/3 from the positive real axis; the corresponding positive anticlockwise angle is 4π/3, outside the principal interval.

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Common mistakes

The reference angle π/3 alone is not the argument of this third-quadrant point.

Solution notes

The printed geometrical description calls the argument an angle between the position vector and positive real axis in the anticlockwise direction, while its numerical answer is −2π/3. The reviewed explanation fixes the orientation explicitly: from the positive real axis to the vector, the principal angle is negative; 4π/3 is the positive anticlockwise equivalent.

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