Arguments determine a triangle

arg⁡z=−π/3\arg z=-\pi/3, ∣w∣=2|w|=2, arg⁡w=π/4\arg w=\pi/4 and arg⁡(w−z)=π/3\arg(w-z)=\pi/3. Let W and Z represent w,zw,z respectively.

(i) Use an Argand diagram with reference to origin O to show ∠OWZ=π/12\angle OWZ=\pi/12 radians. marks

(ii) Hence or otherwise find ∣z∣|z|. marks

[2020/DHS/Prelim/I/3 modified]

Total marks: 4

Solution

(i)

Angles from directions

arg⁡z=−π3\arg z=-\frac\pi3, ∣w∣=2|w|=2, arg⁡w=π4\arg w=\frac\pi4 and arg⁡(w−z)=π3\arg(w-z)=\frac\pi3. W,ZW,Z represent w,zw,z. Use an Argand diagram to show ∠OWZ=π12\angle OWZ=\frac\pi{12}.

Hint

  1. Find the directions of WO→\overrightarrow{WO} and WZ→\overrightarrow{WZ}.

  2. The angle at WW is the difference of those directions.

Triangle OWZ with directed arguments
Triangle OWZ with directed argumentsScroll across to see the full diagram.

Answer · (i)

∠OWZ=π/12\angle OWZ=\pi/12; ∠OZW=π/3\angle OZW=\pi/3.

Takeaway

Directions give angles

The argument of a difference is the direction of a side. Angles of a triangle are differences of those directions.