(i) The modulus of w is calculated as \(\sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1\).

The argument of w is \(\arctan\left(\frac{\sqrt{3}/2}{-1/2}\right) = \arctan(-\sqrt{3}) = \frac{2}{3}\pi\) or 120°.

(ii) The modulus of wz is \(R \times 1 = R\).

The argument of wz is \(\theta + \frac{2}{3}\pi\).

The modulus of \(\frac{z}{w}\) is \(\frac{R}{1} = R\).

The argument of \(\frac{z}{w}\) is \(\theta - \frac{2}{3}\pi\).

(iii) In an Argand diagram, the points z, wz, and \(\frac{z}{w}\) are equidistant from the origin and subtend equal angles of \(\frac{2}{3}\pi\) at the origin, forming an equilateral triangle.

(iv) To find the other vertices, multiply 4 + 2i by w and use \(i^2 = -1\).

\((4 + 2i) \times \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = -2 - 2\sqrt{3}i + 2i + \sqrt{3} = -(2 + \sqrt{3}) + (2\sqrt{3} - 1)i\).

Divide 4 + 2i by w, multiplying numerator and denominator by the conjugate of w.

\(\frac{4 + 2i}{-\frac{1}{2} + i \frac{\sqrt{3}}{2}} \times \frac{-\frac{1}{2} - i \frac{\sqrt{3}}{2}}{-\frac{1}{2} - i \frac{\sqrt{3}}{2}} = -(2 - \sqrt{3}) - (2\sqrt{3} + 1)i\).