(i) To find \(uv\), substitute \(u = -3\sqrt{3} + i\) and \(v = \sqrt{3} + 2i\) into the product:

\((-3\sqrt{3} + i)(\sqrt{3} + 2i) = -3\sqrt{3} \cdot \sqrt{3} + (-3\sqrt{3}) \cdot 2i + i \cdot \sqrt{3} + i \cdot 2i\)

\(= -9 - 6\sqrt{3}i + \sqrt{3}i - 2\)

\(= -11 - 5\sqrt{3}i\)

To find \(\frac{u}{v}\), multiply numerator and denominator by the conjugate of \(v\):

\(\frac{-3\sqrt{3} + i}{\sqrt{3} + 2i} \cdot \frac{\sqrt{3} - 2i}{\sqrt{3} - 2i} = \frac{(-3\sqrt{3} + i)(\sqrt{3} - 2i)}{(\sqrt{3} + 2i)(\sqrt{3} - 2i)}\)

\(= \frac{-3\cdot3 + 6i + \sqrt{3}i - 2i^2}{3 + 4}\)

\(= \frac{-9 + 6i + \sqrt{3}i + 2}{7}\)

\(= \frac{-7 + 7\sqrt{3}i}{7}\)

\(= -1 + \sqrt{3}i\)

(ii) On the Argand diagram, point \(A\) represents \(u = -3\sqrt{3} + i\) and point \(B\) represents \(v = \sqrt{3} + 2i\). To find \(\angle AOB\), calculate \(\arg\left(\frac{u}{v}\right)\):

\(\arctan(-\sqrt{3}) = -\frac{2\pi}{3}\)

Thus, \(\angle AOB = \frac{2}{3}\pi\).