Understand and use multiplication, division of complex numbers in polar form and the geometric interpretation of these. 𝑧1 𝑧2 = 𝑟1 𝑟2 cis(𝜃1 + 𝜃2) 𝑧1 𝑧2 = 𝑟 1 𝑟 2 cis(𝜃1 − 𝜃2)

Consider the complex numbers $z_1 = 6\operatorname{cis}\left(\frac{\pi}{3}\right)$ and $z_2 = 2\operatorname{cis}\left(\frac{\pi}{4}\right)$. (a) Determine the value of $w = z_1 z_2$ in polar form. (2 marks) (b) Determine the value of $u = \frac{z_1}{z_2}$ in polar form. (2 marks)

Consider the complex numbers $z_1 = 3\,\text{cis}\left(\frac{\pi}{4}\right)$ and $z_2 = 2\,\text{cis}\left(\frac{\pi}{3}\right)$. a) Determine $z_1 z_2$ in polar form. (1 mark) b) Determine $\frac{z_1}{z_2}$ in polar form. (1 mark) c) Verify your result from part (b) by converting both $z_1$ and $z_2$ to Cartesian form, performing the division, and converting the result back to polar form. (1 mark)

Use the given polar forms to determine the value of $w$.