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It appears that the image you've uploaded shows a mathematical problem involving complex numbers and an Argand diagram. From the visible content, here's what I can interpret:

The question seems to focus on the complex numbers \( z_1 \), \( z_2 \), and \( z_3 \) and explores their geometric properties on the Argand plane. Specifically, the problem deals with equilateral triangles and their vertices being represented by these complex numbers.

### Problem Breakdown:

1. **Given Complex Numbers:**
   - The complex numbers \( z_1 \), \( z_2 \), and \( z_3 \) are mentioned as vertices of equilateral triangles.
   - \( z_1 \) is given as \( \text{cis} \frac{\pi}{12} \), which means \( z_1 = e^{i \frac{\pi}{12}} \), where \( \text{cis} \theta \) stands for \( \cos \theta + i \sin \theta \).

2. **Part (i):**
   - The question asks to explain why \( z_1 \times z_2 = z_3 \), implying a relationship between the angles or magnitudes of these complex numbers.
   - From the handwritten notes, it seems that the student explained that \( z_1 \times z_2 \) equals \( z_3 \) because all three complex numbers lie on a circle and form an equilateral triangle. The angles between the vertices are likely key to understanding this.

3. **Part (ii):**
   - The task is to write \( z_2 \) in polar form, specifically where \( \theta > 0 \). The handwritten answer appears to state \( z_2 = 2 \text{cis} \frac{\pi}{3} \).

4. **Part (iii):**
   - This asks to write \( z_3 \) in the form \( r \text{cis} \theta \) with a condition on the angle \( \theta \), where the provided answer is \( z_3 = 2 \text{cis}(-\frac{\pi}{3}) \).

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Would you like a detailed step-by-step explanation of these parts, or is there any specific section you'd like to go over in more depth?

### Relative Questions:
1. How do you represent a complex number in polar form and what does the modulus and argument mean?
2. What is the geometric interpretation of multiplying two complex numbers in the Argand plane?
3. Why does an equilateral triangle have angles of \( 60^\circ \) between its sides, and how does this translate to complex numbers?
4. How can you convert from Cartesian form \( x + iy \) to polar form \( r \text{cis} \theta \)?
5. What are the implications of the angle being negative in \( z_3 = 2 \text{cis}(-\frac{\pi}{3}) \)?

### Tip:
When working with complex numbers in polar form, remember that multiplying two complex numbers involves multiplying their magnitudes and adding their arguments (angles).

The alongside Argand diagram shows the complex numbers z1, z2, and z3 as the vertices of two equilateral triangles. Explain why z1 * z2 = z3 and provide polar forms for z2 and z3.

Explore the solution to a complex number problem involving equilateral triangles on the Argand plane. Understand how to multiply complex numbers in polar form and apply geometric reasoning to explain the relationship between z1, z2, and z3.

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