The problem is about two complex numbers $z_1$ and $z_2$, as represented in the diagram. The relationship provided is $z_1 \cdot z_2$ as a product, and the goal is to determine the value of $z_2$ from the given options.

Key Observations from the Diagram:

1. $z_1$ is represented at an angle $\theta$ with respect to the positive real axis.
2. $z_2$ lies at the same angle $\theta$ but in the opposite direction.
   • This indicates a rotation by $90^\circ$ or multiplication by $i$ in the complex plane.

Mathematical Concept:

• In the complex plane:
  • Rotating a complex number $z$ by $90^\circ$ counterclockwise corresponds to multiplying it by $i$.
  • Rotating $z$ clockwise by $90^\circ$ corresponds to multiplying it by $-i$.

Given:

• The product $z_1 \cdot z_2$ implies a specific relationship between $z_1$ and $z_2$.
• Since $z_2$ is positioned at a $-90^\circ$ rotation relative to $z_1$ (based on the diagram), we multiply $z_1$ by $-i$.

Conclusion:

From the options:

• $z_2 = -i$

Thus, the correct answer is (c) -i.

Follow-Up Questions:

1. What is the significance of the $i$ unit in the complex plane?
2. How does multiplying by $i$ or $-i$ affect the angle of a complex number?
3. Can you explain how complex conjugates affect rotations in the complex plane?
4. What is the geometric representation of the product of two complex numbers?
5. How do you determine the modulus and argument of a complex number?

Tip:

When working with rotations in the complex plane, remember that multiplying by $i$ rotates a number counterclockwise by $90^\circ$, while $-i$ rotates it clockwise by $90^\circ$.