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To solve the problem of multiplying the complex number \( z \) by \( 3i \), we need to understand how multiplication by a complex number affects its position on the complex plane.

### Step-by-step approach:

1. **Interpret the multiplication by \( 3i \):**
   - Multiplying a complex number \( z \) by \( i \) rotates the number counterclockwise by 90 degrees on the complex plane. 
   - Multiplying by \( 3i \) not only rotates it by 90 degrees but also scales the magnitude by 3.

2. **Graphical representation:**
   - Let’s say \( z \) is represented by a point in the complex plane, located at a certain distance from the origin (its magnitude) and an angle (its argument).
   - When you multiply \( z \) by \( 3i \), you rotate this point by 90 degrees (counterclockwise) and extend its distance from the origin by a factor of 3.

3. **Analyze the points on the diagram:**
   - The point that represents \( z \) after multiplying by \( 3i \) should be located at the same angle, but rotated by 90 degrees and at a distance 3 times greater from the origin compared to \( z \).

### Conclusion:
- Based on the points marked on the diagram and the explanation of multiplying by \( 3i \), point **C** likely represents the product of \( z \) and \( 3i \). It is three times farther from the origin and rotated 90 degrees counterclockwise relative to the original point \( z \).

### Correct answer: **C**

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Would you like further details or explanations on any step? Here are five related questions that can expand your understanding:

1. How do you multiply complex numbers in polar form?
2. What is the geometric interpretation of multiplying by \( i \) in the complex plane?
3. How does multiplying by real numbers affect a complex number?
4. What is the significance of the argument (angle) of a complex number?
5. How can we use Euler's formula to represent complex number multiplication?

**Tip:** Multiplying by \( i \) always results in a 90-degree counterclockwise rotation in the complex plane.

Which point represents the product of z and 3i?

Explore how to graphically multiply complex numbers, specifically identifying the product of z and 3i. This problem involves understanding rotation and scaling in the complex plane, making it suitable for high school students in Grades 10-12.

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