To convert the given complex number from polar to Cartesian coordinates, let's analyze it step by step.

Problem Statement

We are given: $\frac{3}{2} \cdot e^{i \cdot \frac{3}{2} \pi}$ and asked to express it in the form $z = x + iy$, where $x, y \in \mathbb{R}$.

Step 1: Analyzing the Exponential Form

The expression $e^{i \theta}$ in polar form represents a complex number on the unit circle, with an angle $\theta$ measured from the positive real axis.

Here, we have: $\theta = \frac{3}{2} \pi$

Step 2: Finding Cartesian Coordinates

The polar form $r \cdot e^{i \theta}$ can be expanded using Euler's formula: $r \cdot e^{i \theta} = r (\cos \theta + i \sin \theta)$

In our case:

• $r = \frac{3}{2}$
• $\theta = \frac{3}{2} \pi$

So, $\frac{3}{2} \cdot e^{i \cdot \frac{3}{2} \pi} = \frac{3}{2} \left( \cos \frac{3}{2} \pi + i \sin \frac{3}{2} \pi \right)$

Step 3: Evaluating Trigonometric Components

1. Cosine: $\cos \frac{3}{2} \pi = 0$ (since $\frac{3}{2} \pi$ points along the negative imaginary axis).
2. Sine: $\sin \frac{3}{2} \pi = -1$

Thus: $\frac{3}{2} \cdot \left( 0 + i \cdot (-1) \right) = 0 - \frac{3}{2} i$

Final Answer

The complex number in Cartesian coordinates is: $0 - \frac{3}{2} i$ or simply: $0 + \left( -\frac{3}{2} \right) i$

This means $x = 0$ and $y = -\frac{3}{2}$.

Would you like to see a graphical representation or have further clarification on any step?

Here are 5 additional related questions for further practice:

1. How would you represent $2 \cdot e^{i \pi}$ in Cartesian form?
2. What are the Cartesian coordinates of $e^{i \frac{\pi}{2}}$?
3. If you multiply two complex numbers in polar form, what happens to their magnitudes and angles?
4. How would you convert $1 + i$ into polar form?
5. What is the significance of the angle $\theta = \pi$ in complex numbers?

Tip: When working with angles in radians, remember that $\pi$ represents a half-turn or 180 degrees, which can help visualize points on the complex plane.