To convert cylindrical coordinates $(r, \theta, z) = (1, \frac{\pi}{2}, 2)$ to rectangular coordinates $(x, y, z)$, we use the following formulas:

$x = r \cos(\theta)$ $y = r \sin(\theta)$ $z = z$

Step-by-Step Solution:

1. Given values: $r = 1, \quad \theta = \frac{\pi}{2}, \quad z = 2$

2. Calculate $x$: $x = r \cos(\theta) = 1 \cdot \cos\left(\frac{\pi}{2}\right) = 1 \cdot 0 = 0$

3. Calculate $y$: $y = r \sin(\theta) = 1 \cdot \sin\left(\frac{\pi}{2}\right) = 1 \cdot 1 = 1$

4. Calculate $z$: $z = 2$

Final Answer:

The rectangular coordinates are: $(x, y, z) = (0, 1, 2)$

Would you like further details on this process or have any questions?

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Here are related questions to expand on this topic:

1. How do we convert from rectangular to cylindrical coordinates?
2. What are the conditions for cylindrical coordinates to represent a point uniquely?
3. How does this conversion work in 3D space for spherical coordinates?
4. What happens when $r$ or $\theta$ are negative in cylindrical coordinates?
5. Why is $z$ the same in both cylindrical and rectangular coordinate systems?

Tip: In cylindrical coordinates, $r$ represents the radial distance from the origin in the $xy$-plane, while $\theta$ is the angle from the positive $x$-axis.