To convert a point from spherical coordinates $(\rho, \theta, \phi)$ to rectangular (Cartesian) coordinates $(x, y, z)$, we use the following formulas:

$x = \rho \sin \phi \cos \theta$ $y = \rho \sin \phi \sin \theta$ $z = \rho \cos \phi$

Given: $\rho = 3, \quad \theta = \pi, \quad \phi = \frac{\pi}{2}$

Let's substitute these values into each formula:

Step 1: Calculate $x$

$x = 3 \cdot \sin \frac{\pi}{2} \cdot \cos \pi$ Since $\sin \frac{\pi}{2} = 1$ and $\cos \pi = -1$, $x = 3 \cdot 1 \cdot (-1) = -3$

Step 2: Calculate $y$

$y = 3 \cdot \sin \frac{\pi}{2} \cdot \sin \pi$ Since $\sin \frac{\pi}{2} = 1$ and $\sin \pi = 0$, $y = 3 \cdot 1 \cdot 0 = 0$

Step 3: Calculate $z$

$z = 3 \cdot \cos \frac{\pi}{2}$ Since $\cos \frac{\pi}{2} = 0$, $z = 3 \cdot 0 = 0$

Final Answer

The rectangular coordinates are: $(x, y, z) = (-3, 0, 0)$

Would you like further details or explanations on any of these steps?

Here are some related questions you might find helpful:

1. How do you interpret angles in spherical coordinates?
2. What is the difference between spherical and cylindrical coordinates?
3. How would you convert from rectangular coordinates back to spherical?
4. What do the coordinates $(\rho, \theta, \phi)$ represent in 3D space?
5. How can we visualize spherical coordinates?

Tip: Always remember the role of each angle in spherical coordinates—$\theta$ is usually the azimuthal angle in the xy-plane, and $\phi$ is the polar angle from the z-axis.