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

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

Given spherical coordinates: $(\rho, \theta, \phi) = \left(3, \pi, \frac{\pi}{2}\right)$.

Step 1: Substitute the values into the formulas

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

2. Calculate $y$: $y = 3 \cdot \sin\left(\frac{\pi}{2}\right) \cdot \sin(\pi)$ Since $\sin\left(\frac{\pi}{2}\right) = 1$ and $\sin(\pi) = 0$, $y = 3 \cdot 1 \cdot 0 = 0$

3. Calculate $z$: $z = 3 \cdot \cos\left(\frac{\pi}{2}\right)$ Since $\cos\left(\frac{\pi}{2}\right) = 0$, $z = 3 \cdot 0 = 0$

Conclusion

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

Would you like further details or explanations on any step?

Here are some related questions you might find useful:

1. How do you convert from rectangular to spherical coordinates?
2. What are the applications of spherical coordinates in physics?
3. How do you calculate distance in spherical coordinates?
4. What are cylindrical coordinates, and how do they differ from spherical coordinates?
5. How do you interpret the angles $\theta$ and $\phi$ in spherical coordinates?

Tip: In spherical coordinates, $\theta$ represents the angle in the $xy$-plane from the positive $x$-axis, while $\phi$ is the angle from the positive $z$-axis.