To determine the cylindrical coordinates when the particle is at the position given by the vector $\mathbf{r}_p = 3.25\mathbf{i} + 7.14\mathbf{j} + 12.23\mathbf{k}$, we can convert the Cartesian coordinates to cylindrical coordinates.

Step 1: Identify Cartesian Coordinates

From the vector $\mathbf{r}_p$:

• $x = 3.25$
• $y = 7.14$
• $z = 12.23$

Step 2: Convert to Cylindrical Coordinates

Cylindrical coordinates $(r, \theta, z)$ are related to Cartesian coordinates $(x, y, z)$ as follows:

• $r = \sqrt{x^2 + y^2}$
• $\theta = \tan^{-1}\left(\frac{y}{x}\right)$
• $z = z$

Calculate $r$:

$r = \sqrt{(3.25)^2 + (7.14)^2} = \sqrt{10.5625 + 50.9796} = \sqrt{61.5421} \approx 7.84$

Calculate $\theta$:

$\theta = \tan^{-1}\left(\frac{7.14}{3.25}\right) = \tan^{-1}(2.197) \approx 1.15 \text{ radians}$

The $z$ coordinate remains the same:

$z = 12.23$

Step 3: Summary of Cylindrical Coordinates

The cylindrical coordinates of the particle when it is at $\mathbf{r}_p = 3.25\mathbf{i} + 7.14\mathbf{j} + 12.23\mathbf{k}$ are approximately: $(r, \theta, z) \approx (7.84, 1.15, 12.23)$

Would you like to explore any additional details or have further questions?

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Further Questions to Consider:

1. How would the cylindrical coordinates change if the particle were at a different position along the spiral?
2. What is the physical significance of the angle $\theta$ in cylindrical coordinates?
3. How does the cylindrical coordinate system simplify the analysis of helical or spiral paths?
4. If the particle moved to a new position, how would you update the cylindrical coordinates?
5. How can you find the velocity and acceleration components in cylindrical coordinates?
6. What is the relationship between cylindrical and spherical coordinates for this position?
7. How does the $z$-coordinate in cylindrical coordinates relate to the Cartesian $z$?
8. How does the cylindrical coordinate system help in understanding rotational motion around the z-axis?

Tip: When converting between coordinate systems, ensure that you use the correct quadrant for the angle $\theta$, particularly when working with arctangents.