Let's solve the problem **step-by-step**. ## **Given:** \[ \vec{A} = 2\hat{e}_\rho + \hat{e}_\phi - 4\hat{e}_z \] at point \(P(3, \frac{3\pi}{2}, 2)_{cylindrical}\). --- ## **Step 1: Cylindrical to Cartesian Unit Vectors** The relationships between cylindrical and Cartesian unit vectors are: \[ \begin{align*} \hat{e}_\rho &= \cos\phi\,\hat{i} + \sin\phi\,\hat{j} \\ \hat{e}_\phi &= -\sin\phi\,\hat{i} + \cos\phi\,\hat{j} \\ \hat{e}_z &= \hat{k} \end{align*} \] --- ## **Step 2: Substitute the Point** Given \( (\rho, \phi, z) = (3, \frac{3\pi}{2}, 2) \): - \(\cos\phi = \cos\left(\frac{3\pi}{2}\right) = 0\) - \(\sin\phi = \sin\left(\frac{3\pi}{2}\right) = -1\) --- ## **Step 3: Express Each Unit Vector at This Point** \[ \begin{align*} \hat{e}_\rho &= (0)\hat{i} + (-1)\hat{j} = -\hat{j} \\ \hat{e}_\phi &= -(-1)\hat{i} + (0)\hat{j} = \hat{i} \\ \hat{e}_z &= \hat{k} \end{align*} \] --- ## **Step 4: Express \(\vec{A}\) in Cartesian** \[ \vec{A} = 2\hat{e}_\rho + \hat{e}_\phi - 4\hat{e}_z \] Plug in the above: \[ \vec{A} = 2(-\hat{j}) + (\hat{i}) - 4(\hat{k}) = \hat{i} - 2\hat{j} - 4\hat{k} \] ### **Final Answer for (a):** \[ \boxed{ \vec{A}_{cartesian} = \hat{i} - 2\hat{j} - 4\hat{k} } \] at point \( (x, y, z) = (\rho \cos\phi, \rho \sin\phi, z) = (0, -3, 2) \). --- ## **Step 5: Cylindrical to Spherical Conversion** ### **Spherical Unit Vectors in Terms of Cylindrical** \[ \begin{align*} \hat{e}_\rho &= \sin\theta\,\hat{e}_r + \cos\theta\,\hat{e}_\theta \\ \hat{e}_z &= \cos\theta\,\hat{e}_r - \sin\theta\,\hat{e}_\theta \\ \hat{e}_\phi \text{ is the same in both} \end{align*} \] ### **Find \(\theta\) at the Point** \[ \theta = \arccos\left(\frac{z}{r}\right) \] where \( r = \sqrt{\rho^2 + z^2} = \sqrt{9 + 4} = \sqrt{13} \). \[ \cos\theta = \frac{z}{r} = \frac{2}{\sqrt{13}} \] \[ \sin\theta = \frac{\rho}{r} = \frac{3}{\sqrt{13}} \] --- ### **Rewrite \(\vec{A}\) in Spherical** \[ \hat{e}_\rho = \sin\theta\,\hat{e}_r + \cos\theta\,\hat{e}_\theta \] \[ \hat{e}_z = \cos\theta\,\hat{e}_r - \sin\theta\,\hat{e}_\theta \] Plug in the values: \[ \vec{A} = 2\hat{e}_\rho + \hat{e}_\phi - 4\hat{e}_z \] \[ = 2(\sin\theta\,\hat{e}_r + \cos\theta\,\hat{e}_\theta) + \hat{e}_\phi - 4(\cos\theta\,\hat{e}_r - \sin\theta\,\hat{e}_\theta) \] Expand: \[ = [2\sin\theta - 4\cos\theta]\,\hat{e}_r + [2\cos\theta + 4\sin\theta]\,\hat{e}_\theta + \hat{e}_\phi \] Plug in \(\sin\theta = \frac{3}{\sqrt{13}}\), \(\cos\theta = \frac{2}{\sqrt{13}}\): \[ \begin{align*} \hat{e}_r: &\quad 2\left(\frac{3}{\sqrt{13}}\right) - 4\left(\frac{2}{\sqrt{13}}\right) = \frac{6 - 8}{\sqrt{13}} = -\frac{2}{\sqrt{13}} \\ \hat{e}_\theta: &\quad 2\left(\frac{2}{\sqrt{13}}\right) + 4\left(\frac{3}{\sqrt{13}}\right) = \frac{4 + 12}{\sqrt{13}} = \frac{16}{\sqrt{13}} \\ \hat{e}_\phi: &\quad 1 \end{align*} \] ### **Final Answer for (b):** \[ \boxed{ \vec{A}_{spherical} = -\frac{2}{\sqrt{13}}\hat{e}_r + \frac{16}{\sqrt{13}}\hat{e}_\theta + \hat{e}_\phi } \] at \( (r, \theta, \phi) = (\sqrt{13}, \arccos(2/\sqrt{13}), 3\pi/2) \). --- ## **Summary Table** | Form | Expression | At Point | |-------------|-------------------------------------------------------------|----------------------------------| | Cartesian | \( \hat{i} - 2\hat{j} - 4\hat{k} \) | \( (x, y, z) = (0, -3, 2) \) | | Spherical | \( -\frac{2}{\sqrt{13}}\hat{e}_r + \frac{16}{\sqrt{13}}\hat{e}_\theta + \hat{e}_\phi \) | \( (r, \theta, \phi) = (\sqrt{13}, \arccos(2/\sqrt{13}), 3\pi/2) \) | Let me know if you need any clarifications or further explanations!

**Cartesian Explanation:** In Cartesian coordinates, vectors are expressed as components along x, y, and z axes. To convert from cylindrical components, use unit vector relationships with \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). This allows straightforward calculation of vector components at any point in space. **Spherical Explanation:** Spherical coordinates define points via radius \(r\), polar angle \(\theta\), and azimuthal angle \(\phi\). To convert vectors, decompose cylindrical or Cartesian components into spherical unit vectors \(\hat{e}_r\), \(\hat{e}_\theta\), and \(\hat{e}_\phi\) considering geometric relationships involving angles and distances.