Below are various vectors in cartesian, cylindrical and spherical coordinates.

ossidianaZ 2021-09-22 Answered

Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed
\(a) \overrightarrow{A}(x,y,z)=\overrightarrow{e}_{x}\)
\(d)\overrightarrow{A}(\rho, \phi, z)= \overrightarrow{e}_{\rho}\)
\(g)\overrightarrow{A}(r, \theta, \phi)=\overrightarrow{e}_{\theta}\)
\(j)\overrightarrow{A}(x,y,z)=\frac{-y\overrightarrow{e}_{x}+x\overrightarrow{e}_{y}}{x^{2}+y^{2}}\)

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Expert Answer

Latisha Oneil
Answered 2021-09-23 Author has 24671 answers

Step 1
Given: \(a)\overrightarrow{A}(x,y,z)=\overrightarrow{e}_{x}\)
To express it in other two coordinate systems:
ie., in terms of cartesian, cylindrical and spherical coordinates
Step 2
Given: \(a)\overrightarrow{A}(x,y,z)=\overrightarrow{e}_{x}\)
spherical coordinates can be expressed as
\(\displaystyle{\left({r},\theta,\phi\right)}={\left(\sqrt{{{x}^{{{2}}}+{y}^{{{2}}}+{z}^{{{2}}}}},{{\tan}^{{-{1}}}{\left({\frac{{{y}}}{{{x}}}}\right)}},{{\cos}^{{-{1}}}{\left({\frac{{{z}}}{{\sqrt{{{x}^{{{2}}}+{y}^{{{2}}}+{z}^{{{2}}}}}}}}\right)}}\right)}\)
Therefore,
\(\displaystyle{\left({r},\theta,\phi\right)}={\left({1},{0},{\frac{{\pi}}{{{2}}}}\right)}\)

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