# 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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