# 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.

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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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
$$(r, \theta, \phi)=(\sqrt{x^{2}+y^{2}+z^{2}}, \tan^{-1} (\frac{y}{x}), \cos^{-1}(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}))$$
Therefore,
$$\displaystyle{\left({r},\theta,\phi\right)}={\left({1},{0},{\frac{{\pi}}{{{2}}}}\right)}$$