jernplate8

2021-08-02

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
$a\right)\stackrel{\to }{A}\left(x,y,z\right)={\stackrel{\to }{e}}_{x}$
$d\right)\stackrel{\to }{A}\left(\rho ,\varphi ,z\right)={\stackrel{\to }{e}}_{\rho }$
$g\right)\stackrel{\to }{A}\left(r,\theta ,\varphi \right)={\stackrel{\to }{e}}_{\theta }$
$j\right)\stackrel{\to }{A}\left(x,y,z\right)=\frac{-y{\stackrel{\to }{e}}_{x}+x{\stackrel{\to }{e}}_{y}}{{x}^{2}+{y}^{2}}$

ottcomn

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

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