Expressing a Field in Spherical Coordinates as Cartesian Vectors

[yoamocuy]

Homework Statement

A field is given in spherical coordinates as F=[cos(θ)/r²]∙a_r+[sin(θ)/r]∙a_θ. Express F in terms of x, y, z, a_x, a_y, a_z

Homework Equations

a_r∙a_x=sin(θ)cos(∅)
a_r∙a_y=sin(θ)sin(∅)
a_r∙a_z=cos(θ)
a_θ∙a_x=cos(θ)cos(∅)
a_θ∙a_y=cos(θ)sin(∅)
a_θ∙a_z=-sin(θ)
x=r*sin(θ)*cos(∅)
y=r*sin(θ)*sin(∅)
z=r*cos(θ)
r=√(x²+y²+z² )
cos(θ)=z/r
∅=tan^-1(y/x)

The Attempt at a Solution

cos(θ)/r²*[sin(θ)cos(∅)a_x+sin(θ)sin(∅)a_y+cos(θ)a_z]+sin(θ)/r*[cos(θ)cos(∅)a_x+cos(θ)cos(∅)a_y-sin(θ)a_z]

z/r³*[sin(θ)cos(∅)a_x+sin(θ)sin(∅)a_y+cos(θ)a_z]+sin(θ)/r*[cos(θ)cos(∅)a_x+cos(θ)cos(∅)a_y-sin(θ)a_z]

(z*r)/r⁴*[sin(θ)cos(∅)a_x+sin(θ)sin(∅)a_y+cos(θ)a_z]+sin(θ)/r*[cos(θ)cos(∅)a_x+cos(θ)cos(∅)a_y-sin(θ)a_z]

z/r⁴*[xa_x+ya_y+za_z]+sin(θ)/r*[cos(θ)cos(∅)a_x+cos(θ)cos(∅)a_y-sin(θ)a_z]

z/(x²+y²+z²)³*[xa_x+ya_y+za_z]+sin(θ)/r*[cos(θ)cos(∅)a_x+cos(θ)cos(∅)a_y-sin(θ)a_z]

That's about as far as I've gotten. I'm not even sure if what I've done so far is on the right track or not :/ I'm not sure what to do with the 2nd half of this equation?

 

[vela]

Use the fact that $\vec{F} = (\vec{F}\cdot\hat{a}_x)\hat{a}_x + (\vec{F}\cdot\hat{a}_y)\hat{a}_y + (\vec{F}\cdot\hat{a}_z)\hat{a}_z$.

Calculate $\vec{F}\cdot \hat{a}_x$ using the various dot products you listed above. Then convert from the spherical variables to the Cartesian variables.

 

[yoamocuy]

Use the fact that $\vec{F} = (\vec{F}\cdot\hat{a}_x)\hat{a}_x + (\vec{F}\cdot\hat{a}_y)\hat{a}_y + (\vec{F}\cdot\hat{a}_z)\hat{a}_z$.

Calculate $\vec{F}\cdot \hat{a}_x$ using the various dot products you listed above. Then convert from the spherical variables to the Cartesian variables.

Isn't that what I did above? I didn't originally show the a_x, a_y, a_z in my work but I just added them in there for clarity.

 

[vela]

Oh, OK. I didn't see the unit vectors in your original attempt, so I figured you were doing it all wrong and didn't bother to look too closely.