I How to Write Vectors in Spherical Coordinates for Scalar Product Evaluation

[Ki-nana18]

Here is the problem verbatim:
The polar and azimuthal angles of a vector are Θ₁ and Φ₁. The polar and azimuthal angles of a second vector are Θ₂ and Φ₂. Show that the angle ϒ between the two vectors satisfies the relation:

cos ϒ = cosΘ₁*cos Θ₂+sinΘ₁*sinΘ₂*cos(Φ₁-Φ₂)

Hint: write out the Cartesian components of each vector in spherical coordinates and then evaluate the scalar product.

Where I run into trouble with this problem is when writing out the Cartesian components into Spherical components. I do know that x = ρ*sinΘ*cosΦ, y = ρ*sin Θ*sinΦ, and z=ρ cos Θ, but I'm not sure how to write a vector in the form of B= B_xx-hat+B_yy-hat+B_zz-hat using the x, y, and z spherical components. Someone please point me in the right direction.

 

[Ki-nana18]

I just realized I posted this in the wrong section of the forum. Sorry.