Vector operations

So, for example, to take the dot product of two vectors in spherical coordinates, we would do the following (warning: remember that these are in {ρ,φ,θ} form):

DotProduct[{1, π/2, π/6}, {2, π/2, (2π)/3}, Spherical]

0

If we use Cartesian coordinates, we get:

DotProduct[{1, π/2, π/6}, {2, π/2, (2π)/3}, Cartesian]

2 + (13 π^2)/36

CrossProduct[{1, π/2, π/6}, {2, π/2, (2π)/3}, Spherical]

{2, 0, 0}

CrossProduct[{1, π/2, π/6}, {2, 0, (2π)/3}, Spherical]

{2, π/2, -π/3}

Problem

Problem

Explain why the dot product in spherical coordinates is 0 above.  Why is the second dot product (in Cartesian coordinates, by default) not 0?  

What coordinate system is the answer to the cross product given in?  (This is important.  Don't continue until you have figured this out.)  How do you know?

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