Can You Combine Basis Vectors from Different Coordinate Systems?

[Lancelot59]

While not paying attention in class my friend made a joke that a cube squared was in six dimensions, or something like that. Terrible joke, but now I'm trying to figure out if it is valid to arithmatically combine the basis vectors for two or more coordinate systems to get a new one.

 

[gsal]

the 2-dimensional space has its square...
the 3-dimensional space has its square (cube)...
most probably each n-dimensional space has its "square", too...

I seem to recall a reference to the 4-dimensional space "square" and while impossible to visualize it in its own space, I think somebody figured out what its shade looks like in the 3-dimensional space...I am sure there is a figure somewhere on the net.

 

[SteveL27]

the 2-dimensional space has its square...
the 3-dimensional space has its square (cube)...
most probably each n-dimensional space has its "square", too...

I seem to recall a reference to the 4-dimensional space "square" and while impossible to visualize it in its own space, I think somebody figured out what its shade looks like in the 3-dimensional space...I am sure there is a figure somewhere on the net.

http://en.wikipedia.org/wiki/Tesseract

 

[Lancelot59]

That's not exactly what I had in mind...

For instance, is it valid to take the basis vectors for spherical coordinates, and conical coordinates, then add or multiply them together?