Can You Combine Basis Vectors from Different Coordinate Systems?

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Discussion Overview

The discussion revolves around the validity of combining basis vectors from different coordinate systems, specifically exploring whether it is possible to arithmetically combine these vectors to create a new coordinate system. The context includes theoretical considerations and conceptual clarifications related to dimensional spaces and their representations.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant humorously references a friend's joke about a cube squared being in six dimensions, prompting a question about the arithmetic combination of basis vectors from different coordinate systems.
  • Another participant discusses the concept of "squares" in various dimensions, suggesting that each n-dimensional space has a corresponding "square," and mentions a reference to the 4-dimensional "square" (tesseract) and its projection in 3-dimensional space.
  • A later reply seeks clarification on the specific question of whether it is valid to combine basis vectors from spherical and conical coordinates through addition or multiplication.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the validity of combining basis vectors from different coordinate systems, and the discussion remains unresolved with competing views and questions raised.

Contextual Notes

The discussion includes assumptions about the nature of basis vectors and their operations across different coordinate systems, which may not be universally applicable. There are also references to visualizations of higher-dimensional objects that may depend on specific interpretations.

Lancelot59
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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.
 
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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.
 
gsal said:
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
 
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?
 

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