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?

 

[CellCoree]

Combining coordinate systems is a valid mathematical concept, but it is not quite as simple as arithmetically combining basis vectors. In order to combine coordinate systems, one must first ensure that the basis vectors of each system are orthogonal (perpendicular) to each other. Then, the basis vectors can be combined using a transformation matrix, which takes into account the orientation and scaling of each coordinate system.

This process is commonly used in vector calculus and linear algebra, where multiple coordinate systems may be needed to describe a complex system. It allows for a more efficient and accurate representation of the system, as each coordinate system can capture different aspects of the system.

In terms of your friend's joke, it is not accurate to say that a cube squared exists in six dimensions. A cube is a three-dimensional object, and squaring it simply means multiplying its dimensions by itself. So, a cube squared would still exist in three dimensions. However, in mathematics, we can use higher dimensions to represent and analyze complex systems, but it does not necessarily mean that those systems physically exist in those dimensions.

In summary, combining coordinate systems is a valid mathematical concept, but it requires a proper understanding and application of transformation matrices. It is not as simple as arithmetically combining basis vectors, and it should not be used to make jokes about dimensional concepts.