sorry for this trivial question, but I could not find an answer anywhere. Does all unit vectors of a coordinate system, need to be mutually orthogonal to each other,?
Like, they are in every coordinate system I checked so far, so I presume they have to. But I need to know for sure, because I want to derive a unit vector for elliptical coordinate system, by using that as a restriction.
Edit: I think I misused a term here, what I meant was "Does all basis unit vectors, that define a coordinate system, need to be mutually orthogonal". Let me phrase it in another way: lets say we have a vector space in 2D, that is defined by two vectors $e_\mathrm{1}$ and $e_\mathrm{2}$, and a point in space that is described by a combination: x$e_\mathrm{1}$+y$e_\mathrm{2}$
I want to describe that point by a different coordinate system, with two other vectors: $k_\mathrm{1}$ and $k_\mathrm{2}$ . Do they have to be mutually orthogonal?
A coordinate system need not be orthogonal. It can be defined by unit vectors, but this does not yield any simplification (in particular, the expression of the dot product is cumbersome).
When a coordinate system is orthogonal, it is an asset that the vectors be unit as well.
But nothing is mandated.