Why do we choose to have perpendicular axis?

[Avichal]

Why do we have x-axis perpendicular to y-axis? Why not 45° or something else?
Even if we keep any other angle other than 90° we can still build up all other vectors using them in the plane.

So what is there is 90° that makes it special and simpler?

 

[lurflurf]

You can use any angle (and sometimes it is convenient to do so) and have all the same vectors. Several things make 90° special and simpler.

c²=a²+b--2ab cos(t)
This is most simple if cos(t)=cos(90°)=0

very small t are particularly trouble some as two axis are almost the same

to determine the coordinates we must solve
x.i=x_ii.i+x_jj.i
x.j=x_ii.j+x_jj.j

when t=90° this is easy

x_i=x.i
x_j=x.j

 

[lavinia]

Why do we have x-axis perpendicular to y-axis? Why not 45° or something else?
Even if we keep any other angle other than 90° we can still build up all other vectors using them in the plane.

So what is there is 90° that makes it special and simpler?

lurlurf is correct that other axes are possible. In my opinion right angles allow that Pythagorean theorem to be reduced to the sum of squres ot the coordinates. This makes calculation simpler.

 

[HallsofIvy]

Also the dot product of two perpendicular vectors is 0 so if the x and y axes are perpendicular, and v_x and v_y are unit vectors in the direction of those vectors, then the components of vector v are just v\cdot v_x and v\cdot v_y. If the axes were not perpendicular, those formulas would be more complicated.

In more abstract situations, vector spaces of functions, for example, we typically do that the other way around: choose some convenient basis, then define the inner product to make the basis vectors orthogonal.

 

[lavinia]

Also the dot product of two perpendicular vectors is 0 so if the x and y axes are perpendicular, and v_x and v_y are unit vectors in the direction of those vectors, then the components of vector v are just v\cdot v_x and v\cdot v_y. If the axes were not perpendicular, those formulas would be more complicated.

In more abstract situations, vector spaces of functions, for example, we typically do that the other way around: choose some convenient basis, then define the inner product to make the basis vectors orthogonal.

The inner product derives geometrically from orthogonal projection. If the coordinate axes are perpendicular, then orthogonal projection is just picking coordinates.

 

[mathwonk]

once you pick one axis, what other natural choice is there for a second axis other than perpendicular? i.e. what is more natural and simpler than angles that are equal?

 

[Avichal]

A related question that I don't understand: Why perpendicular axis are independent of each other?

 

[chiro]

A related question that I don't understand: Why perpendicular axis are independent of each other?

It has to do with the inner product of the underlying geometry:

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

 

[HallsofIvy]

We might also note that in "differential geometry", we work with surfaces, such as the surface of a sphere, on which we cannot have coordinate curves that are always perpendicular. That causes all sorts of problems, among them that we now have both "covariant" and "contravarient" components of vectors and tensors. If we stick to "Cartesian tensors" in which we only allow "Cartesian coordinate systems" with coordinate curves that are always perpendicular, the distinction between "covariant" and "contravariant" disappears.

 

[Studiot]

I'm not sure since you have posted this in linear and abstract algebra whether my response is relevant or not.

But does this mean

Under what circumstances do we use perpendicular axes and under what circumstances do we use some other axes?

Or do you think we only use perperdicular axes?

The second is far from the truth.
Many different arrangements are in use and the common theme is a blend of ease of presentation and ease of use.

In mathematics you will find cylindrical polar and spherical coordinates.
In cartography, navigation, surveying and fluid mechanics you will find hyperbolic, 'rho-rho' and perhaps even parabolic coordinates.
Look in some engineering texts you will find many graphs with exotic shaped cooordinates.
In geology, soil mechanics and materials science you will find some strange triangular coodinates. These also appear in colour theory in lighting.

Some coordinates have straight line axes, some do not. Look up 'curvilinear coordinates'

go well