Why are the axes in coordinate geometry perperndicular?

Question

In coordinate geometry, the $x$ and $y$ axis are perpendicular to each other. But is there any special reason for this (other than to make it simple)? Will coordinate geometry have contradictions if the axis are at any other angle? If we take the angle between the $x$ and $y$ axis as $\theta$, will we not be able to find new theorems?

Earlier we had Eucledian geometry in which the surface was taken as a plane, and then we invented non-eucledian geometry, and we found many new theorems. Can the same thing be done to coodrdinate geometry?

Example: for $\theta=60^\circ$

Answer 1

Is there any special reason for this (other than to make it simple)?

No.

Will coordinate geometry have contradictions if the axis are at any other angle?

No. No contradictions. Just extra terms.

If we take the angle between the x and y axis as θ, will we not be able to find new theorems?

No. No new theorems. Just the old ones with extra terms added.

We invented non-Euclidian geometry, and we found many new theorems. Can the same thing be done to coordinate geometry?

No. $\big($There's no comparison between the two ideas$\big)$.

Answer 2

Using perpendicular axes is the same that taking a orthogonal basis.

Answer 3

Taking the axes perpendicular has few advantages. If the axes are not perpendicular, finding coordinates leads to relatively long computations, while for perpendicular axes the coordinates are easy to calculate. This problem becomes clearer when one deals with abstract vector spaces, when the need for an inner product and inner product space becomes clear. Anyhow, "making things easier" is a minor point.

There are few tools in linear algebra, like dot product, cross product for which the connection between the geometric and algebraic formulas are clear in perpendicular axes. If the axes would not be perpendicular, the connection would not be that clear, and most importantly, when one would find the algebraic formula, the formula/proof would basically change the basis to an orthogonal basis and do the computations there, but probably not in an explicit way.

Answer 4

Yes, there is a special reason: axes represent independent magnitudes, and their perpendicular position is the only geometric way to clearly note their independence.

By doing so, we obtain something very useful for analysis: the angle of any straight line we can draw within a plane or a space made by perpendicular magnitudes as axes gives us precise information about the relations of dependence these lines observe in relationship with each one of the magnitudes we use to serve as the base reference of the system through which we analyse the reality we want to analyse.

For instance: if we represent units of space by the y-axis and units of time by the x-axis, the angle of any line we draw within the coordinate system we have will tell us precisely about the speed of the object we are analsying. So it is not "for making it simplier", but "for making it useful".