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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 $\theta=60^\circ$ enter image description here

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    $\begingroup$ It may be of interest to note that Descartes did not use perpendicular axes in his Geometry. He used distances to given lines, the given lines being at whatever angles and sometimes more than two in number. So having coordinate axes at angles other than 90 degrees came first. Descartes used his method to solve a problem of Pappus that the ancients Greeks could not, and thus proved a new theorem. $\endgroup$ – Michael E2 Mar 15 '14 at 16:27
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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$\big)$.

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  • $\begingroup$ +1 Thanks, nice answer. But are "Just the old ones with extra terms added" not new theorems (Like iPhone 4 with new features added is a iPhone 5). $\endgroup$ – Kartik Mar 15 '14 at 16:48
  • $\begingroup$ Not in any meaningful manner, as that to which you were alluding. $\endgroup$ – Lucian Mar 15 '14 at 16:54
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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.

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

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  • $\begingroup$ Thanks for your help, but can you please expand you answer? Why is an orthogonal basis required? $\endgroup$ – Kartik Mar 15 '14 at 16:04
  • $\begingroup$ @Kartik It isn't required, it's just simpler to deal with. $\endgroup$ – llllllllllllllllllllllllllllll Mar 15 '14 at 16:08
  • $\begingroup$ Isn't required, is more comfortable. $\endgroup$ – Martín-Blas Pérez Pinilla Mar 15 '14 at 16:11
  • $\begingroup$ @Martín-BlasPérezPinilla I said in the question "other than to make it simple". Eucledian geometry was more simple than non-eucledian, but does it mean that non-eucledian geometry is not useful? $\endgroup$ – Kartik Mar 15 '14 at 16:13
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    $\begingroup$ @Kartik Euclidean and non-euclidean geometries are not equivalent. Two coordinate geometries are. You just get different formulas with different coordinate systems, formulas for things like distance, translations, rotations. Some of these are simpler in one coordinate system than another. $\endgroup$ – Michael E2 Mar 15 '14 at 16:16

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