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 θ, will we not be able to find new theorems?Example: for θ = 60 ∘

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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   θ, will we not be able to find new theorems?Example: for   θ  =      60    ∘

hocelwsmjc · Accepted Answer

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, &quot;making things easier&quot; 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.

Arendrogfkl · Accepted Answer

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

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