# Why are the axes taken as perpendicular to each other?

• B
• murshid_islam
murshid_islam
TL;DR Summary
Why are the axes taken as perpendicular to each other rather than at some other angle?
Why are the axes taken as perpendicular to each other rather than at some other angle? Is it just a matter of convention? Is there any mathematical reason behind it? Is there some other reason?

Having basis vectors orthogonal to each other is a nice property.

FactChecker
You certainly can make axes non-orthogonal. In most cases, though, it makes the math harder, not easier. Same principle as playing piano wearing mittens. I mean, you could...

SammyS
murshid_islam said:
Is it just a matter of convention?
Yes, I think so. Although it is a pervasive and really convenient convention. Streets in cities, and the walls in your house are mostly 90o angles, etc. It's what we are used to, for good reasons. The equation of a circle still works and you can still call it a circle. But when you display it this way, it won't look like a circle to you. The math will appear to us as more complex, but I'm not sure the people that live in "acute axes world" would see it that way.

It's nearly a pointless philosophical question. More about how we see and think about the world than "pure math". OTOH, I personally can't imagine learning analytic geometry or linear algebra with this picture, it would be sooo much harder for me to visualize anything.

PS: BTW, when/if you get to studying things like eigenvectors and normal forms, used in dynamic systems analysis (among other things). You will learn how to deal with systems that make more sense (i.e. easier) when you use non-orthogonal basis (axes). There are things in this world that look simpler with "tilted" axes.

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mathwonk
I'll say Pythagoras. with orthogonal axes, distances are easy to measure using coordinates by his theorem.

berkeman and Nugatory
mathwonk said:
I'll say Pythagoras. with orthogonal axes, distances are easy to measure using coordinates by his theorem.
@murshid_islam This is pretty much your answer. It is very convenient to be be able to calculate the distance between two points as ##\sqrt{\Delta x^2+\Delta y^2}## instead of the more complicated expression (probably involving trig functions) that we get when the axes aren’t orthogonal. So because we see no reason to make problems harder than they need to be, we draw the axes perpendicular because it’s simpler and clearer that way.
For an example, consider the equation for a circle (all points the same distance from the center point). It’s trivial with perpendicular axes, wildly confusing otherwise.

In other words, Pick the axes such that sin = 0 and cos = 1. Unless there is a good reason to do otherwise.

Nugatory
Nugatory said:
more complicated expression (probably involving trig functions)

The distance is $\sqrt{\Delta x^2 + \Delta y^2 + 2\Delta x \Delta y \cos \theta}$.

Maybe we should put some standards organization to work quantifying how bad “that bad” is?

I left my trig tables in my other briefcase.

Telling you how old I am without telling you...

Shouldn't there be a minus sign in front of the cosine part?

Sure, there isn't a straightforward answer to this seemingly simple question? Axes divide the surface into four quadrants, and that's where everything seems to begin.

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