Why is the representation of the direction of the x and y axes in two dimensions different than in three dimensions?

Question

So I apologize if this question seems a bit nit-picky, but it has bothered me for a while. Usually when a coordinate system is represented in two-dimensions, the x-axis is pointing towards what might be considered "3 o' clock", while the y-axis is at midnight.

However, most representations of 3-dimensional space place the x-axis in the "6 o' clock" direction, and the Y axis in the "3 o'clock" direction.(if you were to lay the clock on the x-y plane).

It does not seem like the intuitive way of representing things. Of course a person has the liberty to arrange the axes in whatever fashion they see fit, but the standard seems to be the way shown above. I wonder who thought it was a good idea to arrange the axes in this fashion, and why they chose to do so.

Answer 1

So I think I know the answer to this.

First, note that both coordinate systems relegate the vertical axis to the last coordinate (y in 2D, z in 3D); that this feels 'natural' is perhaps a result of Latin script being written horizontally. Now while this fully defines the 2D coordinate system, there are still two possibilities for the 3D system: a so-called left-handed system and a right-handed system.

The reason (I believe) that the right-handed system is standard in both mathematics and physics is that it agrees with the definition of the vector cross product, which by convention follows the right-hand rule: $\mathbb{x} \times \mathbb{y} = \mathbb{z}$. This definition is fundamental to vector algebra and much of physics.

This definition, in turn, follows directly from Hamilton's 1843 formulation or quaternions, defined by $ij=k$, $jk=i$ and $ki=j$, by identifying $i, j, k$ with $x, y, z$. Had Hamilton instead defined them by $ji=k$, $kj=i$ and $ik=j$, there would likely have been a left-handed rule instead.

Now the final question of why representations of right-handed 3D coordinates typically show the y axis at "3 o'clock" and the x axis "somewhere past 6 o'clock" (like the picture below, but unlike the first picture), I expect that's simply to separate the axes on the page as far as possible.

Answer 2

I'd say that the arrangement of axes in 3d is due to the conventions we have for the cross product, as the cross product U x V is normal to the plane spanned by the vectors $u$ and $v$; normal, suggests vertical or perpendicularity and hence we have the z-axis pointing towards 12 O'Clock.

It is worth noting that the diagrams you've drawn are all for spatial diagrams, when one of the axes is time then diagrams modelling classical mechanics usually have time along the 3 O'clock direction and height at the 12 O'clock direction.

However this convention is reversed in special and general relativity.

I suspect, the first convention is simply due to the fact that when we model the trajectory of a thrown stone then this trajectory follows a parabola, and this is the same curve that is drawn on the diagram using this convention. Obviously this is of no relevance to SR/GR.

A final observation is that axes actually define charts and whilst the coordinate frame is special in that the axes are all perpendicular, the notion of a manifold uses charts in a democratic sense - all charts are equally important.