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In all the graphs I see of the 3d cartesian system, $x$ and $y$ are "horizontal", although they can change positions, but $z$ is always the vertical one.

Even if we had a choice to graph the $yz$ plane, the $z$ will always be the vertical axis, in almost every textbook and example. Why is this?

Is this convention?

I even see in a few youtube videos that $z$ is always the vertical axis, no matter what. How come this is the case?

K Split X
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    It doesn't have to be I think. It's just like you say, kind of a convention. – Karn Watcharasupat Dec 15 '17 at 02:16
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    It's just a convention. What about solids of revolution? They're usually depicted by sweeping a curve around the $x$- or $y$-axis. The $z$-axis isn't drawn, but it points out of the paper. – Matthew Leingang Dec 15 '17 at 02:16
  • Not always. Try Mathematica. You can put the $z$-axis in whatever direction you want. –  Dec 15 '17 at 02:26
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    In computer graphics, people usually take the $y$-axis to be vertical. It is just a convention, either way. –  Dec 15 '17 at 02:46
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    In aviation, it's not uncommon to use a "North-East-Down" coordinate system where the $z$-axis points downwards: https://en.m.wikipedia.org/wiki/North_east_down In navigation, one often uses a vehicle coordinate system where the $z$-axis points downwards out of the bottom of the vehicle: https://www.mathworks.com/help/aeroblks/about-aerospace-coordinate-systems.html?requestedDomain=www.mathworks.com#f3-23354 – littleO Dec 15 '17 at 02:54

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Besides as you said it is convention, the direction of the $z$-axis can be determined by using the "Right Hand Rule". When we take unit vectors $\hat{i}$ and $\hat{j}$ from $x$-axis and $y$-axis respectively, we have $\hat{i} \times \hat{j} = \hat{k}$, where $\hat{k}$ is the unit vector on $z$-axis. Since the direction of cross-product will be determined by the Right Hand Rule, whether it is vertical or not is all about how the directions for $x$-axis and $y$-axis are determined (And since they generally are lying on a horizontal plane, your result follows).

ArsenBerk
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The $z$-axis doesn't have to be the vertical direction. It's just a convention people like to use to aid visualization. You can point the axes in whichever direction you want, as long as they obey the right-hand rule (which is also another convention). In practice, which way the axis points has no effect on the math. All your results will be the same.

As to why people prefer to have the $z$ axis pointed up (which seems to be what you're actually asking), I believe it comes from transitioning from 2D to 3D geometry. People are used to looking at the $xy$ plane on paper, so in adding another dimension it's intuitive to imagine "looking down" onto it, as if it's on the ground, and the third axis points towards the sky (in our direction).

You can, however, make the argument as to why it's not the case to have the $xy$ plane stay the same as before, with $y$ vertical, and $z$ points out of the page; maybe it's harder to depict, I don't know. Again, there's no objective answer to this. A convention is a convention.

Dylan
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It is the same reason why we call some cars as righthand drive, one could have equally well have called it rigthfoot drive (we use a our foot a lot to drive)

P Vanchinathan
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    What does "right hand drive" have to do with the use of hands in driving? And what does this answer have to do with the question? – David K Dec 15 '17 at 02:48
  • I wanted to make the point that one should not confuse plain conventions with mathematical statements. Why counterclockwise direction is called positive? I can say say right eye side, righ ear side, instead of righthand side. Same way it is a convention to use the letter $z$ as vertical axis . Nothing more to it. – P Vanchinathan Dec 15 '17 at 04:48
  • My choice of car driving was careful and deliberate. To emphasize there is no mathematics behind it. If I try to to explain it in mathematical context it might give the impression that the question is mathematical in nature. It is not. This is my explanation to all the people who have taken interest in my answers and read it and wondering why I wrote that way. – P Vanchinathan Dec 15 '17 at 04:53