Why does $z$-axis have to point out of the page for a typical orientation of the $y$-axis and $x$-axis? In class, when I constructed my plot of a $3$D solid, I thought that it was arbitrary which directions I chose as my positive $x$-, $y$-, and $z$- axes. But when I got the paper back, I was told that the $z$-axis must be the cross product of the $x$- and $y$ axes (in that order of course, because cross product is not commutative). Why is this? What does it affect if this rule is not followed?
 A: *

*If we define

*

*the vectors $\vec i, \vec j, \vec k$ as pointing in the $x$-, $y$- and $z$- directions, respectively,

*and the cross product's direction according to the right-hand rule,

*and $\vec i \times \vec j = \vec k,$
then adopting the convention of a right-handed Cartesian coordinate system,
rather than a left-handed one, is consistent with this.


*On the other hand, from this article (3-D Coordinate
Systems), for example:

Typically, 3-D graphics applications use two types of Cartesian coordinate systems: left-handed and right-handed.
Microsoft Direct3D uses a left-handed coordinate system. If you are porting an application that is based on a right-handed coordinate
system, you must make two changes to the data passed to Direct3D.

A: It doesn't really break anything. It is a convention, just like it is a convention that in a planar coordinate system, the positive $y$-axis lies $90^\circ$ counterclockwise of the positive $x$-axis. Conventions are nice, and it is within reason for a teacher to require that you work within a given set of conventions. Such as the order of operations (BEMDAS or whatever you call it), or writing the function name on the left-hand side of the brackets with arguments (so $f(x)$ rather than $(x)f$).
It is entirely fine to let the unit $z$-vector be the negative of the cross product of the $x$ and $y$ unit vectors. Right-hand and left-hand are the terms given to distinguish between these two coordinate systems.
How this affects things depends on your situation:

*

*You have algebra and sets of coordinates and want to use a coordinate system to visualize it. In this case, choosing one over the other will mirror the resulting figures

*You have points in space and want to use a coordinate system to get coordinates and do calculations. In this case, choosing one over the other will flip the sign of the $z$ coordinate.

Note that many 3D computer frameworks, like the 3D modeling software Blender and the game engine Unity, work in this "mirrored" coordinate system, and they work entirely without flaws (at least in this respect). So this is not as universal a convention as some of the other conventions I mention in this answer.
