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Say something has cylindrical symmetry so I align it on an axis to take advantage of that symmetry (this will make things like calculating the volume of a cylinder much easier if the $z$ axis pierces through its center). I can do this because space is space and coordinate axes are arbitrary as long as we value the importance of the fact that up and down are independent to left and right and in and out spacially (and I think generally in physics?).

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Let's say I choose going up to be in the $x$ direction. When I typically draw axes like this, I put $x$ along the horizontal direction, increasing to the right of the page, $y$ along the vertical increasing up, and $z$ into and out of the page increasing out. However, are these directions (up, right and out being increasing) convention? Does me making $x$ go up and down and increase up and $z$ go left and right and increasing at the right specify the direction where $y$ ought to increase by the right hand rule or something? If so, why does this violate the arbitrary-ness of coordinate axes I was talking about before?

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As for the orientation of the axes between each other I believe it is dictated by the right hand rule. Let us denote with $\vec{u}_z$ the unit vector ($|\vec{u}_z| = 1$) directed along the positive verse of $z$. We define similarly $\vec{u}_x$ and $\vec{u}_y$ for the other two axes (they are unitary and oriented towards the positive direction of $x$ and $y$).
According to the right hand rule if you were to fold with the palm of your right hand $\vec{u}_x$ towards $\vec{u}_y$ your thumb will point to the direction of $z$. Imagine to do that on this picture:

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This is done to be consistent with the cross product. The vectors returned by the cross product follow the same rule. We could also write indeed: \begin{equation} \vec{u}_z = \vec{u}_x \times \vec{u}_y. \end{equation} We could also choose to define our axes in a left-handed system of course, but then the cross product should be also calculated with the left hand.

As for the reason why the height is identified with the $z$-axis I guess it's just a progressive use of notation. You start to describe the motion in 1 dimension, you have 1 unknown: $x$. Then you consider motion in the plane and you get to use the next variable in alphabetical order $y$. Finally you consider a motion in three dimension and you get to use $z$.

So for the next system you draw, define the $z$-axis as the altitude (growing upwards) and define the $x$ and $y$ axes such that their cross product is $z$ (whether you choose a left-handed or right-handed one).

Cheers!

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