Find the Y-Axis in a Coordinate System with Given X and Z-Axis Values

[Philosophaie]

MENTOR note: moved from General Math hence no template

What would be the Y-Axis if:

X-Axis: theta=266.4 phi=-28.94
Z-Axis: theta=192.85 phi=27.13

where:
theta=atan(Y/X)
phi=asin(Z/R)

My thinking, theta is +90 from X-Axis and phi is -90 from the Z-Axis.
Is the Y-Axis theta=356.4 phi=-62.87?

 

[Ray Vickson]

MENTOR note: moved from General Math hence no template

What would be the Y-Axis if:

X-Axis: theta=266.4 phi=-28.94
Z-Axis: theta=192.85 phi=27.13

where:
theta=atan(Y/X)
phi=asin(Z/R)

My thinking, theta is +90 from X-Axis and phi is -90 from the Z-Axis.
Is the Y-Axis theta=356.4 phi=-62.87?

Are these supposed to be angles in a spherical coordinate system? If so, please specify precisely which convention you are using. There are two common, but different conventions: (1) $\theta =$ angle between the $z$-axis and the vector $(x,y,z)$, $\phi =$ angle from the positive $x$-axis, with counterclockwise angles being positive (so $\phi$ = longitude, measured west to east and $\theta$ = latitude, measured down from the North pole); and (2) the roles of $\theta$ and $\phi$ are swapped from the previous use. Convention (1) is most common in Physics, while (2) is used a lot (but not universally) in Math.

 

[Philosophaie]

I thought:

theta=atan(Y/X)
phi=asin(Z/R)

explained it.

In spherical coordinates:
theta is measured on the x-y plane from the x-axis.
phi is measured upward from the x-y plane to the z-axis.

 

[Ray Vickson]

I thought:

theta=atan(Y/X)
phi=asin(Z/R)

explained it.

In spherical coordinates:
theta is measured on the x-y plane from the x-axis.
phi is measured upward from the x-y plane to the z-axis.

The usual convention for a $\phi$ like yours would be $\phi = \arccos(z/r)$, so $\phi$ would be latitude as measured down from the north pole; see, eg., the second figure in https://en.wikipedia.org/wiki/Spherical_coordinate_system or the diagram in http://mathworld.wolfram.com/SphericalCoordinates.html or http://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx .

In any case, if you can determine (in $(x,y,z)$-space) the vectors $\vec{e_X}$ and $\vec{e_Z}$, which are the unit vectors along the $X$ and $Z$ axes, you can take $\vec{e_Y} = \vec{e_Z} \times \vec{e_X}$ as the unit vector along the $Y$ axis. (Here, $\times$ denotes the vector cross-product.)