Short question about spherical coordinates

Question

If I have a vector orthogonal to the $x$-$y$ plane of an $xyz$ axis system, I mean, a vector with just $z$ component:

How can I express it in spherical coordinates?

Answer 1

Spherical coordinates of a point $P$ are defined as :

The radius $=r$,i.e. the distance of $P$ from the origin $O$.

The azimuth $=\phi$ , i.e. the angle between the positive $x$ axis and the orthogonal projection of the vector $ \overrightarrow{OP}$ on the $(x,y)$ plane, measured anticlockwise. Its range is usually $0\le \phi < 2 \pi$.

The polar angle $=\theta$ (also called zenith angle), i.e. the angle between the positive $z$ axis and the vector $ \overrightarrow{OP}$ measured clockwise. Its range is usually $0\le \theta \le \pi$.

So, for a point $(0,0,P_z)$ on the $z$ axis: $\theta=0$, $\phi$ can take any value ( usually $0$) and $r=P_ Z$, and his polar coordinates are $P=(P_z,0,0)$.

The simbols $\phi$ and $\theta$ are not so fixed, and in some contest they are interchanged. Note that in a geographical contest is used the latitude $\lambda$(the angle from the equator) insted of the polar angle , and we have $\lambda=\pi/2-\theta$.