I am working in a right-handed orthonormal coordinate system with the bases $\hat e_1,\hat e_2,\hat e_3$. I am trying to work out the dot product of the unit position vector with the $\hat e_2$ axis. I have acknowledged that the dot product of $\hat r\cdot \hat e_3$ is equal to $\cos\theta$ by inspection as in the picture below, as the angle $\theta$ is between the two vectors.
However, there is no direct angle between the $\hat e_2$ and the $\hat r$ vector. I have reasoned that the dot product of $\hat r\cdot \hat e_2$ will be the equivalent to the unit parametrisation of the y axis, being $\sin\phi\cos\theta$. I believe this to be the likely result as the dot product of $\hat r\cdot \hat e_3$ is equivalent to the unit parametrisation of the z axis.
In short:
- Is $\hat r\cdot \hat e_2=\sin\phi\cos\theta$?
Thanks
Examine again the right triangle in your illustration. Its hypotenuse has length $r$ and as you’ve already found, the vertical leg has length $\lvert r\cos\theta\rvert$. By the Pythagorean theorem, if for no other reason, the length of the other leg, which lies in the $x$-$y$ plane, must have length $r\sin\theta$. That line segment is itself the hypotenuse of a right triangle with one leg along the $x$-axis. Can you take it from there?
N.B.: This is summarized in the formulas for converting from spherical to Cartesian coordinates that you can find here.