From polar to heliocentric ecliptic coordinates

[Shukie]

I'm currently studying how to calculate planetary orbits and I'm stuck. I have calculated the polar coordinates of the planet, with the at the origin and the x-axis as the striped line in this picture:

Now I have to transform the polar coordinates into heliocentric ecliptic coordinates. To do so, I first have to convert them into cartesian coordinates (simple enough) and then I have to switch the plane of reference so that the x-axis will point towards $$\Upsilon$$. How do I do this? The answer is found below, but could anyone tell me the actual process of getting this answer?

 

[lippyka]

The process of transforming the polar coordinates into heliocentric ecliptic coordinates involves rotating the plane of reference by an angle equal to the argument of perihelion. This angle is determined by the longitude of perihelion, which is measured relative to the vernal equinox. The rotation of the plane of reference is done in a counter-clockwise direction and can be expressed mathematically as the following transformation matrix: $\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix}\cos(\omega) & -\sin(\omega) & 0 \\ \sin(\omega) & \cos(\omega) & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}$Where $\omega$ is the argument of perihelion, $x'$, $y'$, and $z'$ are the heliocentric ecliptic coordinates, and $x$, $y$, and $z$ are the cartesian coordinates derived from the polar coordinates.