I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused:
Is the top part saying: if $o_z(t) = 0$ then $r(t)$ = the big vector next to it, otherwise $r(t) = [ -Ω ; -i ; -ω] * o(t)$?
This is a transformation between the Cartesian vectors in the frame of the orbit itself (which are pretty easy to get from orbital elements) to the Cartesian vectors in the frame you want (probably J2000 or another Earth frame). The reason $\mathbf{o}_z(t) = 0$ is because the orbit is planar in its own frame (by Kepler's laws), so there is, by assumption, no $z$ component (similarly no velocity component in $z$). The product $R(-\Omega)R(-i)R(-\omega)$ is a 313 rotation sequence of Euler angles which are identified with the angular Keplerian elements, depending on how you define the signs of various elements and whether you make the rotation passively (I'm guessing this case) or actively (rarely done). What the equation is saying is you can use the elements themselves as Euler angles to rotate Cartesian vectors in the orbit frame to the Earth frame like any other rotational transformation. Those big vectors are the result of multiplying out the 313 sequence to get the general form of the direction cosine matrix for that parameterization, then multiplying this DCM by the orbit frame Cartesian vector. This is, by definition, the Earth frame Cartesian vector.