How to Find Orbital Radius Vector with Inclination I'm trying to figure out how to calculate the orbital radius vector of a satellite orbiting around a planet. The calculations are easy with an inclination of 0 degrees, the radius vector, r, with a true anomaly of t, has the following vector in the i, j, k frame of reference: $$r = < |r| \cos(t), |r| \sin(t), 0 >$$ The problem arises when the inclination is not 0 degrees. A solution that sounded reasonable to me, given an inclination i was: $$r = < |r| \cos(t) \cos(i), |r| \sin(t) \cos(i), |r| \sin(i) >$$ But since inclination is constant, the satellite will have a generally uniform z-value. I came to the conclusion that there must be a term that replaces i that is dependent on it, say (i / t), but I don't know. Is there a formula that is used to determine radius vectors? Thank you.

The problem with your solution is that a constant $z$ means that the satellite is not moving around the center of the planet, but around the point $z$. What you need is to rotate the orbit. Suppose that the new axis of rotation is in the $xz$ plane making an angle $\theta$ with respect to the $z$ axis. To rotate the vertical axis into this axis, you rotate around the $y$ axis by the angle $\theta$. You can write the rotation matrix. You can now multiply this matrix with your initial $r$ to get $$< |r|\cos t \cos\theta, |r| \sin t, -|r|\cos t\sin \theta>$$