Gravity in the x, y and z directions

[Philosophaie]

I am working in classical mechanics. A planet is orbiting a star. The planet has a given velocity and a position vector from the star. How do I find the magnitude of the gravity in the x, y and z directions.

positionvector = (x_0, y_0, z_0)
velocityvector = (v_x, v_y, v_z)
x = \frac{1}{2}*g_x*t^2 + v_x*t + x_0
y = \frac{1}{2}*g_y*t^2 + v_y*t + y_0
z = \frac{1}{2}*g_z*t^2 + v_z*t + z_0
r = \sqrt{x^2 + y^2 + z^2}
\theta = atan(\frac{y}{x})
\phi = acos(\frac{z}{r})
g = \sqrt{g_x^2 + g_y^2 + g_z^2} = -\frac{GM}{r^2}

Any hints on how to find (g_x, g_y, g_z)

 

[mfb]

x = \frac{1}{2}*g_x*t^2 + v_x*t + x_0
y = \frac{1}{2}*g_y*t^2 + v_y*t + y_0
z = \frac{1}{2}*g_z*t^2 + v_z*t + z_0

That is not true, those formulas would require a constant acceleration. Acceleration in a gravitational field is not constant (even if that can be a good approximation in some cases).

Any hints on how to find (g_x, g_y, g_z)

Newton's law of gravity in its vector form gives that. Alternatively, use your g, and let it point from the planet to the central object.

 

[Astrum]

You're playing with the two-body problem, right? Why don't you use polar coordinates, it simplifies things greatly.

I'm not exactly sure what you're trying to do.

 

[Philosophaie]

I am looking for an object that is in freefall and its path towards a star from an initial velocity and position.

How do you formulate the acceleration of non-constant acceleration?

 

[mfb]

How do you formulate the acceleration of non-constant acceleration?

In the general case with more than 2 objects, there is no useful, closed formula to calculate the position for all times.
With just 2 objects, this is known as Kepler problem and has exact solutions.

 

[Electric Red]

Seems like you are a bit confused with are these formula, like Astrum said, you could try to use polar coördinates. Have a look at chapter 6 of Fowles and Cassiday's "analytical mechanics".
Or just at this, hope you will be able to figure it out by now:
https://en.wikipedia.org/wiki/Kepler_problem#Solution_of_the_Kepler_problem

 

[tfr000]

What is generally done is something like multiplying Gm_a / r_ab² by (r_a-r_b) / r_ab, where m_a is the mass of object a and r_ab is the distance between objects a and b.
(r_a-r_b) / r_ab forms "direction cosines" when you resolve the vectors with appropriate x, y, z coordinates.