I am writing a program for a computer science class in which I am doing an n-body simulation in 3-dimensional space. Currently, I have figured out the gravitational force along the hypotenuse between two bodies. Now I have to split these up into the x, y, and z components, and this is where I am having trouble.

Force between two objects (force\_t)=G\*m1\*m2/r^2

Typically, force\_x=force\_t\*cos(angle between x-axis and hypotenuse)
force\_y=force\_t\*sin(angle between x-axis and hypotenuse)

So, I can figure out my triangle. I know that my two bodies are located at (x1, y1, z1) and (x2, y2, z2), so I have my triangle. And I am fairly certain that the above equations will hold for x and y, even though I'm in three dimensions, but I'm just not sure what the corresponding equation for the z component will be. My thought was that the combination of the three forces has to equal the total force, so perhaps I solve for the z component algebraically using the pythagorean theorem, but I have no way to check whether or not this is true.

Anyway, any help would be appreciated. Thanks!