Acceleration of a ball on a plane I have a plane ($ax+by+cz+d=0$) in a 3D world, and a gravity vector $\vec{g}$ (say it's $[0, 0, -9.81]$.) How would I find the acceleration vector of an object on this plane, ignoring friction?

you are looking to find the component of the gravity vector in the direction of the line of greatest slope of the plane.
Taking the gravity vector as vertically downwards, the vector parallel to the plane which is horizontal is $$\underline{n}\times\underline{\hat{g}}=\left(\begin{matrix}-b\\a\\0\end{matrix}\right)$$
Then the line of greatest slope is $$\left(\begin{matrix}-b\\a\\0\end{matrix}\right)\times\underline{n}=\left(\begin{matrix}ac\\bc\\-a^2-b^2\end{matrix}\right)$$
Now calculate the component of $$\left(\begin{matrix}0\\0\\-g\end{matrix}\right)$$ in this direction, which is $$\frac{g(a^2+b^2)}{\sqrt{a^2c^2+b^2c^2+(a^2+b^2)^2}}$$
