So, I'm working in 3D space. I have a frictionless particle sitting on an inclined plane. There's gravity (pushing down on the Y axis), so the particle will slide down the slope.
If I know the normal of the plane, how can I figure out the direction (as a 3D vector) that the particle will slide in? I feel like this should be very obvious, but I'm just stuck trying to figure it out.
On
Take a plane that contains both the normal and gravity vectors, then the intersection of that plane and your inclined plane will contain the direction you seek.
Please remember that this solution assumes that your plane is inclined, i.e. the normal and gravity vectors are not parallel, and so there exists exactly one plane that contains them both.
I hope this helps $\ddot\smile$
You need to project the normal direction $\vec n$ out of the vertical direction $\vec e_y$; the sliding direction is
$$\vec e_y-(\vec e_y\cdot\vec n)\vec n$$
(if the normal $\vec n$ is given as a unit vector; else you need to divide the right-hand term by $\vec n\cdot\vec n$).