Ok, so it's been a while since college and I'm trying to add something to our software that requires so math I'm a bit rusty on. I'm also in software and not a math superhero, so please dumb it down a little for me.
I need to get the equation to find a point along a plane at a constant distance away from an orthogonal vector.
Just for reference on what I'm using this for, I have a camera looking in some general direction. I want to find a point some set distance away from the center of the camera's view on it's focal plane.
That said, this is what I have so far:
I have n=[a b c] and I have (x0, y0, z0) where x0 = longitude, y0 = latitude and z0 = altitude (above ground) in the world.
I need a simplified formula that I can use to move any arbitrary x or y distance on that plane to get (x, y, z).
The condition you are after is that the vector $(x-x_0, y-y_0, z-z_0)$ is orthogonal to $(a,b,c)$, i.e. their dot product is $0$. This boils down to:
$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$
or:
$$ax+by+cz=ax_0+by_0+cz_0$$