How to project a point onto a plane in 3D?

1. Make a vector from your orig point to the point of interest:

v = point-orig (in each dimension);

2. Take the dot product of that vector with the unit normal vector n:

dist = vx*nx + vy*ny + vz*nz; dist = scalar distance from point to plane along the normal

3. Multiply the unit normal vector by the distance, and subtract that vector from your point.

projected_point = point - dist*normal;

Edit with picture:
I’ve modified your picture a bit. Red is v. dist is the length of blue and green, equal to v dot normal. Blue is normal*dist. Green is the same vector as blue, they’re just plotted in different places. To find planar_xyz, start from point and subtract the green vector.