Ok. I am trying to translate a point between 2 different 3d planes.
For example, I have a plane .623x - .33y -.7z = 0 and the point ~(233,181,122). I want to translate this point onto a another plane. Aka, the point with the same (vector translation?) onto a second plane, let's say, .35x + .37y - .85z = 0. How would I find the "same point" on the second plane. I'm a bit confused how this might work, so my assumption is that the "same point" would have a similar vector. I figured this would be possible if we assume the origin at (0,0,0) since the planes intercept there.
I was thinking change of basis? But not sure.
You can find the coordinates of a point relative to a plane if and only if you know the basis vectors $x'$ and $y'$ in the new plane.
When you say "the coordinates are $(233,181,122)$" I have no idea where that is. If it's in the standard $x,y,z$ Cartesian system, then OK, but does your origin stay the same? And in what directions do the $x',y'$ axes point in either plane? That will let you determine how many units in each of $x',y',z'$ your point is away from that origin.
So, in other words, you have to define a perpendicular set of $x',y'$ in each of your planes (and then take $z'$ to be normal to that plane, and directed such that you get a right-handed coordinate system) to define points relative to those planes. The planes themselves can't do that for you.