Author 
Comment/Response 
Bill Simpson

12/16/12 5:12pm
If you search your university library catalogs for "Mathematica Graphics" by Tom WickhamJones then on pages 431432 he shows how to find the point in the plane closest to another point. The distance between those two points will be what you are looking for.
Suppose your plane is defined by a point in the plane and a point on a vector orthogonal to that point. Thus the xy plane might be {0,0,0} in the plane and {0,0,1} a normal vector to that point.
Now suppose you are given point p and you want to find the closest point q. The book shows
q=p(pc).n/(n.n) n
with c your point in your plane and n your point normal to your plane and where . is Mathematica syntax for dot product between vectors.
Now a concrete example.
In[1]:= n={0,0,1};c={0,0,0};p={3,8,5};p(pc).n/(n.n) n
Out[4]= {3,8,0}
Thus the point {3,8,0} is the point in the xy plane that is closest to {3,8,5}. Not surprising.
Now to get the distance from p to the point in the plane we subtract and use Norm to get the length of the resulting vector
In[5]:= Norm[p(p(pc).n/(n.n) n)]
Out[5]= 5
Check all this carefully to make certain that I haven't included any little mistakes. Test this by hand on some examples. Then test this with Mathematica to see that you consistently get correct results.
I find the index in the book doesn't usually immediately lead me to the page with the equation that I need for some geometry problem, but most of the equations are in there. Every time I keep saying I should make an index for just those equations, but I never get that done.
If you are new to Mathematica and plan on using this for geometry problems then you might see if you can get an inexpensive used copy from a book supplier. It would be a fairly good introduction for you.
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