Question 977660
<pre>
If we look at it in 2D, you'll get the idea:

{{{drawing(200,2800/9,-1,8,-1,6,
locate(3,5.5,u), locate(7.3,.3,v), locate(1.6,0,p),
line(0,0,3,5), line(2.99159003,4.50007073,3,5), line(2.56284357,4.75731861,3,5),
line(0,0,3,0), line(2.5669873,-0.25,3,0), line(2.5669873,0.25,3,0),
line(0,0,7,0), line(6.5669873,-0.25,7,0), line(6.5669873,0.25,7,0),

green(line(3,0,3,5))

 )}}}
                               _                  _
The green line from the tip of u perpendicular to v cuts off the
           _    _      _
projection p of u onto v.
                                    _                                _
We want the vector perpendicular to v that has its tip at the tip of u. 

That has to be the vector coinciding with the green line! Like this:

{{{drawing(200,2800/9,-1,8,-1,6,
locate(3,5.5,u), locate(7.3,.3,v), locate(1.6,0,p),
line(0,0,3,5), line(2.99159003,4.50007073,3,5), line(2.56284357,4.75731861,3,5),
line(0,0,3,0), line(2.5669873,-0.25,3,0), line(2.5669873,0.25,3,0),
line(0,0,7,0), line(6.5669873,-0.25,7,0), line(6.5669873,0.25,7,0),

green(line(3,0,3,5),line(3.2,4.7,3,5))

 )}}}
 
All I did was simply to put an arrowhead on that green line!
I made that green line that cut off the projection vector p into a vector!
I put an arrowhead on the tip of it at the tip of <span style="text-decoration: overline">u</span>!
                                            _   _
Now what is that green vector?  it's simply u - p.   See now?

So here's what we do:
                             _    _      _                              _
We first find the projection p of u onto v.  Then we subtract that from u
        _ _                    
1. Find u•v: <0,5,3>•<0,-2,-5> = 0-10-15 = -25
         _                               
2. Find &#8741;v&#8741;²  = 0²+(-2)²+(-5)² = 29
          _ _     _
3. Divide u•v by &#8741;v&#8741;² = {{{-25/29}}}
                       _      
4. Multiply that by by v:   
{{{matrix(1,7,""<"",0,",",50/29,",",125/29,"">"")}}}  <-- that's <span style="text-decoration: overline">p</span>, the projection of <span style="text-decoration: overline">u</span> onto <span style="text-decoration: overline">v</span>.
             _   _
5. Subtract  u - p:

{{{matrix(1,7,""<"",0,",",5,",",3,"">"")}}}{{{""-""}}}{{{matrix(1,7,""<"",0,",",50/29,",",125/29,"">"")}}} {{{""=""}}} {{{matrix(1,7,""<"",0,",",95/29,",",-38/29,"">"")}}} 
                                         _
That's the green vector perpendicular to v.

Better check my fraction math.

Edwin</pre>