SOLUTION: If I have the two vectors u = <0,5,3> and v = <0,-2,-5>, how can I find the projection of u orthogonal to v? I understand regular projections, but I'm not sure how to work out a pe

If we look at it in 2D, you'll get the idea:


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The green line from the tip of u perpendicular to v cuts off the
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projection p of u onto v.
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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:


 
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 u!
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Now what is that green vector?  it's simply u - p.   See now?

So here's what we do:
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We first find the projection p of u onto v.  Then we subtract that from u
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1. Find u•v: <0,5,3>•<0,-2,-5> = 0-10-15 = -25
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2. Find ∥v∥²  = 0²+(-2)²+(-5)² = 29
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3. Divide u•v by ∥v∥² = 
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4. Multiply that by by v:   
  <-- that's p, the projection of u onto v.
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5. Subtract  u - p:

   
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That's the green vector perpendicular to v.

Better check my fraction math.

Edwin