From: Wayne Piekarski <waynep@cleese.apana.org.au>
Subject: Re: Solution to a vector problem
To: rusin@math.niu.edu (Dave Rusin)
Date: Tue, 22 Nov 1994 15:06:00 +1523847 (CST)

> >I have an equation  p = o + Xu + Yv    where p, o, u, and v are all three
> >dimensional vectors, and X and Y are numerical values. I want to be able
> >to solve for X and Y. I cannot see any normal way to do this, so I
> 
> If I understand you right, you're trying to find real numbers  X  and  Y
> such that the linear combination   Xu+Yv  of vectors comes out to the
> vector  p-o,  right? First off, this may not be possible. You need to

Yes, basically a virtual screen is defined in 3D with the origin at 'o'
and the u and v vectors defining up and right.

> verify that  p-o  lies in the plane spanned by  u   and  v. One way to
> do this is to take the cross product of the vectors  u  and  v, then
> take the dot product of that with  p-o. If you don't get zero, quit -- no
> X's and  Y's can be found. If you do get zero proceed as follows. Take
> a vector w perpendicular to  u  but not to  v  -- (u x v) x u  will do --
> and then take dot products with both sides of the equation. The  Xu will
> drop out and you can solve for  Y =  (w . (p-o)) / (w. v). Similarly,
> you solve for   X.
> 
> Is that what you wanted?

Thanks for that. It is exactly what I wanted. I'll try it out and see what
happens....

thanks,
Wayne
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