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From: "Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Angle between two vectors
Date: Fri, 28 Dec 2007 21:52:57 +0000 (UTC)
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"baris kazar" <mbkazar.nospam@gmail.com> wrote in message <fl3nhq$3tc
$1@fred.mathworks.com>...
> Hi Roger,-
>   yes, this is one step closer to what i need but not 
> exactly. Let's take a numeric example:
> x=(1,0,0); y=(1,0,1) and z=(1,0-1)
> let's call the angle between x and y theta.
> Then i wanna get 2pi-theta for the angle between x and z.
> i dont have access y and z at the same time. 
> hope that this problem statement is clear.
> Thanks much for your reply
> Best regards
--------
  You will have to try harder to explain your problem, if I am to understand 
you, Baris.  The example you gave has x, y, and z all in the same plane.  You 
didn't state previously that all your vectors are coplanar.  Are they?  But 
whether they are or not, this doesn't explain how you would define the angle 
theta between x and y.  It could be plus pi/4 or it could be minus pi/4 (or 
7/4*pi.)  Which one would you choose and according to what criterion?  It 
would depend on which side of the plane, in this case the x-z plane, is 
regarded as its positive side - along the plus y-axis, or along the negative 
side of the y-axis.  Also it depends on whether you are moving from x 
towards y or from y towards x if you are adhering to right-hand cross 
product direction conventions.

  Select two arbitrary vectors in three-dimensional space (x1,y1,z1) and 
(x2,y2,z2) and try to think of a consistent way of defining the angle between 
them without reference to any other vector that would allow this quantity to 
range over the full four quadrants, 0 to 2*pi.  I think you will find this a 
difficult thing to do in any way that could reasonably be considered canonical.  
In two dimensions, there is a clearly defined counterclockwise direction from 
vector x to vector y which would give you the range you desire.  In three 
dimensions, you lose the sense of what is a "counterclockwise" direction.  You 
can go from x to y along either of two great circle paths and one direction will 
give the supplement angle to the opposite direction.  If you always select the 
shortest path, then your angle range is restricted to [0,pi].

  If one vector of each pair is restricted to a particular fixed vector, as your 
previous wordage seemed to imply, I still don't see what criterion you wish to 
use to define these angles.  For example, you can move from the fixed vector 
by one degree in all possible directions giving a cone, but which half of these 
angles should be adjusted so as to be the supplements (that is 359 degrees,) 
of those on the opposite side?  If you are restricting your vectors to all be 
coplanar, then which side of such a plane is to be considered its "positive" 
side?

Roger Stafford