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 March 1st, 2008, 07:26 PM #1 Newbie   Joined: Mar 2008 From: Canada Posts: 2 Thanks: 0 orthogonal unit vectors i am having a lot of trouble with the following question: Find two vectors of norm 1 that are orthogonal to the three vectors u=(2,1,-4,0) v=(-1,-1,2,2) w=(3,2,5,4) i have figured out (but could be wrong ) that the orthogonal vectors must be unit vectors if they have a norm of 1. also, the dot products of the unit vector and u,v, and w must all equal zero. however, i can't figure out a unit vector that makes the dot products equal to zero with u,v, and w! if someone could help me out i would really appreciate it
 March 2nd, 2008, 06:11 AM #2 Member   Joined: Aug 2007 Posts: 93 Thanks: 0 have you tried converting u,v & w to unit vectors (i.e. dividing by their length)?
March 2nd, 2008, 07:20 AM   #3
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Quote:
 Originally Posted by STV have you tried converting u,v & w to unit vectors (i.e. dividing by their length)?
That shouldn't help at all. As it stands, the OP just needs to find 2 vectors that are perpendicular; after that, getting a unit vector is trivial.

Intuitively, I've gotten so far, which may be of use:
If these three vectors are neither co-linear (which they obviously aren't), nor co-planar, (thus, co 3-planar, by definition), then there is only one vector in 4-space that is perpendicular. This means the 3 points need to be co-planar, right? Unless, the two unit vectors are z and -z, which would be a ridiculous requirement.

So, since the three vectors must be co-planar (I hope), each one intersects each of the others at some point. so u=v, v=z, and u=z at 3 distinct points, or all 3 cross at a single point (but this point wouldn't be enough to define a plane, so hopefully this isn't the case).

I hope this will help, or I may be misunderstanding some subtle difference between 3-space and 4-space, but there's only one degree of freedom for a 3-plane in 4-space, so there should be only one line unless they create a plane.

Anyway, DMY Sommerville has a lot of work on higher dimensions; you may want to look into his work.

March 2nd, 2008, 01:19 PM   #4
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Re: orthogonal unit vectors

Quote:
 Originally Posted by hrc i am having a lot of trouble with the following question: Find two vectors of norm 1 that are orthogonal to the three vectors u=(2,1,-4,0) v=(-1,-1,2,2) w=(3,2,5,4) i have figured out (but could be wrong ) that the orthogonal vectors must be unit vectors if they have a norm of 1. also, the dot products of the unit vector and u,v, and w must all equal zero. however, i can't figure out a unit vector that makes the dot products equal to zero with u,v, and w! if someone could help me out i would really appreciate it
Let z=(a,b,c,d) be the vector you are looking for.

z.u=z.v=z.w=0 gives you three equations in four unknowns. Solving this linear system will give you three of the unknowns in terms of the fourth. Finally using z.z=1 can give you the solution you want. There will be a sign ambiguity, which gives two possibilites.

 March 2nd, 2008, 05:40 PM #5 Member   Joined: Aug 2007 Posts: 93 Thanks: 0 My thought was that the cross product formula in 4-space would work (I do not know for sure) then you would have to calculate determinants on a 4 x 4 matrix. My idea was to that as a unit vector is a vector of cosines of direction angles,calculating the unit vectors for u,v & w one could set up an easier system of equations that would just add pi/2 to each vector. If I find the time I will see if this works.
 March 2nd, 2008, 06:51 PM #6 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 I was thinking that, and it may work, but I don't know enough linear algebra to know if that's valid.
 March 3rd, 2008, 01:59 PM #7 Newbie   Joined: Mar 2008 From: Canada Posts: 2 Thanks: 0 thanks for your help!

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