Thread: Finding a vector perpendicular to three vectors in R^4

I'm trying to find all vectors in R^4 that are perpendicular to the following three vectors:

[ 1 1 1 1 ]
[ 1 2 3 4 ]
[ 1 9 9 7 ]


I'm not sure exactly how to set up a matrix that would help me to find these vectors. Any help setting this up would be appreciated. Thanks!

Here's what I would do: set up a system of three equations in four unknowns. You've got your three known vectors, call them The vector you're trying to produce that's perpendicular to all of them, we'll call Then you can get three equations by setting



and



Use normal Gaussian elimination to proceed. Does that make sense?

Another way of saying the same thing: call the vector <w, x, y, z>. Saying that is orthogonal to each of the given vectors means that
w+ x+ y+ z= 0
w+ 2x+ 3y+ 4z= 0
w+ 9x+ 9y+ 7z= 0

That is three equations in 4 unknow so you should be able to solve for three of them in terms of the fourth. That is, you should be able to find, say, w= az, x= bz, y= cz, for specific numbers a, b, and c, so that <w, x, y, z>= <az, bz, cz, z>= z<a, b, c, 1> and so the set of all such vectors is the one dimensional subspace, the set of all multiples of <a, b, c, z>.

Or you can just do a 4-dimensional cross-product!

(Note that while you can do the cross product of two vectors in three dimensional space, you need three vectors in four dimensional space.)