How to obtain a vector perpendicular to a set of vectors (with Matlab)?

Question

Let suppose I want to build a set of N orthonormal vectors. Actually I know N-1 orthonormal vectors and I just want to learn the last one.

I should to solve a linear system, setting to zero each scalar product between the N-1 known vectors and that one to find. But how could I do that with Matlab?

EDIT: note that this problem can be seen as a system of linear equations. If U is the (N-1xN) matrix whose rows contain the known vectors, then I should solve the equation Ux = 0, with x the vector to find and 0 the vector of zeros. To do that I used this simple code:

   x = U\0

But, in this way, it obviously returns the banal solution 0. I need to the non trivial solution, any idea?

Answer 1

I'm quite sure you want a non-trivial solution to a linear system that satisfies Ax = 0 without x being a vector of zeros.

This can be done using Singular value decomposition like this:

A = [2 -1 1; 2 -1 1; 3 2 1];
[U S V] = svd(A);
x = V(:,end)

x =

  -0.39057
   0.13019
   0.91132

A*x = 

   0
   0
   0