# How to construct a vector perpendicular to a bunch of known vectors?

1. Jul 18, 2014

### jollage

Hi,

Given several vectors, which may be or not be orthogonal to each other, how to construct a vector perpendicular to them? In a sense of inner production being zero.

To be specific, I have $n$ vectors $v_{N}$ of length $N$, where $n<N$. So the maximum rank for these vectors is $n$, which leaves space for new vectors perpendicular to all of them. How to construct such a vector? I know Gram-Schmidt process, but it seems it's not what I want.

Thanks.

2. Jul 18, 2014

### micromass

Staff Emeritus
Why isn't the Gram-Schmidt process what you want?

3. Jul 18, 2014

### WWGD

4. Jul 18, 2014

### jollage

Thanks. I see. I should use Gram-Schmidt process.

5. Jul 18, 2014

### WWGD

You can, but you can also write a matrix M using your vectors, row-reduce, calculate the nullspace of M and then use the fact , by the fundamental theorem of algebrar ,that the nullspace of the matrix is the ortho complement of the row space, and then you can find a basis for the nullspace. Just an alternative.

Last edited: Jul 18, 2014