# How to construct a vector orthogonal to all but one?

1. Sep 21, 2015

### td21

Given n linearly independent vectors, v1, v2, v3, .......vn.
How to find construct a vector that is orthogonal to v2, v3, ....., vn (all v but not v1)?
Is Gram Schmitt process the way to do this? or just by brute force?

2. Sep 21, 2015

### BvU

I take it this is in Rn ? Then: yes !
Gram Schidt is brute force as far as I am concerned.

3. Sep 21, 2015

### td21

Thanks for the reply. But why is Gram Schmidt process needed? My original brute force idea is to solve n-1 equations.

4. Sep 21, 2015

### td21

I also wonder if there is a neat expression for such vector?

5. Sep 21, 2015

### BvU

Ah, sorry, I didn't read carefully enough. You have no prior knowledge of the vv ... vn to exploit, so orthogonalizing seems to me the only way to get rid of the components of v1 that are in the subspace spanned by vv ... vn ... But by now I'm not all that certain any more....

6. Sep 21, 2015

### FactChecker

Gram-Schmidt is not necessary. People usually think of Gram-Schmidt as making an entire set of n orthogonal vectors. Is that what you mean?

Since you only care about finding the orthogonal part of v1, I think you can modify the Gram-Schmidt process to work on only v1. Just keep subtracting the projection of the next vi on what remains of v1.

Your "brute-force" method of solving equations also works. Just like the modified Gram-Schmidt, it does nothing to make the v2, ..., vn orthodonal. I don't know which approach would be less "brute-force".

PS. Don't you mean n equations? You want the dot product of x with v1 to be nonzero (set it = 1) and all the other dot products to be =0.

7. Sep 21, 2015

### FactChecker

I think that would be w = [v1; v2; ... ; vn]-1(1, 0, 0, ..., 0)