How to construct a vector orthogonal to all but one?

[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?

 

[BvU]

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

 

[td21]

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

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

 

[td21]

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

 

[BvU]

Ah, sorry, I didn't read carefully enough. You have no prior knowledge of the v_v ... v_n 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 v_v ... v_n ... But by now I'm not all that certain any more...

 

[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 v_i 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, ..., v_n 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.

 

[FactChecker]

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

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