Orthogonal basis to two vectors in R4

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1. The problem statement, all variables and given/known data

Find all vectors that are perpendicular to (1,4,4,1) and (2,9,8,2)

3. The attempt at a solution

Create matrix A = [[1,4,4,1],[2,9,8,2]]
Set Ax = 0
Reduce by Gauss elimination
Produces basis of (-4,0,1,0) and (-1,0,0,1)

I don't know what the correct solution to this problem is, but as far as I understand it, it would seem that the basis should be one dimensional as the two given vectors form a plane and only a line is orthogonal to a plane, not a plane.


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You are in FOUR dimensions. The original vectors span a 2 dimensional subspace. The orthogonal subspace ought to also be 2 dimensional. Only in three dimensions would the space orthogonal to a two dimensional space be a line.
But these are vectors in R^4, so your basis will have to have four vectors. True, the given vectors form a plane, but it's a plane in four-dimensional space. There are two more dimensions that aren't in this plane.

Start with the vector (x, y, z, w). It has to be perpendicular to (1, 4, 4, 1) and (2, 9, 8, 2), so find the dot the first vector with each of the two others. That will give you two equations in four unknowns, so there will be two variables that are free. Choose convenient values to get two more vectors for your basis.

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