Solution

The plane is the column space of the matrix whose columns are the two vectors given and the

row space of its transpose . The orthogonal complement then is the nullspace of this matrix A.

All we really need to know to answer the question is the dimension of the nullspace of this matrix. Since it has rank 2 we know that the nullspace has dimension 4-2 = 2 so the complement will be of dimension 2. We can of course find a basis for for this space by finding one for the nullspace of A.

The given vectors are clearly independent (we see a 2 by 2 nonzero subdeterminant ). So the complement must have dimension . We simply make the nullspace calculation for the transposed matrix of the two vectors.