Question

find the orthogonal complement of the plane spanned by the vectors (1,1,2) and (1,2,3), by taking these to be the rows of A and solving Ax=0

Answer 1

The two vectors are independent. Elimination will lead to two pivot columns hence you will have one free variable. Assign the free variable a value of '1' and solve for the pivot (basic, non-free) variables. This will give you the line of solutions to Ax=0. If V and W are orthogonal complements of n-dimensional space, then every vector in n-dimensional space that is orthogonal to every vector in V, belongs to W. The orthogonal complement of the row space is the null space - Ax=0 implies that x multiplies every row of A to produce zero, hence solving for the nullspace gives you the orthogonal complement of the plane spanned by the two vectors.