Find the basis for the following vector space. Please state the dimension of the vector space. S consists of all x in R3 such that x is orthogonal to n=(2,3,2)

Answer 1

The dimension of vector space R3 is 3.
n=(2,3,2) occupies one of the three dimensions, so the subspace orthogonal to n has a dimension of two, i.e. two vectors span the remaining subspace.

The basis of the remaining subspace can be in many forms. One way is to start with an arbitrary vector, say a1=(1,1,0) and apply Gram-Schmidt process to transform it into a vector orthogonal to n. After normalization, we get A1=(7,2,-10)/(3sqrt(17)).
Note that A1.n=0.

Similarly, if we start with a2=(1,0,1), and apply Gram-Schmidt process, we get A2=(1,0,-1)/sqrt(2).
Note also that A2.n=0, and A1.A2=0.

Thus we have a basis for R3 {n,A1,A2} of which two are orthogonal to n. (n being one of the three vectors of the basis).

Answer 2

... and the basis for the subspace orthogonal to n is {A1,A2}.