Linear algebra: orthogonal complements?

Let W be the subspace of R3 spanned by the vector w= [2; -3; 1]. Find a basis for W-perp.

How am I supposed to know how many vectors there are in W-perp? The answer has 2 vectors. I thought that since it's R3, it would have 3?

Question

Linear algebra: orthogonal complements?

Let W be the subspace of R3 spanned by the vector w= [2; -3; 1]. Find a basis for W-perp.

How am I supposed to know how many vectors there are in W-perp? The answer has 2 vectors. I thought that since it's R3, it would have 3?

zarkon · Answer

dim(span(w))+dim(w-perp)=3

he66666 · Answer

but the answer only has 2 vectors :/

zarkon · Answer

correct 1+2=3

dim(w-perp)=2

he66666 · Answer

Oh, so span(w) would be 1 because w has only one vector, right?

zarkon · Answer

correct..the dimension of the span of w is 1

he66666 · Answer

I understand. THanks so much zarkon!