$M$ be the subspace of $\Bbb R^3$ spanned by $(1,0,-1).$

Question

I am stuck on the following problem:

Consider $\Bbb R^3$ with the standard inner product. Let $M$ be the subspace of $\Bbb R^3$ spanned by $(1,0,-1).$ Which of the following is a basis for the orthogonal complement of $M\,\,?$

1. $\{(2,1,2),(4,2,4)\}$

2. $\{(2,-1,2),(1,3,1),(-1,-1,-1)\}$

3. $\{(1,0,1),(0,1,0)\}$

4. $\{(1,2,1),(0,1,1)\}$

Can someone explain it?

Answer 1

You know that the complement has dimension 2. You also know that each vector in the complement is orthogonal to $(1,0,-1)$, so the basis vectors are too. In other words, you need to check

a) Which set of vectors are linearly independent and span a two dimensional subspace.

b) Which set consists of vectors orthogonal to $(1,0,-1)$.