Let V be an inner product space, and let W be a finite-dimensional subjspace of V. If x is not in W, prove that there exist a vector y in V such that y is in the orthogonal complement of W, but <x,y> is not equal to 0.
Proof. By a theorem, there exist a unique vector u in W and z in orthogonal complement of W such that x = u + z.
<x,z> = <u+z,z> = <u,z> + <z,z>. Now, z is not 0, therefore <z,z> cannot equal to 0, so z is the vector y that we are looking for.
Is that right?