Consider the vector space V=R3V = \mathbb{R}^3V=R3 with the inner product ⟨u,v⟩=u1v1+2u2v2+3u3v3\langle u, v \rangle = u_1 v_1 + 2u_2 v_2 + 3u_3 v_3⟨u,v⟩=u1v1+2u2v2+3u3v3. Are u=(0,1,0)u = (0, 1, 0)u=(0,1,0) and v=(1,0,0)v = (1, 0, 0)v=(1,0,0) orthogonal?
Yes
No
Only if they are unit vectors
Depends on the basis