Question: is inner product always positive?
Answer: The inner product is positive definite if it is both positive and definite, in other words if ‖x‖2>0 whenever x≠0. The inner product is negative semidefinite, or simply negative, if ‖x‖2≤0 always. The inner product is negative definite if it is both positive and definite, in other words if ‖x‖2<0 whenever x≠0.