Why Isn't Bloch's Theorem Reduced to Unity?

[toqp]

This is not any homework problem but just something I don't understand. The Bloch theorem states that
$$\psi(\textbf{r}+\textbf{R})=e^{i\textbf{k}\cdot \textbf{R}}\psi(\textbf{r})$$

Now the k is a vector in the reciprocal lattice (usually in the first Brillouin zone), which is defined as the set of vectors K that satisfy
$$e^{i\textbf{K}\cdot\textbf{R}}=1$$

Now, if k points to a point in the reciprocal lattice, then why isn't the Bloch theorem
$$\psi(\textbf{r}+\textbf{R})=e^{i\textbf{k}\cdot \textbf{R}}\psi(\textbf{r})$$
just
$$\psi(\textbf{r}+\textbf{R})=1\psi(\textbf{r})$$?

 

[turin]

AFAIK, k is not limited to a reciprocal lattice vector. I think that would only be for standing waves. The complex phase gives the wave a direction.