Vector Algebra

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}$ be two vectors such that $\mathbf{a} \cdot \mathbf{b}=1$, $\cos (\mathbf{a} \cdot \mathbf{b})=\frac{1}{3}$ and the components of $\mathbf{b}$ w.r.t. $(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}})$ be integers. Then, the number of possible vectors that represent $\mathbf{b}$ is