Vector Algebra

\(\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) are three vectors. For a vector \(\mathbf{r}\) with \(\mathbf{r} \times \mathbf{a}=\mathbf{b}\) and \(\mathbf{r} \cdot \mathbf{c}=3,|\mathbf{r}|\) is