Vector Algebra: JEE Main 2023 Mathematics question

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j}, \hat{i}+\hat{k}$ and $\hat{i}-\hat{j}, \hat{j}-\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}$ and $\vec{a} \cdot \vec{b}=6$, then the ordered pair $(\theta,|\vec{a} \times \vec{b}|)$ is equal to :