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 \( \left(\theta, \left|\vec{a}\times\vec{b}\right|\right) \) is equal to:
