Let \( \vec{a}= \hat{i}-2\hat{j}+3\hat{k} \), \( \vec{b}=2\hat{i}+\hat{j}-\hat{k} \), \( \vec{c}= \lambda \hat{i}+\hat{j}+\hat{k} \) and \( \vec{v}=\vec{a}\times \vec{b} \). If \( \vec{v}\cdot \vec{c}=11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then \( 9p^{2} \) is equal to:
