Let \(\vec{a}=\hat{i}-\hat{j},\text{ }\!\!~\!\!\text{ }\vec{b}=\hat{i}+\hat{j}+\hat{k}\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }\vec{c}\) be a vector such that \(\vec{a}\times \vec{c}+\vec{b}=\vec{0}\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }\vec{a}\cdot \vec{c}=4,~\text{then }\!\!~\!\!\text{ }{{\left| {\vec{c}} \right|}^{2}}\) is equal to:

• A. \(\frac{19}{2}\)
• B. 9
• C. 8
• D. \(\frac{17}{2}\)