Vector Algebra

Let three vectors ,\(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}\) form a triangle such that \(\vec{c}=\vec{a}-\vec{b}\) and the area of the triangle is \(5 \sqrt{6}\). If \(\alpha\) is a positive real number, then \(|\vec{c}|^2\) is equal to: