Let three vectors a =α i +4 j +2 k, b =5 i +3 j +4 k, c =x i +y j +z k form a triangle such that c = a - b and the area of the triangle is 5 √ 6. If…

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