Let a vector c be coplanar with the vectors a =- i + j + k and b =2 i + j - k. If the vector c also satisfies the conditions c ·[( a + b ) ×( a × b…

Let a vector $\displaystyle \overrightarrow{\mathrm{c}}$ be coplanar with the vectors $\displaystyle \overrightarrow{\mathrm{a}}=-\hat{i}+\hat{j}+\hat{k}$ and $\displaystyle \overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\hat{k}$. If the vector $\displaystyle \vec{c}$ also satisfies the conditions $\displaystyle \vec{c} \cdot[(\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})]=-42$ and $\displaystyle (\overrightarrow{\mathrm{c}} \times(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})) \cdot \hat{k}=3$, then the value of $\displaystyle |\overrightarrow{\mathrm{c}}|^2$ is equal to: