Let x be a vector in the plane containing vectors a =2 i - j + k and b = i +2 j - k. If the vector x is perpendicular to (3 i +2 j - k ) and its…

Let $\displaystyle \vec{x}$ be a vector in the plane containing vectors $\displaystyle \overrightarrow{\mathrm{a}}=2 \hat{i}-\hat{j}+\hat{k}$ and $\displaystyle \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-\hat{k}$. If the vector $\displaystyle \vec{x}$ is perpendicular to $\displaystyle (3 \hat{i}+2 \hat{j}-\hat{k})$ and its projection on $\displaystyle \overrightarrow{\mathbf{a}}$ is $\displaystyle \frac{17 \sqrt{6}}{2}$, then the value of $\displaystyle |\vec{x}|^2$ is equal to \_\_\_\_.