Let $\overrightarrow x$ be a vector in the plane containing vectors $\overrightarrow a = 2\widehat i - \widehat j + \widehat k$ and $\overrightarrow b = \widehat i + 2\widehat j - \widehat k$. If the vector $\overrightarrow x$ is perpendicular to $\left( {3\widehat i + 2\widehat j - \widehat k} \right)$ and its projection on $\overrightarrow a$ is ${{17\sqrt 6 } \over 2}$, then the value of $|\overrightarrow x {|^2}$ is equal to __________.

Let, $\overrightarrow x = k(\overrightarrow a + \lambda \overrightarrow b )$<br><br>$\overrightarrow x$ is perpendicular to $3\widehat i + 2\widehat j - \widehat k$<br><br><b>I.</b> k{(2 + $\lambda$)3 + (2$\lambda$ $-$ 1)2 + (1 $-$ $\lambda$)($-$1) = 0<br><br>$\Rightarrow$ 8$\lambda$ + 3 = 0<br><br>$\lambda = {{ - 3} \over 8}$<br><br><b>II.</b> Also projection of $\overrightarrow x$ on $\overrightarrow a$ is ${{17\sqrt 6 } \over 2}$ therefore<br><br>$${{\overrightarrow x .\overrightarrow a } \over {|\overrightarrow a |}} = {{17\sqrt 6 } \over 2}$$<br><br>$$ \Rightarrow k\left\{ {{{(\overrightarrow a + \lambda \overrightarrow b ).\overrightarrow a } \over {\sqrt 6 }}} \right\} = {{17\sqrt 6 } \over 2}$$<br><br>$$ \Rightarrow k\left\{ {6 + \left( {{3 \over 8}} \right)} \right\} = {{17 \times 6} \over 2}$$<br><br>$\Rightarrow k = {{51} \over {51}} \times 8$<br><br>k = 8<br><br>$\therefore$ $$\overrightarrow x = 8\left( {{{13} \over 8}\widehat i - {{14} \over 8}\widehat j + {{11} \over 8}\widehat k} \right)$$<br><br>$= 13\widehat i - 14\widehat j + 11\widehat k$<br><br>$|\overrightarrow x {|^2} = 169 + 196 + 121 = 486$