Let O be the origin. Let $\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$ and $\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$, x, y$\in$R, x > 0, be such that $\left| {\overrightarrow {PQ} } \right| = \sqrt {20}$ and the vector $\overrightarrow {OP}$ is perpendicular $\overrightarrow {OQ}$. If $\overrightarrow {OR}$ = $3\widehat i + z\widehat j - 7\widehat k$, z$\in$R, is coplanar with $\overrightarrow {OP}$ and $\overrightarrow {OQ}$, then the value of x² + y² + z² is equal to :

$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k\,$ <br/><br/>$\overrightarrow {OP} \bot \overrightarrow {OQ}$<br><br>$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$<br><br>$$\overrightarrow {PQ} = \left( { - 1 - x} \right)\widehat i + \left( {2 - y} \right)\widehat j + \left( {3x + 1} \right)\widehat k$$<br><br>$$\left| {\overrightarrow {PQ} } \right| = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}} $$ <br><br>$$\sqrt {20} = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}} $$<br><br>20 = 1 + x² + 2x + 4 + y² $-$ 4y + 9x² + 1 + 6x<br><br>20 = 10x² + y² + 8x + 6 $-$ 4y<br><br>20 = 10x² + 4x² + 8x + 6 $-$ 8x<br><br>14 = 14x² $\Rightarrow$ x² = 1 <br><br>Also, $\overrightarrow {OP} .\,\overrightarrow {OQ} = 0$<br><br>$- x + 2y - 3x = 0$<br><br>$4x = 2y$<br><br>y = 2x <br><br>$\therefore$ y² = 4x² $\Rightarrow$ y² = 4<br><br>x = 1 as x &gt; 0 and y = 2<br><br>$\therefore$ $$\left| {\matrix{ x &amp; y &amp; { - 1} \cr { - 1} &amp; 2 &amp; {3x} \cr 3 &amp; z &amp; { - 7} \cr } } \right| = 0$$<br><br>$\Rightarrow$ $$\left| {\matrix{ 1 &amp; 2 &amp; { - 1} \cr { - 1} &amp; 2 &amp; 3 \cr 3 &amp; z &amp; { - 7} \cr } } \right|$$ = 0<br><br>$\Rightarrow$ 1($-$14 $-$3z) $-$ 2(7 $-$ 9) $-$ 1($-$z $-$6) = 0<br><br>$\Rightarrow$ $-$14 $-$3z + 4 + z + 6 = 0<br><br>$\Rightarrow$ 2z = $-$4 $\Rightarrow$ z = $-$2<br><br>$\therefore$ x² + y² + z² = 9