Let \( O \) be the origin. Let \( \overrightarrow{OP} = x \hat{i} + y \hat{j} - \hat{k} \) and \( \overrightarrow{OQ} = -\hat{i} + 2 \hat{j} + 3x \hat{k} \), \( x, y \in \mathbb{R}, \, x > 0 \), be such that \( |\overrightarrow {PQ}| = \sqrt{20} \) and the vector \( \overrightarrow{OP} \) is perpendicular to \( \overrightarrow{OQ} \). If \( \overrightarrow{OQ} = 3 \hat{i} + z \hat{j} - 7 \hat{k} \), \( z \in \mathbb{R} \), is coplanar with \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \), then the value of \( x^{2} + y^{2} + z^{2} \) is equal to:
