Let O be the origin. Let OP =x i +y j - k and OQ =- i +2 j +3 x k, x, y ∈ R, x>0, be such that | PQ |=√ 20 and the vector OP is perpendicular to OQ.…

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