Let $O$ be the origin. Let $\overrightarrow{O P}=x \hat{i}+y \hat{j}-\hat{k}$ and $O Q=-\hat{i}+2 \hat{j}+3 x \hat{k}, x, y \in R, x0$, be such that $|\overrightarrow{P Q}|=\sqrt{20}$ and the vector $\overrightarrow{O P}$ is perpendicular to $\overrightarrow{O Q}$. If $\overrightarrow{O R}=3 \hat{i}+z \hat{j}-7 \hat{k}, z \in R$, is coplanar with $\overrightarrow{O P}$ and $\overrightarrow{O Q}$ , then the value of $x^2+y^2+z^2$ is equal to: [JEE Main 2021, 17 Mar (Shift 2)]

Question

Let $O$ be the origin. Let $\overrightarrow{O P}=x \hat{i}+y \hat{j}-\hat{k}$ and $O Q=-\hat{i}+2 \hat{j}+3 x \hat{k}, x, y \in R, x>0$, be such that $|\overrightarrow{P Q}|=\sqrt{20}$ and the vector $\overrightarrow{O P}$ is perpendicular to $\overrightarrow{O Q}$. If $\overrightarrow{O R}=3 \hat{i}+z \hat{j}-7 \hat{k}, z \in R$, is coplanar with $\overrightarrow{O P}$ and $\overrightarrow{O Q}$ , then the value of $x^2+y^2+z^2$ is equal to: [JEE Main 2021, 17 Mar (Shift 2)]

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Let \(O\) be the origin. Let \(\overrightarrow{O P}=x \hat{i}+y \hat{j}-\hat{k}\) and \(O Q=-\hat{i}+2 \hat{j}+3 x \hat{…

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