Three particles P, Q and R are moving along the vectors A = i + j, B = j + k and C =- i + j respectively. They strike on a point and start to move in…

Three particles $\displaystyle P, Q$ and $\displaystyle R$ are moving along the vectors $\displaystyle \vec{A}=\hat{i}+\hat{j}, \vec{B}=\hat{j}+\hat{k}$ and $\displaystyle \vec{C}=-\hat{i}+\hat{j}$ respectively. They strike on a point and start to move in different directions. Now particle $\displaystyle P$ is moving normal to the plane which contains vector $\displaystyle \vec{A}$ and $\displaystyle \vec{B}$. Similarly particle $\displaystyle Q$ is moving normal to the plane which contains vector $\displaystyle \vec{A}$ and $\displaystyle \vec{C}$. The angle between the direction of motion of P and Q is $\displaystyle \cos ^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of $\displaystyle x$ is \_\_\_\_.