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

Force $F$ applied on a body is written as $F =(\hat{ n } \cdot F ) \hat{ n }+ G$, where $\hat{ n }$ is a unit vector. The vector $G$ is equal to

Three vectors $\overrightarrow a ,\,\overrightarrow b $and $\overrightarrow c $ satisfy the relation $\overrightarrow a \,.\,\overrightarrow b = 0$ and $\overrightarrow a \,.\,\overrightarrow c = 0.$ The vector $\overrightarrow a $ is parallel to

$\vec A$ and $\vec B$ are two vectors and $\theta$ is the angle between them, if $|\vec A \times \vec B|=\sqrt 3(\vec A \cdot \vec B) $ the value of $\theta$ is ......... $^o$

For any two vectors $\overrightarrow A $ and $\overrightarrow B $, if $\overrightarrow A \,.\,\overrightarrow B = \,\,|\overrightarrow A \times \overrightarrow B |,$ the magnitude of $\overrightarrow C = \overrightarrow A + \overrightarrow B $ is equal to

If $\overrightarrow A \times \overrightarrow B=\overrightarrow B \times \overrightarrow A$ then the angle between $\overrightarrow A$ and $\overrightarrow B$ is