A vector $\overrightarrow{ A }$ points vertically upward and $\overrightarrow{ B }$ points towards north. The vector product $\overrightarrow{ A } \times \overrightarrow{ B }$ is

Vectors $a \hat{i}+b \hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+4 \hat{k}$ are perpendicular to each other when $3 a+2 b=7$, the ratio of a to $b$ is $\frac{x}{2}$. The value of $x$ is $..............$

Let $\vec{A}=2 \hat{i}-3 \hat{j}+4 \hat{k}$ and $\vec{B}=4 \hat{i}+j+2 \hat{k}$ then $|\vec{A} \times \vec{B}|$ is equal to ...................

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 ...... .