Two vector $A$ and $B$ have equal magnitudes. Then the vector $\mathop A\limits^ \to + \mathop B\limits^ \to $ is perpendicular 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 ...... .

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 $..............$

The area of the parallelogram represented by the vectors $\overrightarrow A = 2\hat i + 3\hat j$ and $\overrightarrow B = \hat i + 4\hat j$ is.......$units$

The angle between the vectors $(\hat i + \hat j)$ and $(\hat j + \hat k)$ is ....... $^o$

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