The area of the parallelogram whose sides are represented by the vectors $\hat j + 3\hat k$ and $\hat i + 2\hat j - \hat k$ is

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 ...... .

What is the angle between $\vec A\,\,$ and  $\vec B\,\,$ if $\vec A\,\,$ and  $\vec B\,\,$ are the adjacent sides of a parallelogran drawn from a common point and the area of the parallelogram is $\frac {AB}{2}$

If $\overrightarrow P .\overrightarrow Q = PQ,$ then angle between $\overrightarrow P $and $\overrightarrow Q $ is ....... $^o$

Find the angle between two vectors with the help of scalar product.

The angle between vectors $(\vec{M} \times \vec{N})$ and $(\bar{N} \times \vec{M})$ is ................