If $\theta$ is the angle between vectors $\vec{A}$ and $\vec{B}$,then what is the value of the product $(\vec{B} \times \vec{A}) \cdot \vec{A}$?

$\vec A$ 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 ......... $^\circ$.

Two vectors $\vec{A}$ and $\vec{B}$ are at right angles to each other,when

The edges of a parallelepiped are represented by the vectors $\hat{i} + 2\hat{j}$,$4\hat{j}$,and $\hat{j} + 3\hat{k}$. Find its volume.

If $\vec{P} = (k, 2, 3)$ and $\vec{Q} = (0, 3, k)$ and $\vec{P} \perp \vec{Q}$,then what is the value of $k$?

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