The vectors $\overrightarrow P = a\hat i + a\hat j + 3\hat k$ and $\overrightarrow Q = a\hat i - 2\hat j - \hat k$ are perpendicular to each other. The positive value of $a$ is

If $\overrightarrow{F}=(4 \hat{i}-10 \hat{j}) \text{ N}$ and $\overrightarrow{r}=(-5 \hat{i}-3 \hat{j}) \text{ m}$,then $(\overrightarrow{r} \times \overrightarrow{F})$ is

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

The angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\theta$. The value of the triple product $\overrightarrow{A} \cdot (\overrightarrow{B} \times \overrightarrow{A})$ is

If for two vectors $\vec{A}$ and $\vec{B}$,$\vec{A} \times \vec{B} = 0$,then the vectors:

State and explain the characteristics of the vector product of two vectors.