If the vector $2\hat{i} + 3\hat{j} + 8\hat{k}$ is perpendicular to the vector $4\hat{j} - 4\hat{i} + \alpha\hat{k}$,then what is the value of $\alpha$?

If the vectors $\vec P = a\hat i + a\hat j + 3\hat k$ and $\vec Q = a\hat i - 2\hat j - \hat k$ are perpendicular to each other,then the positive value of $a$ is

If $\vec{A}$ and $\vec{B}$ are two vectors satisfying the relation $\vec{A} \cdot \vec{B} = |\vec{A} \times \vec{B}|$,then the value of $|\vec{A} - \vec{B}|$ will be:

If $a + b + c = 0$,then $a \times b$ is equal to:

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors given by $\overrightarrow{A} = 2\widehat{i} + 3\widehat{j}$ and $\overrightarrow{B} = \widehat{i} + \widehat{j}$. The magnitude of the component (projection) of $\overrightarrow{A}$ on $\overrightarrow{B}$ 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 ......... $^\circ$.