$A$ vector $\vec{A}$ points vertically upward and $\vec{B}$ points towards north. The vector product $\vec{A} \times \vec{B}$ is

If diagonals of a parallelogram are $(5\hat i - 4\hat j + 3\hat k)$ and $(3\hat i + 2\hat j - \hat k)$,then its area is

The dot product of two mutually perpendicular vectors is:

If $\overrightarrow{P} = 3\hat{i} + \sqrt{3}\hat{j} + 2\hat{k}$ and $\overrightarrow{Q} = 4\hat{i} + \sqrt{3}\hat{j} + 2.5\hat{k}$,then the unit vector in the direction of $\overrightarrow{P} \times \overrightarrow{Q}$ is $\frac{1}{x}(\sqrt{3}\hat{i} + \hat{j} - 2\sqrt{3}\hat{k})$. The value of $x$ is:

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 two vectors $\vec{P} = \hat{i} + 2m \hat{j} + m \hat{k}$ and $\vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k}$ are perpendicular to each other,then the value of $m$ will be: