If the diagonals of a parallelogram are represented by the vectors $3\hat{i} + \hat{j} - 2\hat{k}$ and $\hat{i} + 3\hat{j} - 4\hat{k}$,then its area in square units is:

The number of vectors of unit length perpendicular to vectors $a = (1, 1, 0)$ and $b = (0, 1, 1)$ is

If $A, B, C, D$ are any four points in space, then $|\overrightarrow{AB} \times \overrightarrow{CD} + \overrightarrow{BC} \times \overrightarrow{AD} + \overrightarrow{CA} \times \overrightarrow{BD}|$ is equal to (where $\Delta$ denotes the area of $\Delta ABC$) (in $\Delta$)

If $\vec{a}=\hat{i}+\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ is a unit vector perpendicular to $\vec{a}$ and coplanar with $\vec{a}$ and $\vec{b}$, then the unit vector $\vec{d}$ perpendicular to both $\vec{a}$ and $\vec{c}$ is

If $\vec{a}=\hat{i}-\hat{j}+\hat{k}$,$\vec{b}=\hat{i}+\hat{j}-2 \hat{k}$,$\vec{c}=2 \hat{i}-3 \hat{j}-\hat{k}$,and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$ are four vectors,then find the value of $(\vec{a} \times \vec{c}) \times(\vec{b} \times \vec{d})$.

$A$ unit vector perpendicular to the vectors $\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}$ and $\vec{b} = 3 \hat{j} + 2 \hat{k}$ is