The vector of magnitude $6$ units and perpendicular to vectors $2 \hat{i}+\hat{j}-3 \hat{k}$ and $\hat{i}-2 \hat{j}+\hat{k}$ is

The area of the parallelogram whose diagonals are represented by the vectors $\bar{a}=3 \hat{i}-\hat{j}-2 \hat{k}$ and $\bar{b}=-\hat{i}+3 \hat{j}-3 \hat{k}$ is

The position vectors of the points $A, B$ and $C$ are $i + j, j + k$ and $k + i$ respectively. The vector area of the $\Delta ABC = \pm \frac{1}{2} \vec{\alpha}$ where $\vec{\alpha} = $

Let $a=2 \hat{i}-3 \hat{j}+4 \hat{k}$, $b=7 \hat{i}+2 \hat{j}-3 \hat{k}$, and $c=\hat{i}+\hat{j}+\hat{k}$. The vector $x$ such that $x \cdot c=60$ and $x$ is perpendicular to both $a$ and $b$ is:

$A$ non-zero vector $\vec{a}$ is parallel to the line of intersection of the plane defined by $\hat{i}$ and $\hat{i} + \hat{j}$,and the plane defined by $\hat{i} - \hat{j}$ and $\hat{i} + \hat{k}$. Find the angle between $\vec{a}$ and $\hat{i} - 2\hat{j} + 2\hat{k}$.

Let $\overline{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\overline{b}=\hat{i}+\hat{j}$. If $\overline{c}$ is a vector such that $\overline{a} \cdot \overline{c}=|\overline{c}|$,$|\overline{c}-\overline{a}|=2 \sqrt{2}$ and the angle between $(\overline{a} \times \overline{b})$ and $\overline{c}$ is $30^{\circ}$,then $|(\overline{a} \times \overline{b}) \times \overline{c}|$ is equal to