If $\bar{a}=\hat{j}-\hat{k}$ and $\bar{c}=\hat{i}-\hat{j}-\hat{k}$,then the vector $\bar{b}$ satisfying $\bar{a} \times \bar{b}+\bar{c}=\vec{0}$ and $\bar{a} \cdot \bar{b}=3$ is

The unit vector perpendicular to both $i + j$ and $j + k$ is

The locus of the point $P(\vec{r})$ which forms a triangle $ABP$ of area $1$ sq. unit with the fixed points $A(\hat{i})$ and $B(\hat{j})$ is

Let $\overrightarrow{OA}=\overrightarrow{a}$,$\overrightarrow{OB}=12 \overrightarrow{a}+4 \overrightarrow{b}$,and $\overrightarrow{OC}=\overrightarrow{b}$,where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$,then the ratio of the area of the quadrilateral $OABC$ to the area of $S$ is:

If $\vec{a} = 2 \hat{i} + 2 \hat{j} + \hat{k}$, $|\vec{b}| = 6$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$, then the area of the triangle (in square units) with $\vec{a}$ and $\vec{b}$ as two of its sides is

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31}$,$4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi}{3}$,then $\left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$ is equal to $............$.