If $\bar{a} = 4\hat{i} + 3\hat{j} + \hat{k}$ and $\bar{b} = \hat{i} - 2\hat{j} + 2\hat{k}$,then find the value of $\bar{a} \times (\bar{a} \times (\bar{a} \times (\bar{a} \times \bar{b})))$.

Let $\overrightarrow{c}$ be a vector perpendicular to the vectors $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}.$ If $\overrightarrow{c}\cdot(\hat{i}+\hat{j}+3\hat{k})=8,$ then the value of $\overrightarrow{c}\cdot(\overrightarrow{a}\times\overrightarrow{b})$ is equal to ...... .

The area of the quadrilateral $ABCD$ with vertices $A(0,4,1)$,$B(2,3,-1)$,$C(4,5,0)$,and $D(2,6,2)$ is equal to

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c})$. If $|\vec{a}| = 1, |\vec{b}| = 4, |\vec{c}| = 2$, and the angle between $\vec{b}$ and $\vec{c}$ is $60^{\circ}$, then $|\vec{a} \cdot \vec{c}|$ is:

If $a, b, c$ are position vectors of vertices of a triangle $ABC$,then the unit vector perpendicular to its plane is:

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