Let $2\hat{a} = \hat{b} \times \hat{c} + 2\hat{b}$. Then the sum of possible value$(s)$ of $\left| 2\hat{a} + \hat{b} + \hat{c} \right|$ is:

Using vectors,find the area of the triangle $ABC$ with vertices $A(1, 2, 3)$,$B(2, -1, 4)$ and $C(4, 5, -1)$.

If $P=3 \hat{i}+5 \hat{j}-\hat{k}$ and $Q=\hat{i}+2 \hat{j}+3 \hat{k}$ are two sides of a triangle,then its area is equal to . . . . . . sq units.

If $u = 2i + 2j - k$ and $v = 6i - 3j + 2k$,then find the unit vector perpendicular to both $u$ and $v$.

If the area of the parallelogram with $\bar{a}$ and $\bar{b}$ as two adjacent sides is $16$ sq. units,then the area of the parallelogram having $3 \bar{a}+2 \bar{b}$ and $\bar{a}+3 \bar{b}$ as two adjacent sides (in sq. units) is

The moment of a force represented by $\overrightarrow{F} = i + 2j + 3k$ about the point $P(2, -1, 1)$ is given by the cross product of the position vector $\overrightarrow{r}$ and the force vector $\overrightarrow{F}$. If the origin is $O(0, 0, 0)$,then $\overrightarrow{r} = 2i - j + k$. Calculate the moment $\overrightarrow{M} = \overrightarrow{r} \times \overrightarrow{F}$.