If $\hat{a}$ is a unit vector and projection of $\overrightarrow{x}$ along $\hat{a}$ is $2$ units and $\left(\hat{a}\times \overrightarrow{x}\right)+\overrightarrow{b}=\overrightarrow{x}$, then $\overrightarrow{x}$ is equal to

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}, \overrightarrow{d}$ be vectors such that $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=2$ and $\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \left(\overrightarrow{c}\times \overrightarrow{d}\right)+\left(\overrightarrow{b}\times \overrightarrow{c}\right)\times \left(\overrightarrow{a}\times \overrightarrow{d}\right)+\left(\overrightarrow{c}\times \overrightarrow{a}\right)\times \left(\overrightarrow{b}\times \overrightarrow{d}\right)=-\mu \overrightarrow{d},$ then the value of $\mu$ is

Given the vectors

$\overrightarrow{\mu }=2\hat{i}-\hat{j}-\hat{k}$

$\overrightarrow{v}=\hat{i}-\hat{j}+2\hat{k}$

$\overrightarrow{w}=\hat{i}-\hat{k}$

If the volume of the parallelepiped having $–c\overrightarrow{\mu },\overrightarrow{v}$ and $c\overrightarrow{w}$ as concurrent edges, is $8$, then $c$ is equal to

If $\overrightarrow{a}$ and$\overrightarrow{b}$ are any two vectors, then possible integers(s) in the range of $\frac{3|\overrightarrow{a}+\overrightarrow{b}|}{2}+2|\overrightarrow{a}-\overrightarrow{b}|$ is

If in triangle $ABC$, $\overrightarrow{AB}=\frac{\overrightarrow{u}}{\left|\overrightarrow{u}\right|}\mathrm{ }–\mathrm{ }\frac{\overrightarrow{v}}{\left|\overrightarrow{v}\right|}$ and  $\overrightarrow{AC}\mathrm{ }=\frac{2\overrightarrow{u}}{\left|\overrightarrow{u}\right|}$, where $\left|\overrightarrow{u}\right|\ne \left|\overrightarrow{v}\right|,$ then