If $C$ is the mid-point of line segment $AB$ and $P$ is any point not on the line $AB$,then

The vector which is parallel to the resultant vector of $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} - \hat{k}$ and having a magnitude of $5$ units is . . . . . . .

If $(\alpha, \beta, \gamma)$ is a triad of real numbers satisfying $\hat{i}-2 \hat{j}+5 \hat{k}=\alpha(\hat{i}+\hat{j}+\hat{k})+\beta(\hat{i}+2 \hat{j}+3 \hat{k})+\gamma(2 \hat{i}-\hat{j}+\hat{k}),$ then $\alpha^2-\beta^2+\gamma^2=$

If $a = 2i + 2j - k$ and $|xa| = 1$,then $x =$

$OABCD$ is a pentagon in which the sides $OA$ and $CB$ are parallel and the sides $OD$ and $AB$ are parallel. Also,it is given that $\frac{OA}{CB}=2$,$\frac{OD}{AB}=\frac{1}{3}$. If $\vec{OA}=\vec{a}, \vec{OD}=\vec{d}$,then $\vec{AD}+\vec{OC}+\vec{DC}=$

If $\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $|\bar{a}+\bar{b}|^2+|\bar{a}+\bar{c}|^2=8$,then $|\bar{a}+3\bar{b}|^2+|\bar{a}+3\bar{c}|^2=$