If $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$ are vectors such that $|\vec{a}+\vec{b}|=25$,then the sum of the values of $t$ is

The number of values of $m \in R$ for which the vectors $\hat{i}+2 \hat{j}+m \hat{k}$ and $\hat{i}+m \hat{j}+2 \hat{k}$ are collinear is

If $ABCDEF$ is a regular hexagon and $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = p\vec{AD} = q\vec{AO}$, where $O$ is the center of the hexagon, then the values of $p$ and $q$ respectively are

The perimeter of a triangle with sides represented by vectors $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$,$\vec{b} = 4\hat{i} - 3\hat{j} - 5\hat{k}$,and $\vec{c} = 7\hat{i} + \hat{j}$ is:

Let $A$ and $B$ be points with position vectors $a$ and $b$ with respect to the origin $O$. If the point $C$ on $OA$ is such that $2AC = CO$,$CD$ is parallel to $OB$ and $|\overrightarrow{CD}| = 3|\overrightarrow{OB}|$,then $\overrightarrow{AD}$ is equal to

The unit vector in the direction of $\vec{x} = (2, 3, \sqrt{3})$ is . . . . . . .