Find the position vector of the point on the line passing through the point $\hat{i} - \hat{j} + 2\hat{k}$ and parallel to the vector $3\hat{i} + \hat{j} + \hat{k}$,which is at a distance of $3\sqrt{11}$ units from the point $\hat{i} - \hat{j} + 2\hat{k}$.

Let $O$ be the origin,$A$ and $B$ be two points with position vectors $-3 \hat{i}-3 \hat{j}+4 \hat{k}$ and $4 \hat{i}-4 \hat{j}-3 \hat{k}$ respectively. Let $P$ be a point such that the line drawn through $P$ parallel to $\overrightarrow{OB}$ meets $OA$ in $L$ and another line through $P$ parallel to $\overrightarrow{OA}$ meets $OB$ in $M$. If $L$ divides $OA$ in the ratio $2:3$ and $M$ divides $OB$ in the ratio $3:2$,then the distance from $O$ to $P$ is

If $ABCDEF$ is a regular hexagon with $\vec{AB} = \vec{a}$ and $\vec{BC} = \vec{b}$,then $\vec{CE}$ equals to

The position vector of a point $C$ with respect to $B$ is $i + j$ and that of $B$ with respect to $A$ is $i - j$. The position vector of $C$ with respect to $A$ is

Let $\overrightarrow{OA} = \hat{i} - 3\hat{j} + \hat{k}$, $\overrightarrow{OB} = \hat{i} + 3\hat{j} - 2\hat{k}$, and $\overrightarrow{OC} = 4\hat{i} + 3\hat{j} + 5\hat{k}$ be the position vectors of three points $A$, $B$, and $C$. Let $P$ be the point which divides $AB$ in the ratio $2:1$. If $l, m, n$ are the direction cosines of the vector $\overrightarrow{PC}$, then $l + 3m + 2n =$

Answer the following as true or false.
Two collinear vectors having the same magnitude are equal.