The unit vector in the direction of the vector $\vec{a} = (2, 2, -1)$ is $......$

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let $S$ be any point on the vector $\vec{R}$. The distance between the point $P$ and $S$ is $b|\vec{R}|$. The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is

The position vectors of the points $A$ and $B$ are $\vec{a}$ and $\vec{b}$ respectively. If the position vector of the point $C$ is $\frac{\vec{a}}{2} + \frac{\vec{b}}{3}$,then:

Let $H$ be the orthocentre of an acute angled $\triangle ABC$ and $O$ be its circumcenter. Then,$\vec{HA} + \vec{HB} + \vec{HC}$

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 $l + 3m + 2n =$

If $|\vec{a}| = 2$,$|\vec{b}| = 3$ and $|2\vec{a} - \vec{b}| = 5$,then $|2\vec{a} + \vec{b}|$ equals