The perimeter of the triangle whose vertices have the position vectors $\hat{i}+\hat{j}+\hat{k}$,$5\hat{i}+3\hat{j}-3\hat{k}$,and $2\hat{i}+5\hat{j}+9\hat{k}$ 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

If the vector $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ makes angles $\alpha, \beta, \gamma$ with vectors $\hat{i}, \hat{j}, \hat{k}$ respectively,then:

$2 \hat{i}-3 \hat{j}+\hat{k}$ and $\hat{i}+2 \hat{j}-3 \hat{k}$ are the position vectors of two points $A$ and $B$ respectively and $C$ divides $AB$ in the ratio $3:2$. If $3 \hat{i}-\hat{j}+2 \hat{k}$ is the position vector of a point $D$, then the unit vector in the direction of $\overrightarrow{CD}$ is

If $ABCDEF$ is a regular hexagon,where two adjacent sides $\vec{AB}$ and $\vec{BC}$ are $\vec{a}$ and $\vec{b}$ respectively. Then $\vec{CD}$ is

$A$ vector of magnitude $14$ lies in the $xy-$ plane and makes an angle of $60^\circ$ with the $x-$ axis. The components of the vector in the direction of the $x-$ axis and $y-$ axis are: