The value of $x$ if $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector 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 $l + 3m + 2n =$

Let $ABC$ be a triangle and $\bar{a}, \bar{b}, \bar{c}$ be the position vectors of $A, B, C$ respectively. If $D$ divides $BC$ in the ratio $2:3$ internally and $E$ divides $CA$ in the ratio $2:1$ internally,then the position vector of the point $P$ which divides $DE$ in the ratio $3:5$ internally is

If $\vec{a}=(p, -2, 5)$ and $\vec{b}=(1, q, -3)$ are collinear vectors,then:

Classify the following physical quantity as a scalar or a vector:
$10 \, g/cm^3$

If the position vectors of the vertices $A, B, C, D$ of a quadrilateral are $7 \hat{i}-4 \hat{j}+7 \hat{k}, \hat{i}-6 \hat{j}+10 \hat{k}, -\hat{i}-3 \hat{j}+4 \hat{k}$,and $5 \hat{i}-\hat{j}+5 \hat{k}$ respectively,then $ABCD$ is