Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively,in the ratio $2: 1$ externally.

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

The vertices of triangle $ABC$ are $A \equiv (3, 0, 0)$,$B \equiv (0, 0, 4)$,and $C \equiv (0, 5, 4)$. Find the position vector of the point $D$ where the angle bisector of $\angle A$ meets $BC$.

Two adjacent sides of a parallelogram are $2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\hat{i}-2 \hat{j}-3 \hat{k}$. Find the unit vector parallel to its diagonal.

Let $A$ be the set of vectors $a = (a_1, a_2, a_3)$ satisfying $\left(\sum_{i=1}^3 \frac{a_i}{2^i}\right)^2 = \sum_{i=1}^3 \frac{a_i^2}{2^i}$. Then,

$I$. Two non-zero, non-collinear vectors are linearly independent.
$II$. Any three coplanar vectors are linearly dependent.
Which of the above statements is/are true?