If the points with position vectors $10\hat{i} + 3\hat{j}$,$12\hat{i} - 5\hat{j}$,and $a\hat{i} + 11\hat{j}$ are collinear,then the value of $a$ is:

Let $A(3 \hat{i}+\hat{j}-\hat{k})$ and $B(13 \hat{i}-4 \hat{j}+9 \hat{k})$ be two points on a line $L$. $C$ and $D$ are points on $L$ on either side of $A$ at distances of $9$ and $6$ units respectively,and $C$ lies between $A$ and $B$. Then the position vectors of $C$ and $D$ are respectively:

If $PQRST$ is a pentagon,then the resultant of forces $\overline{PQ}, \overline{PT}, \overline{QR}, \overline{SR}, \overline{TS}$ and $\overline{PS}$ is

If the position vectors of $P$ and $Q$ are $(i + 3j - 7k)$ and $(5i - 2j + 4k)$,then $|\overrightarrow{PQ}|$ is

If $\vec{PO} + \vec{OQ} = \vec{QO} + \vec{OR}$,then

Three points whose position vectors are $a + b$,$a - b$,and $a + kb$ will be collinear if the value of $k$ is: