If the points $P(4, 5, x)$,$Q(3, y, 4)$ and $R(5, 8, 0)$ are collinear,then the value of $x + y$ is

The position vectors of the points $P$ and $Q$ are respectively $-2 \bar{i}-3 \bar{j}+\bar{k}$ and $3 \bar{i}+3 \bar{j}+2 \bar{k}$. The ratio in which the point having position vector $\frac{-9}{2} \bar{i}-6 \bar{j}+\frac{1}{2} \bar{k}$ divides the line segment joining $P$ and $Q$ is

If $\vec{a}$ is a non-zero vector of modulus $|\vec{a}|$ and $m$ is a non-zero scalar,then $m\vec{a}$ is a unit vector if:

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec{a} + 3\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 2\vec{c}$ is collinear with $\vec{a}$,then find the value of $\vec{a} + 3\vec{b} + 6\vec{c}$.

If $P, Q, R$ are the mid-points of the sides $AB, BC$ and $CA$ of $\triangle ABC$ respectively,then $PC - BQ =$

$R$ 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. $S$ divides $PQ$ internally in the ratio $2: 1$. Then,the position vector of the midpoint of the line joining $R$ and $S$ is