If $P$ and $Q$ are the midpoints of the sides $BC$ and $CD$ of the parallelogram $ABCD$,then $\overrightarrow{AP} + \overrightarrow{AQ} = $

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and the points represented by position vectors $\bar{a}-2 \bar{b}+3 \bar{c}$, $-4 \bar{a}+5 \bar{b}-6 \bar{c}$, and $x \bar{a}-9 \bar{b}+z \bar{c}$ are collinear, then $2x-z=$

If the position vectors of the points $A, B, C$ are $i + j$,$i - j$,and $a i + b j + c k$ respectively,then the points $A, B, C$ are collinear if

Show that the points $A (-2 \hat{i}+3 \hat{j}+5 \hat{k})$,$B (\hat{i}+2 \hat{j}+3 \hat{k})$ and $C (7 \hat{i}-\hat{k})$ are collinear.

If $a = \alpha \hat{i} + 3 \hat{j} - 6 \hat{k}$ and $b = 2 \hat{i} - \hat{j} + \beta \hat{k}$,then the values of $\alpha, \beta$ so that $a$ and $b$ may be collinear are

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 $l + 3m + 2n =$